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Full text of "A short course of experiments in physical measurement"

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SCIENCE CENTER LIBRARY 




PHYSICAL MEASUREMENT. 



^ A SHORT COURSE 

••• OF 



EXPERIMENTS 



IN 



PHYSICAL MEASUREMENT. 



By HAROLD WHITING, 

IN8TRUCT0B IN PHYSICS AT HARVARD UNIVBBSITT. 

In jFour Ij^wcin. 
Pabt III. 

PRINCIPLES AND METHODS. 

H0TE8 AUD EXPLAKATIOKS FOS THE TT8E OF STUDENTS. 
KATHEICATIGAL AHB PHTSIGAL TABLES. 



CAMBRIDGE: 
JOHN WILSON AND SON. 

1891. 




"* ^vri,VP.D COO 



V^ 



[ '-N - 18 



Copyright, 1891, 
By Habold Whiting. 



TABLE OF CONTENTS. 



PRINCIPLES AND METHODS. 

Chaptsb Paqi 

Introduction 585 

I. General Definitions 602 

11. Observation and Error 614 

III. General Methods 626 

IV. Reduction of Results 658 

v. Hydrostatics 669 

VI. Heat 679 

VII. Sound and Lioht 691 

VrU. Force and Work 703 

IX. Electricity and Magnetism 718 

X. Electromotive Force and Resistance . . 729 

Addenda 739 

Arrangement of Tables 746 

Explanation of Tables 761 

Sources of Authority 794 

MATHEMATICAL AND PHYSICAL TABLES. 
X^BLB MATHEMATICAL TABLES. 

1. Proportional Parts 797 

2. Powers, Circular Properties, ktc 798 

3. Trigonometric Functions (3-place) .... 800 

3A. Reciprocals 802 

30. Squares 804 

3D. Cubes 808 

3F. Circumferences of Circles 808 

3G. Areas of Circles 810 

3H. Volumes of Spheres 812 

4. Natural Sines 814 

4 A. Logarithmic Sines 816 

5. Natural Tangents 818 



VI TABLE OF CONTENTS. 

6A. Logarithmic Tangents 820 

0. Logarithms 822 

7. Probability of Errors 842 

PHYSICAL TABLES. 
General Properties of Solids, Liquids, and Gases, 

8. Properties of Elementary Substances . . 843 

9. Properties of Solids — Building Materials, 

ETC 846 

9a. Properties of Solids — Chemical Materials, 

ETC 848 

10. Properties of Solids — Optical Materials, 

ETC 852 

11. Properties of Liquids 856 

12. Properties of Gases and Vapors 861 



HYPSOMETRIC, HYGROMETRIO, AND BAROMETRIC 
TABLES. 

13a. Maximum Pressure of Vapors 864 

14. Boiling Points op Water, 68 cm. to 80 cm. . 807 
14A. Dew-point, Temperature, and Humidity . . 867 

15. Hygrometric Table 868 

15A. Specific Heat of Moist Air 868 

15B. Velocity of Sound . 869 

15C. Coefficients of Interdiffusion of Gases . . 869 

16. Reduction of Barometric Readings to cm. . 870 
16 A. (Reduction of Barometric Readings) g = 980 871 
16B. ( TO Megadynes per sq. cm. . . J G = 981 871 

17. Estimation of Heights by the Barometer . 871 
17A. Correction for Temperature in Table 17 . 871 
17B. Correction for Humidity in Table 17 . . . 871 
18a. Barometric Corrections for Expansion . . 872 
18b. Barometric Corrections for Capillarity. . 872 
18c. Barometric Corrections for Mercurial 

Vapor 872 

18d. Reductions of Density to 76 cm 873 

18e. Reductions of Density to 0° 873 

18f. Reductions of Volume to 76 cm 873 

18g. Reductions of Volume to 0® 873 



TABLK OF CONTENTS. vii 



REDUOTION OF WEIQHINQ8 TO VACUO. 

19. Density of Air (0^-30°; 72-77 cm.) .... 874 

20. Corrections i«ok Humidity in Table 19 . . 874 
20A. Buoyancy of Air on Brass Weights . . . 874 

21. Reduction of Apparent Weighings to Vacuo 875 



8PEOIFIO VOLUMES AND DENSITIES. 

22. Apparent Specific Volumes of Water . . . 875 

23. True Specific Volume of Water 876 

23A. True Specific Volume of Mercury .... 876 

2'SB. Apparent Specific Volume of Mercuky . . 876 

21. Density of Mercury 877 

25. Density of Water 877 

26. Density of Commercial Glycerine ^ . • . 877 



PROPERTIES OF SOLUTIONS DEPENDING UPON 
THEIR strength. 

27. Density OF Alcohol (0%-100%; 15**- 22^) . . 878 

28. Density of Acids and Solutions at 15® . . 880 

29. Boiling-points of Solutions 882 

30. Specific Heats of Solutions 883 

31 A. Electrical Conductivity of Solutions at 18** 884 

31 B. Refractive Indices of Solutions 885 

31 C. Table for Mixing Solutions 885 

31D. Coefficients of Saline Diffusion 885 

3 IE. ( Rotation of the Plank of ) in Solutions . 886 



R OF ) 

A-H ) 



31 F. ( Polarization, Lines A-H ) in Solids . . 886 

MISCELLANEOUS DATA. 

31 G. Magnetic Rotation of Polarization. . . . 887 

31H. Magnetic Susceptibility, etc 887 

311. Coefficients of Hydraulic Friction . . . 887 

31J. Coefficients of Friction between Solids. . 887 

31 K. Relative Radiation, Absorption, etc. . . . 887 

31L. Constants of Radiation 887 

32a. Heats of Combustion in Oxygen 888 

32b. Heats of Combustion in Chlorine .... 888 

33. Heats of Combination 888 



viii TABLE OF CONTENTS. 

ELECTROMOTIVE FORCE AND RESISTANCE. 

84. Contact Differkncbs op Potential .... 889 

85. Electromotive Forces of Voltaic Cells . . 890 

86. Electromotive Force and Length of Spark. 890 
37a. Specific Resistances of Conductors . . . . 891 
37b. Specific Resistances of Insulators .... 891 
38. Specific Resistances of Electrolytes . . . 891 

arbitrary scales. 

89. Fahrenheit and Centigrade Thermometers 892 

40. Hydrometer Scales 892 

41. WaveJ^engths 892 

42a. Board of Trade (Imperial) Wire Gauge . . 893 

42b. Birmingham Wire Gauge 893 

43. Musical Pitch 893 

ASTRONOMICAL AND GEOGRAPHICAL TABLES. 

44A. Reduction of (') and (") to (°) . . . . . . 894 

44B. Equation of Datbs in Different Years . . 894 

44C. Gain of Sidereal Time 894 

44D. Sidereal Time at Greenwich, Noon .... 894 

44E. Semi-diameter of the Sun 894 

44F. Declination of the Sun ........ 895 

44G. Equation of Time 895 

44H. The Solar System 896 

45. Right Ascensions and Declinations of Stars 896 

46. Geographical Latitudes, Longitudes, and 

Elevations 897 

47. Acceleration of Gravity 897 

48. Lengths of Seconds Pendula 897 



reduction of MEASURES TO AND FROM THE 
C. G. S. SYSTEM. 

49a. Reductions Independent of "g" 898 

49b. Reductions for "g''= 980 AND "G*'= 981 . . 899 



50. Constants Frequently Required . . . . . 900 



PHYSICAL MEASUREMENT. 



PRINCIPLES AND METHODS. 



INTRODUCTION. 

The fii*st step in all scientific progress consists in 
a classification of different objects based upon simi- 
larities and differences. The distinguishing charac- 
teristics of solids and liquids, minerals, metals, 
crystals, &c., were undoubtedly observed long before 
history began. The necessity for shelter and cloth- 
ing must have drawn attention to the difference be- 
tween insulating substances and conductors of heat ; 
and in the same way all physical properties of im- 
portance to mankind cannot have failed to receive 
early recognition. The manner in which different 
branches of science have been developed is perhaps 
best illustrated in the case of electricity, the phe- 
nomena of which were virtually unknown^ before the 

1 The development of electricity from amber was known to Tlialee 
sereral years before Christ. It would appear, however, that at this 
time little or nothing else was known about electricity. Ganot's 
Physics, § 723. 

1 



686 INTRODUCTION. 

end of the sixteenth century. WtB find in very early 
writings tables like the following : — 





CONDUCTORS OF ELECTRICITY. 




Metals. 


Animal Substances. 


Sea Water. 


Charcoal. 


Vegetable Substances. 
NON-CONDUCTORS. 


Vinegar, &c. 


Resins. 


Glass. 


Wax. 


Sulphur. 


Silk. 


Oils, &c. 



A division of substances into two classes may in 
certain cases be exceedingly useful. The reactions 
which take place in chemical solutions are, for in- 
stance, frequently determined bj^ the solubility or 
insolubility of the compounds which may be formed. 
It is rarely necessary to make fine distinctions in the 
statement of chemical solubilities.^ The term " spar- 
ingly soluble " must occasionally be employed ; and, 
again, comparisons must be made between different 
solubilities. Most substances, however, are either 
very soluble, or else very insoluble, in a given liquid ; 
and a single word, " soluble " or " insoluble," conveys 
to the chemist a valuable piece of information. 

In the construction of electrical instruments, on 
the other hand, it became important to distinguish 
both good conductors and good non-conductors from 
a large class of substances called '' semi-conductors " 
(Ganot's Physics, § 725) ; and with the growing im- 
portance of electricity came the necessity of still 
further distinctions. Substances were finally ar- 

^ See Storer*8 Dictionary of Solubilities. 



INTRODUCTION. 687 

ranged in a list in the order of their power to con- 
duct or to insulate electricity (Deschaners Natural 
Philosophj', § 409). In the same way certain bodies, 
at first classed simply as positive or negative with re- 
pect to the charges of electricity which they receive 
when rubbed together, are in latQi: works arranged as 
follows (Deschanel, § 411) : — 



Fur of Cat. 


Feathers. 


Silk. 


Polished Glass. 


Wood 


Shellac. 


Wooden Stuffs. 


Paper. 


Rough Glass. 



If any of the substances in this list be rubbed with 
one following it, it will generally become "posi- 
tively electrified ; " but if rubbed with one preceding 
it, it will be '^ negatively electrified." Such an ar- 
rangement is evidently more useful than a simple 
division into two classes. 

Mohs' scale of hardness consists of 10 substances : ^ 

1. Talc. 3. Calc Spar 5 Apatite. 7 Quartz. 9. Sapphire. 

2. Gypsum. 4. Fluor Spar 6 Feldspar. 8 Topaz. 10. Diamond. 

Each substance contained in this list will scratch the 
one above it. If, accordingly, a piece of steel which 
will scratch feldspar is scratched by quartz, its hard- 
ness must be represented by a number between 6 
and 7 (let us say 6.5) on this arbitrary scale. 

The distinction between any two substances in such 
a list is purely qualitative ; that is, we know only 
that each possesses a certain quality or property more 
than the one below it. We do not know whether the 

1 Cooke's Chemical Physics, p 209. 



688 INTRODUCTION. 

gaps in the list are great or small, equal or unequal. 
We have no idea even of the relative values which 
the numbers (1-10) represent. Still, the assignment 
of numbers to the different substances may be con- 
sidered as a first attempt to obtain precise results ; and 
in the case of physical quantities which admit of no 
more exact estimation, the value of an arbitrary scale 
like that of Mohs must not be overlooked. , 

The next step in the accurate representation of 
results is to make the intervals between different 
scale-numbers equ^l, — or, at least, to make them fol- 
low in regular progression. Among the earliest ap- 
plications of this principle may be 
mentioned the arbitrary hydrometer 
scales of Beaum^, Beck, Cartier and 
Twaddell. A mark was made upon a 
hydrometer (see Fig. a) to show how 
deep it sank in water ; and this mark 
was numbered or 10, as the case might 
be. Then the hydrometer was floated 
in some other liquid of known compo- 
sition, and another mark was made to 
show how deep it sank in that liquid. 
The second mark was also numbered 
arbitrarily — 60 or 80, for instance 
(see Table 40). The distance between 
the two marks was then subdivided. 
The scale of an ordinary thermome- 
ter (see Fig. b) is constructed in a similar way. 
A mark is made to show where the mercury stands 
when surrounded with melting ice, and another 




INTRODUCTION. 689 

mark is made to show where it stands in steam 
(see Exp. 25). The distance between the two 
marks is divided by Fahrenheit into 180 parts; 
by Celsius, into 100 parts; by Reaumur, into 80 parts. 
Fahrenheit called the freezing-point of water 82®, 
without any scientific reason ; Celsius and Reaumur 
called it 0°. Their scales are accordingly simpler 
than Fahrenheit's, but none the less arbitrary. The 
Celsius scale is still in use in the ordinary centigrade 
thermometer (§ 4) ; the other scales, t(^ether with 
the hydrometer scales of Baum^, Beck, Cartier, and 
Twaddell, are going out of use. The gradual disap- 



Fio. 6. 

pearance of arbitrary scales is in general an indication 
of scientific progress. 

It is obviously desirable that the numbers in 
a scale should be proportional to the quantities 
which they represent. With the advance of sci- 
ence in the earlj'^ part of the present century, we 
find an abundance of physical tables showing the 
relative values of different quantities (§ 3). Spe- 
cific gravities of solids and liquids compared with 
water, specific gravities of gases and vapors com- 
pared with air or with hydrogen, specific heats 
compared with water, &c., were all more or less 
accurately determined. 

At the same time that the physical properties of 



690 



INTRODUCTION. 



different bodies were compared together, the changes 
which take place in a given substance under varying 
conditions were carefully studied. The expansion of 
solids, liquids, and gases due to heat were, for instance, 
observed and tabulated. We find in Biot's "Phy- 
sique ' (1821, vol. i., page 320) a table showing the 
relative densities of water at different temperatures, 
some of which are compared below with the best re- 
sults of modern observers, as given by Everett in 
§ 34 of his "Units and Physical Constants." Calling 
the density of water at 4® equal to 1, these results 
become ^ — 





Biot. 


Everett. 


Differenoe. 




Biot. 


ETerett. 


Differenc 


Qo 


.99993 


.99987 


+ 6 


50O 


.98778 


.98820 


- 42 


40 


1.00000 


1.00000 




6O0 


.98251 


.98338 


- 87 


100 


.99973 


.99975 


- 2 


70*^ 


.97652 


.97794 


-142 


200 


.99832 


.99826 


+ 6 


8O0 


.96998 


.97194 


-196 


SO*' 


.99579 


.99577 


+ 2 


90° 


.96286 


.96556 


-271 


40O 


.99226 


.99285 


-10 


1000 


.95537 


.95865 


-328 



This is but one of the many fairly accurate determi- 
nations dating back even into the last century. Most 
of our modern physical laws and principles were 
known in the early part of the nineteenth centurj^ 
and a great number of physical properties had been 
investigated. The results of this early period are, 
however, characterized by the absence of all data 
by which it is possible to find anything more than the 
relative values of different quantities. The powers 

1 The results quoted by Biot, though creditable for his time, were 
generally inaccurate in the fourth and sometimes even in the third 
place of decimals. They were, nevertheless, carried out, according 
to the custom of early observers, to 7 and 8 decimal placea 



INTRODUCTION. 691 

of di£Ferent metals to conduct heat were, for instance, 
given by Despretz as follows, counting gold as 1,000 
(Ganot's Physics, § 404) : — 





Despreti. 


Wiedemann and Fiani. 


Platinum 


081 


158 


Silver 


973 


1880 


Copper 


897 


1384 


Iron 


374 


202 


Zinc 


363 


874 


Tin 


304 


273 


Lead 


179 


160 



That these results were not particularly accurate 
may be inferred by comparing them with those of 
Wiedemann and Franz (1853), reduced in the right- 
hand column to the same system.^ Thus platinum, 
which is the best conductor of heat according to 
Despretz, is the worst according to Wiedemann and 
Franz. Even, however, if we assume the accuracy 
of either set of results, it is still impossible to apply 
them unless we know, in a single case, how much 
heat flows from one place to another through a bar 
or plate of given length, breadth, thickness, and 
material, and the difference of temperature to which 
this flow of heat corresponds. 

The determination of relative values (such as are 
contained in the table above) is in general a much 
easier task than the determination of absolute values 
(see Table 8, et aeq.} ; and has the advantage that 
gross errors are not so likely to be made. 

Relative measurements are, however, to a certain 

^ Wiedemann and Franz counted silver aa 100 See Deschanel'a 
Natural Philosophy, § 388. 



592 INTRODUCTION. 

extent non-committal, and hence justly unpopular 
with scientific men. The highest end of physical 
measurement is not attained unless every quantity 
with which it has to deal is compared directly or 
indirectly with the so-called absolute units (§8) which 
lie at the base of the system. Quantities subjected 
to such comparisons are said to be determined in ab- 
solute measure. 

We have seen that, historically, in various branches 
of science, the absolute system of physical measure- 
ment has been approached by a series of stages. The 
first stage may be called classification * the second, 
ordination ; the third, numbering ; the fourth, gradu- 
ation ; the fifth, comparison ; the sixth and last, de- 
termination. The first two stages deal with qualities, 
and involve only qualitative experiments. Physical 
measurement is properly confined to the last two 
stages. It deals exclusively with the numerical rela- 
tions between diflferent physical quantities. Meas- 
urements are, accordingly, quantitative in their 
nature. 

It is unnecessary to distinguish physical measure- 
ment from measurement in general, as the term is 
usually employed. It is only physical quantities 
which are capable of being measured. Measurement 
implies observation; exact measurement implies ac- 
curate observation. The observation required in 
physical measurement is, it is true, exceedingly lim- 
ited in its character (see § 23). In the natural sci- 
ences, the powers of observation have their widest 
application. In physical measurement the sharpest 



INTRODUCTION. 698 

use of this faculty is required. The student is apt 
to imagine that an increase of precision in the in- 
struments at his disposal would relieve the contin- 
ual tax which he feels upon his power of observation. 
Quite the reverse is generally true. The better the 
instrument, the harder it is to do justice to it. One 
must learn to obtain the best possible results with 
rough instruments before one is fitted to use instru- 
ments of precision. The habit of accurate observa- 
tion is an important object to be gained by a course 
of physical measurement. 

The most accurate results in physical measurement 
often require practice, not only in observation, but 
also in manipulation. The skill acquired in a course 
of quantitative determinations is an advantage by 
no means to be overlooked. 

The principal benefit to be expected from a course 
of laboratory instruction is, however, familiarity with 
the experimental method and the processes of induc- 
tive reasoning which it involves. Certain of these 
processes belong especially to quantitative determi- 
nations. The results of physical measurement fre- 
quently depend, not only upon a long series of 
observations, but also upon a more or less compli- 
cated chain of reasoning, including the mathematical 
calculations by which the observations are reduced. 
A single error in any one of the data, or in any step 
in the process of reduction, will in most cases entirely 
change the result. The student is not, however, in 
physics as in philosophy, necessarily misled by such 
an error. Physical measurement abounds in what 



594 INTRODUCTION. 

are called " check methods " (§ 45), by which errors 
either in observation or in reasoning may generally 
be detected. Having once discovered the sources of 
error into which he has fallen, the student is less 
likely to commit the same errors in the future. The 
result of a course of physical measurement should be 
to give him a just confidence in what he has seen 
with his own eyes, and in what he has reasoned out 
in his own mind. 

The student should learn, as early as possible, to 
distinguish between real and apparent accuracy. A 
kilogram of wood may, for instance, be weighed to 
a milligram on a good balance. Such a weighing 
would be called precise. The true weight would, 
however, be very inaccurately determined, if no ac- 
count were taken of the buoyancy of the atmosphere, 
which may amount to several thousand milligrams. 

A given degree of accuracy implies an equal degree 
of precision ; but precision does not necessarily imply 
accuracy. Exact results are those which are both 
accurate and precise. 

When a measurement, however inaccurate, is re- 
peated several times in exactly the same manner^ more 
or less concordant results are usually obtained. The 
object of the scientific observer is not to make his de- 
terminations look more accurate than they really are, 
but, on the contrary, to bring to light the errors by 
which they are aflFected. He seeks accordingly every 
possible variation of the conditions under which an 
experiment is tried, in order to bring out discordances^ 
if possible, between methods which ought (as far as 



INTRODUCTION. 696 

he knows) to give exactly the same result. The 
simplest changes — the manner, for instance, of sup- 
porting an instrument — have frequently a most un- 
expected effect, and lead to the disclosure of unknown 
sources of error. 

The student must not be discouraged by the dis- 
covery that his results are^ less accurate than he ex- 
pected. He will find by comparing together the 
determinations of distinguished scientific men, that 
great discrepancies frequently exist between them. 
He must not be deceived by the number of decimal 
places to which their work is carried out. According 
to a custom prevalent, especially in the early part of 
this century, 3, 4, and even 5 figures, having little or 
no significance (§ 66) are often appended to results 
(see footnote, page 690). Within the last twenty 
years, the physical constants have acquired certain 
conventional values. There is an undoubted tendency 
to publish determinations by which these values are 
confirmed, and to suppress others equally good, lead- 
ing to different results. The concordance of modern 
determinations is therefore, to a certain extent, appar- 
ent rather than real. 

From time to time (as every one knows who follows 
scientific proceedings) inaccuracies in the accepted 
values of the physical constants force themselves 
upon our attention. In view of these facts, the 
student should return with increased confidence to 
his own determinations. When an investigation has 
been completed, and all sources of error, in so far 
as possible, allowed for, the facts should be made 



696 INTRODUCTION. 

known, no matter who has arrived at a different 
result. 

The student should learn to value different deter- 
minations for what they are worth. It is a very- 
rough weighing that is not accurate within one part 
in a thousand ; but some of the best electrical meas- 
urements are subject to much greater errors. 

The results of some observers in determining the 
conductivity of different substances for heat are twice 
as great as the results of others ; these results are 
however, useful. They show, for instance, that it 
would be impracticable to heat a house by a system 
of conducting rods radiating from a common centre ; 
but that the thin metallic coatings of a furnace offer 
a comparatively slight resistance to the passage of 
heat. A knowledge even of the number of ciphers 
necessary to express the magnitude of certain quanti- 
ties, — as, for instance, the weight of molecules, — may 
be useful in certain calculations. The fact that some 
measurements are necessarily inexact should not pre- 
vent the student from doing his best where accurate 
work is possible. 

The results of physical measurement can, from 
their nature, never be, like those of mathematics, 
perfectly exact. Errors of greater or less magnitude 
are not only possible, but we may say almost certain 
to occur. Herein lies an important distinction be- 
tween mathematical and physical problems. A mathe- 
matical solution is either right or wrong. In regard 
to the results of physical investigations, we have to 
consider how far each is likely to be in error. The 



INTRODUCTION. 597 

quantitative methods which characterize physical 
measurement are extended even to the errors com- 
mitted in these measurements. The treatment of 
such problems forms an important branch of the 
mathematical theory of probability, upon which all 
inductive methods are founded. It is not easy, from 
a philosophical standpoint, to regard the probable ac- 
curacy of results obtained by observation in exactly the 
right attitude. One cannot strictly afSrm the accu- 
racy of any figure in a result ; but, as concerns some 
figures, it is difficult if not impossible to formulate 
the slightest doubt without enormously exaggerat- 
ing the real uncertainty. Discussions of '* probable 
error'' (§§ 60-62) are characteristic of physical meas- 
urement, and teach a species of reasoning which, in 
problems of insurance, has assumed great practical 
importance. 

One of the principal advantages derived from a 
course of physical measurement is, as has been said, 
the acquisition of habits of accurate thinking. When 
two quantities have been compared together, it is evi- 
dent that, if the magnitude of one is known, that of 
the other must be determined. It is not, however, al- 
ways clear what is determined by a given observation. 
It must be borne in mind that a physical determination 
consists, essentially, in the comparison of a quantity 
with one better known than itself. At the beginning 
of this century, the density of water at high temper- 
atures was known only within a few tenths of 1 %. 
To-day, the density of water is one of the best 
known physical constants. The same experiment 



698 INTRODUCTION. 

(Exp. 19) which one hundred years ago constituted 
a determination of the density of water, now fur- 
nishes data only for calculating the volume of a solid, 
or the rate of expansion of the material of which it is 
composed. Great care must be taken to make a 
proper use of the results of physical measurement. 
One may, for instance, measure the circumference and 
radius of a circle, and from the results calculate the 
ratio which one bears to the other. It would, how- 
ever, be incorrect to speak of this experiment as a 
determination of the ratio in question i since this 
ratio, being capable of exact mathematical calculation, 
is better known than the scale readings upon which 
the result depends. Physical measurement may be 
occasionally employed as a check upon mathematical 
calculations, particularly when (as in certain applica- 
tions to physics) there is any doubt as to the validity 
of the assumptions upon which the calculations de- 
pend. Any attempt, however, to establish mathe- 
matical principles by data obtained from observation 
is an obvious abuse of the experimental method. 

The so-called "proofs" of well-known physical 
laws and principles founded upon rough and insuffi- 
cient data are hardly less objectionable.^ The use of 
the experimentsil method as an illustration of such 
laws is not denied. One of the objects, however, of 
a course of physical measurement is to teach a stu- 



^ It may be remarked that the Law of Boyle and Mario tie (§79) 
was thus taiiglit and implicitly believed in for more than a century, 
before more exact observation showed that this law is only approxi- 
mately fulfilled. 



INTRODUCTION. 599 

dent how to make the best use of the tools at his 
command. The laws and principles which have been 
most carefully studied by scientific men should be 
made the instruments, not the objects of elementary 
research. The teacher should avoid, in so far as pos- 
sible, experiments whose ostensible object is to estab- 
lish well-known facts, — like the conservation of en- 
ergy,— the truth of which is not really in question. 

Among the habits of accurate thinking which it is 
the object of physical measurement to teach, may be 
mentioned those involved in a diligent and methodi- 
cal search after the errors which are likely to be 
committed in one's work. It is hoped that the classi- 
fication of errors in Chapter II. may be of assistance 
to the student who is thrown more or less upon his 
own responsibility. It is of couree impossible to an- 
ticipate in any such classification all errors which 
may arise ; but there are certain, kinds of errors of 
such frequent occurrence that one must always be 
on one s guard against them. The student should 
ask himself, for instance, in respect to every scale 
reading, Have errors of parallax been guarded against 
(§ 25) ? Have errors been committed in the estima- 
tion of tenths (§ 26) ? Are there mechanical devices 
by which such errors could be diminished (§ 27) ? 
Has the zero of the scale been carefully adjusted 
(§ 32) ? Has the scale been carefully tested (§§ 31, 
37)? 

In addition to these considerations, by which errors 
may be frequently avoided, there are certain general 
methods, considered in Chapter III., by which (when 



600 INTRODUCTION. 

they can be applied) the accuracy of a result is 
always increased. The student who is planning ^or 
himself the details of a physical measurement should 
consider these general methods one by one. He 
should ask himself, for instance, Is the method pro- 
posed the most direct (§ 36) ? Could not more ac- 
curate results be obtained by dealing with larger 
quantities (§§ 38, 39) ? or quantities which happen 
to be more nearly coincident (§ 40) ? Could not pre- 
cision be gained by the use of differential instruments 
(§§ 41, 42) ? or accuracy by the check methods (§§ 43- 
45) ? Would it be possible to reverse or interchange 
the quantities compared (§ 44) ? or to obtain and 
average results from several determinations (§ 46) ? 
These and similar questions must occur habitually to 
every successful observer. 

A course in physical measurement is not especially 
suited to students who wish to become acquainted 
with a wide range of physical phenomena. Dealing, 
however, with quantities of nearly every description, 
and with the numerical relations which exist between 
them, it affords numerous examples of the application 
of physical laws and principles. It is only through the 
aid of definite examples that most persons can arrive 
at an understanding of physics. It has been assumed 
in the experimental course described in Parts I. and 
II. of this book, that the student is already familiar 
with the statements of physical phenomena contained 
in ordinary text-books. If this is the case, he must 
expect to gain definiteuess rather than scope in his con- 
ceptions from a course of quantitative determinations. 



INTRODUCTION. 601 

It would be impossible, in the limited space which 
can be devoted to the subject in the present volume 
to describe or explain in full more than a very small 
part of the principles which underlie physical meas- 
urement. The brief notes contained in Chapters V.- 
X. are intended simi)ly to recall to the student (who 
has already taken a course in general physics) the 
laws and principles which he has to employ, and the 
proofs upon which they rest They may also be use- 
ful to the instructor as a basis for his lectures, or to 
the student who is just beginning the study of phys- 
ics as a *' syllabus " of what he should read in oi*der 
to follow intelligently the course of physical measure- 
ment described in Parts I. and II. For a full explan- 
ation of the physical principles involved in this 
course, the student is referred to the standard works 
of Daniell, Deschanel, and Ganot 

The advantages of a course in physical measure- 
ment have been considered chiefly from an educa- 
tional standpoint. It is hardly necessary to point 
out that Physical Measurement is a science of great 
practical importance. The nice adjustments of the 
different parts of a machine would, for instance, be 
impossible without accurate measurements. Success 
in Chemistry, in Astronomy, in Surveying, in fact 
in all branches of Civil and Electrical Engineering, 
depends to a gi-eat extent upon a thorough under- 
standing of the Principles and Methods of Physical 
Measurement. 



CHAPTER I. 

GENERAL DEFINITIONS. 

§ 1. Nature of Measarement. — Measurement con- 
sists in finding out by observation how many things 
of one sort correspond in magnitude to a given num- 
ber of another sort. When 10 spaces on a measure 
divided into inches are found to reach through the 
same distance as 254 spaces on a millimetre scale, the 
length of the inch is said to be measured in milli- 
metres, and conversely the millimetre may be said to 
be measured in inches. Either the millimetre or the 
inch may be used as a standard of comparison. When 
a quantity of known magnitude is compared with 
one of unknown magnitude, the latter is said to 
be measured in terms of the former. Thus, if a load 
is found to be equal in weight to a given number ot 
grams, its weight in grams is said to be measured. 
It is obviously impossible to compare, in general, 
magnitudes of different sorts, — as, for instance, 
length and volume ; but under certain circum- 
stances, correspondences or relations exist between 
such quantities. When a stream of water, for in- 
stance, striking an obstacle with a velocity between 
2 and 3 miles per minute is found to warm itself 1 
Fahrenheit degree, a certain relation between tem- 
perature and velocity is said to be established. Such 
relations are properly objects of physical measure- 



§2.J THE METRIC SYSTEM. 603 

ment. Measurements are either relative or absolute 
(§ 8), and may be classed, accordingly, as compaii- 
sons or determinations.^ 

§ 2. The Metric Bystem. — The metric system is 
now generally adopted in scientific work. It is so 
called from the metre, or standard of length upon 
which it is founded (§ 5). The metre is equal to 
about 39.37 English inches. A cubic metre of ice- 
water weighs 1 "tonne" (1,000,000 grams) or 2205 
lbs. nearly. There are, accordingly, 15.432 grains, 
or about 15 drops of water in one gram (§ 6). In the 
metric, as in other systems, the unit of time is the sec- 
ond (§ 7). The chief advantage of the metric system 
consists in the simplicity of the relations which exist 
between the standards of length and mass, and in 
the use of units each of which is some decimal mul- 
tiple or sub-multiple of the others in the same series. 

These units are distinguished, in the metric sys- 
tem by the aid of prefixes, which have the follow- 
ing significations : mega^ one million \ kiloy one thou- 
sand ; heeto^ one hundred ; deka^ ten ; deci^ one tenth ; 
centi^ one hundredth ; milli^, one thousandth, and micro 

1 The word ** absolute " must not be confounded with the word 
"exact." Measurenienta are said to be "absolute*' only when/uM- 
damental standards or units are employed (see § 8). We speak of the 
measurement rather than the determination of variable quantities, as 
for instance the strength of an electric current We speak also of the 
measurement of accidental quantities, like the length or weight of a 
body, especially when, as in measurements of length, dire^ methods 
can be employed. (See Chap. III.) On the other hand, a magnitude 
is said to be " determined '' rather than ** measured *' by an arbitrary 
scale, and measurements of invariable quantities, like the physical 
constant!, are customarily called *' determinations " 



604 GENERAL DEFINITIONS. [§3. 

one millionth. Thus a kilometre means a thousand 
metres ; a microvolt a millionth part of a volt. When 
the unit begins with a vowel, the last vowel of the 
prefix is generally omitted; thus a million ohms is 
called a megolim. 

§ 3. Relative magnitades. — There are certain quan- 
tities which can be defined without reference to any 
particular system of measurement, such for instance 
as include simply a ratio between two things. Thus 
specific gravity is the proportion which the weight of 
a substance bears to that of an equal bulk of water ; 
specific heat the proportion of heat it absorbs as com- 
pared to that absorbed by an equal weight of water ; 
and specific electrical resistance is sometimes, though 
not generally, used in a similar sense.^ Again, strains 
are defined as the proportion of the distortion which 
is produced to the whole quantity acted upon. Thus 
if a body has been stretched or sheared by an amount 
equal to -j-^ of its length, or compressed by ^^ of 
its volume, it is said to have suffered a strain of -j^. 
Angles too are determined ^ by the ratio of the arc 
which they subtend to the radius; and the sine, cosine, 
or tangent of any angle ^ is simply the ratio between 
two of the three sides of a right-angled triangle in 
which the given angle occurs. Another instance is 
the index of refraction, or ratio of the velocity of a 
wave outside of a medium to its velocity in it. It 



1 See Experiment 88 ; also Trowbridge, New Physics, Experiment 
120. 

3 See Table 3, columns a and c. 
* See Table 3, columns b, e, and/ 



S 5.] UNIT OF LENGTBL 605 

is clear that when only a ratio is concerned, the results 
from all systems must agree. 

§ 4. Scale of Temperature. Our present scale of 
temperature, though recently introduced, is equally in- 
dependent of any particular system of units by which 
other physical quantities are measured. 

The temperature of melting ice is defined as 0"" on 
the centigrade scale ; that of condensing steam as lOO"" 
under a standard atmospheric pressure, or that which 
sustains at Paris a column of mercury 76 cm. long, and 
at 0°.i At other points temperature is measured pro- 
visionally by the indications of a mercurial thermom- 
eter made of ordinary glass, the tube being divided into 
100 parts of equal capacity between 0° and 100.° 

It is assumed that a thermometer reaches, after a 
time, the same temperature as the bodies with which 
it is in contact.^ 

§ 5. Unit of Length. — The unit of length adopted 
in nearly all scientific work is the centimetre, or hun- 
dredth part of the length, at 0° centigrade, of a stand- 
ard metre still preserved in the French Archives. This 
metre was intended to be the ten-millionth part of the 
distance along a meridian from the equator to the 
poles, but it was made about f of a millimetre too 
short, the earth's quadrant being now supposed to 
lie between 10,007 and 10,008 kilometres ; being, 
moreover, subject to shrinkage, though the amount 
has never been measured. The only absolute deter- 
mination of the centimetre which we possess is in 

1 See § 6 below ; also Table 14. 

^ For a further discussion of temperature see § 74. 



606 GENERAL DEFINITIONS. [§ 8. 

wave-lengths of light. It contains, for instance, 
16,972 waves of sodium light in air. 

§ 6. Unit of Mass. — Our unit of mass is the gram, 
or thousandth part of the standard kilogram of the 
French Archives, which was intended to be equal to 
the weight in a vacuum of a cubic decimetre of dis- 
tilled water at its temperature of maximum density 
(very near 4° centigrade). In addition to the error in 
the metre already noticed, the standard kilogram was 
made about 13 milligrams too light ; but if this is 
taken into account, the gram can easily be reproduced 
from a given standard of length which has been com- 
pared either with the original metre or with wave- 
lengths of light. (See § 152.) 

§ 7. Unit of Time. — The unit of time which we use 
is the second, of which there are 86,400 in a mean 
solar day. The second depends therefore on the ro- 
tation of the earth with respect to the sun. As no 
change has been detected in the rotation of the earth 
by comparing it with other astronomical motions, the 
second would seem to be practically constant. In 
one second, sound passes through 83,220 centimetres 
of dry air at 0° centigrade; light through 30 thou- 
sand million centimetres of empty space, as nearly as 
we can tell. From any of these data the second 
could be reproduced independently of the rotation of 
the earth. 

§ 8. Absolute Bystein. — The system followed in this 
work is that recommended by the British Association, 
and is known from its fundamental units as the centi- 
metre-gram-second system, often abbreviated C. G. S. 



§11] ACCELERATION. 607 

The three units of length, mass, and time are called 
fundamental, because all other units of this system are 
derived from them ; and they may be called absolute, 
because they can be reproduced (without the use of 
any standard) from the general properties of such uni- 
versal substances as salt, water, and air. It is in this 
sense only that any system of measurement may be 
called absolute. 

§ 9. Surface, Volume, and Density. — Surface or area 
is measured in square centimetres ; volume or capac- 
ity in cubic centimetres ; density in grams per cubic 
centimetre. Density in general is defined as the ratio 
of mass to volume. (See § 154.) 

§ 10. Velocity. — Velocity is expressed in centime- 
tres per second. It is well to remember that a veloc- 
ity of one hundred centimetres per second or one 
metre per second corresponds to a very slow walk, 
only a little over two miles per hour. It is incorrect 
to speak of a velocity of so many centimetres, or of 
so many miles. A railway train may move at the rate 
of one mile per minute, while a steam roller makes 
only one mile per hour. Both the distance traversed 
and the time occupied in so doing are necessary to 
specify a velocity. 

§ 11. Acceleration. — Acceleration is defined as the 
rate of change of velocity,^ or the change of velocity 
per unit of time. If a steamer starting from a wharf 
acquires in one minute a velocity of three miles per 
hour, in two minutes a velocity of six miles per hour, 

^ For a diBCUBsion of what is meant by a change of velocity, see 
f 105. 



608 GENERAL DEFINITIONS. [§ 1». 

in three minutes a velocity of nine miles per hour, 
etc., increasing its velocity every minute by three 
miles per hour, we should say that its acceleration 
amounts to three miles per hour per minute. It 
would be incorrect to speak of its acceleration as 
three miles per hour, for a horse and carriage might 
acquire the same velocity in one second. 

It is necessary to state not only the magnitude of 
the velocity acquired but also the time it takes to 
acquire it. Since velocity is measured in centimetres 
per second, and time in seconds, acceleration is ex- 
pressed in centimetres per second per second. The 
repetition of the words "per second" in scientific 
works is not therefore, as is commonly supposed, 
simply a printer's favorite mistake. 

§ 12. Force. — The dyne or unit of force is defined 
as that force which acting on a gram for a second 
would give it a velocity of one centimetre per second. 

A dyne is almost too small a force to be felt. It 
may be thought of as the weight of a piece of very 
thin tissue-paper a centimetre square; meaning by 
weight the force with which, for instance, it presses 
against the hand. In the same sense a drop of water 
weighs from 50 to 100 dynes ; a man from 60 to 100 
millions of dynes. 

The dyne can be best represented by means of a 
delicate spring-balance. The weight of a gram in 
latitude 40°-45° is shown by such an instrument to 
be about 980 dynes; at the equator, however, it is 
only 973 dynes, and at the poles nearly 984. The 
weight at the centre of the earth would be nothing. 



§ 14.] WORK. 609 

On the other hand a given number of dynes as above 
defined always stretches the balance to a given mark, 
whether at the equator or at the poles. Hence we 
say that the weight of a gram varies,^ but the dyne, 
in terms of which we measure it, remains always the 
same. Force in general is measured as the product of 
mass and acceleration. (See § 106 and § 153.) 

§ 13. Couple. — The tinit couple is a force of 1 dyne 
acting on an arm 1 centimetre long, at right angles to 
it, with an equal and opposite force at the other 
end of the arm. A couple consists in general of 
two equal forces acting in opposite directions, not in 
the same straight line but in two parallel lines, and 
is measured by multiplying together either force in 
dynes by the arm, or perpendicular distance between 
the two lines of action. Anything which can twist a 
body or make it spin contains a couple ; anything 
which can push it or pull it or shove it to one side 
contains a force. All motions originate either in 
forces or in couples or in combinations of forces and 
couples. (See § 113.) 

§ 14. "Work. — The unit of work is the erg, defined 
as the amount of work done in moving through a dis- 
tance of one centimetre against a resistance of one 
dyne. It makes no difFerence how long it takes to 
complete the motion ; but we assume that there has 
been no gain or loss of velocity on the part of the 

^ By the weight of a gram is here meant the varying force with 
which gravity attracts it. This is the proper signification of weight. 
Some writers, however, use weight in the sense of mass, or quantity of 
matter. The mass of a gram is by definition constant. See " Elemen- 
tary Ideas, etc.," by E. H. Hall (published by Sever, Cambridge). 



610 GENERAL DEFINITIONS. [§ 16 

moving body, since that would also have to be taken 
into account. (See § 121.) Work in general is 
measured as the product of the force in dynes, and 
the motion in centimetres ; considering of course only 
the effect or component of the force in the direction 
of the motion. (See § 119.) When the force acts on 
a body in the direction in which it is moving, it is said 
to do work upon the body ; when the force opposes 
the motion, the body is said to do work against the 
force. 

Those who have been accustomed to measure work 
in foot-pounds (multiplying the motion in feet by the 
number of pounds which have been raised), may no- 
tice that the erg or dyne-centimetre naturally replaces 
the foot-pound in a system in which all forces are 
measured in dynes and all distances in centimetres. 

While three hundred foot-pounds in England are 
the same thing as three hundred and one foot-pounds 
in Brazil, the erg has one great advantage in that it 
is the same all the world over. Ten million ergs are 
sometimes called a joule. 

§ 15. Power. — The practical unit of power is the 
watt, or ten million ergs per second. A man can 
easily do the work of 100 watts. One horse-power 
is rated at 746 watts. It takes about 4.166 watts to 
generate, through friction, one unit of heat per second. 
(See below.) A common paraffine candle is equiv- 
alent in heating power to 60 or 70 watts ; 10 or 12 
candles represent a horse-power. 

§ 16. Unit of Heat. — The unit of heat is the quan- 
tity required to raise a gram of water from O'' to V 



§20] THE OHM. 611 

centigrade. It takes about forty-two million ergs to 
bring this about ; more exactly, 41,660,000 ; hence this 
number is said to represent the mechanical equivalent 
of heat. Other substances take more or less (gen- 
erally less) heat than water to raise 1 gram of them 
1° in temperature, and more or less work in propor- 
tion. This proportion determines the specific heat of 
the substance in question. (See also § 86.) Specific 
heat is strictly defined as the number of units of heat 
necessary to raise 1 gram of a given substance V in 
temperature. 

§ 17. Unit of MagnetiBm. — A unit quantity of mag- 
netism is one which attracts or repels an equal quan- 
tity at a centimetre's distance with the force of 1 
dyne. There are two kinds of magnetism, positive 
and negative. Two positives or two negatives repel 
each other, while positives and negatives attract. 

§ 18. Unit of ZUectrical Ciirrent. — The absolute 
C. G. S. unit of electrical current is one which in 
flowing through a centimetre of wire acts with a 
force of 1 dyne upon a unit of magnetism, distant 
1 cm. from every point of the wire. 

§ 19. The Ampere. The practical unit of current is 
the ampdre or tenth of an absolute unit. A common 
quart Daniell cell will give a current of about 1 ampere 
under favorable conditions. 

§ 20. The Ohm. — The practical unit of resistance is 
the ohm. It was intended to be the electrical resis- 
tance of a wire in which a current of 1 ampere would 
generate in one second an amount of heat equivalent 
to 10,000,000 ergs. That is, an engine of 1 watt 



612 GENERAL DEFINITIONS. [§21. 

power would keep up a current of 1 ampfere through 
such a resistauce. In point of fact the standard ohm 
prepared by the British Association is a little more 
than 1% too small, and as this error has been kept in 
our copies, we have to allow for it in our calculations. 

The ohm may be remembered as the resistance of 
about fifty metres of copper wire 1 mm. in diameter, 
or as that of a column of mercury 106 cm. long aud 
1 sq. mm, in cross section. The value of the latter 
resistance at 0° is adopted in France and elsewhere 
as the legal definition of the ohm. The liquids of a 
quart Daniell cell usually offer a resistance of about 
1 ohm. 

The resistance of a conductor in general is numer- 
ically equal to the power necessary to maintain a unit 
of current through it. 

§ 21. The Volt. — The practical unit of electromo- 
tive force is the volt, or that which is required to 
maintain a current of 1 ampere through a resistance 
of 1 ohm. A Daniell cell has an electromotive force 
of about 1 volt. 

Electromotive force in general is defined as the 
ratio of the power (§ 15) to the current. We have 
seen that it takes one watt to maintain a cun*ent of 
1 ampk-e through a resistance of 1 ohm ; and that it 
takes 1 volt to do the same. It will not do to con- 
clude that one volt is the same thing as one watt ; 
two volts will keep up a current of two ampferes 
through one ohm, but four watts will be required. 
Electromotive force corresponds not to power but to 
hydrostatic pressure. (See §§ 137-139.) 



S 22.] INTENSITY. 613 

§ 22. Intensity. — There are various other terms a 
definition of which might be useful here, but it has 
been thought better to explain each as the necessity 
arises. The use of the word "intensity" in the sense 
of concentration is, however, important. By intensity 
is meant the proportion of one quantity per unit of 
some different quantity. The force in dynes (about 
980) with which gravity attracts each gram of matter 
is sometimes called the intensity of gravity. Intensity 
of pressure, generally called simply pressure^ is ex- 
pressed in dynes per square centimetre, corresponding 
to the ordinary use of pounds per square inch. The 
pressure of the atmosphere is, for instance, about one 
megadyne per sq. cm,^ averaging in this latitude about 
1.3% more than this. Intensity of stress, or simply 
9tre%8 is measured in the same units ; as when we say 
that steel bars break under a stress of eight thousand 
megadynes per %q. cm. In the same way intensity of 
illumination ought to be expressed, not as it often is, 
in candle power, but in candle power per square centi- 
metre of surface illuminated. Intensity should always 
be distinguished from quantity in this way. Like rate 
with respect to time, or the word per'^ with respect 
to quantities in general, intensity signifies a ratio or 
proportion. 

1 Everett's Units and Physical Constants, page 10. 



CHAPTER IL 

OBSERVATION AND ERROR. 

§ 23. Coincidence. — Almost every physical meas- 
urement involves the reading of a scale of some sort, 
by means of what may be called an index or pointer. 
Temperature, for instance, is measured by a thermo- 
meter, consisting of a tube of glass with a scale marked 
upon it, let us say in degrees, and an index of mercury 
or some other liquid moving up and down the tube. 
Aneroid barometers, pressure-gauges, clocks, com- 
passes, and galvanometers are read by a hand or pointer 
of some sort moving over a dial. An ordinary balance 
has an index, and a small scale behind it to show, 
when the weights are nearly adjusted, which pan is 
the heavier, and how much. Spring balances are read 
by the position of a small index. When the length of 
a body is measured by the scale on a metre rod, one 
end of the body is used as the index ; or, again, a mark 
on a sliding scale is used as an index with respect to 
a fixed scale, and conversely. The above list contains 
H small part of the various instruments used in physi- 
cal measurement; but a great part of those from which 
numerical results are actually obtained. Most obser- 
vations therefore* consist in reading scales of various 



§24.] CLASSIFICATION OF ERRORS. 615 

sorts, by noticing the point with which the index ap- 
parently coincides. 

The coincidence of two objects in position may be 
determined with great delicacy by the touch, or the 
coincidence of two sounds in time by the ear ; but 
most observations relate to the coincidence or agree- 
ment of two phenomena both in space and in time^ and 
can be made conveniently only by the eye. 

§ 24. ClaBsiflcation of Errors. — It is obvious that 
mistakes are likely to arise in observation, as when 
we take a figure 3 for a figure 8 ; but mistakes of this 
sort should be distinguished from erroi-s proper. A 
reasonably small error is more likely than a large one ; 
but a mistake in the thousands is as probable as in 
the units. (See § 156.) 

Errors may be divided into two classes : constant 
errors, or those which always tend to increase or to 
diminish a result by a definite amount ; and accidental 
errors, or those which tend sometimes to increase it 
and sometimes to diminish it. Constant errors can be 
allowed for if we have sufficient information about 
them ; but no correction can be applied for accidental 
errors. 

For instance, in measuring length, the temperature 
of a tape, the moisture which it may have absorbed, 
the strain upon it, and the curvature of the surface 
measured, all affect the result. It is impossible to 
predict whether the temperature will be higher or 
lower, the dampness greater or less, the strain more 
or less intense than when the tape was graduated. 
We study accidental errors as we would combinations 



616 OBSERVATION AND ERROR. [§25. 

of "heads and tails" in tossing coins. No result is 
entirely free from them. Their influence may be 
indefinitely reduced (§ 46), but never completely 
eliminated. 

Errors may further be distinguished into three 
classes : first, errors of observation (§§ 25-30) ; second, 
instrumental errors (§§ 31, 32) ; and third, errors of in- 
ference (§§ 33, 34). The various methods of avoiding 
errors of observation are considered below in connec- 
tion with the sources from which they arise, the com- 
monest of which are as follows : uncertainty in a point 
of view (§ 25), the coarseness of a scale (§ 26), the 
minuteness of the object observed (§ 27), the neces- 
sity of observing two diflferent things at the same 
time (§ 28), the unequal rates at which different sen- 
sations are transmitted (§ 29), and the effect of mental 
impressions (§ 30). 

§ 25. ParaUaz. — In many scales where the index is 
between the graduation and the eye, the apparent posi- 
tion of the pointer is affected by the point of view. 
The index seems to slide along the scale as the eye 
moves from one end to the other. This phenomenon 
is called parallax (from irapd^ along, and aWao-o-o), 
to alter). Clearly to avoid errors from parallax, the 
eye must be held in a fixed position so as, for in- 
stance, to look perpendicularly upon the scale. To 
this end one of the simplest devices is to use a mir- 
ror parallel to the scale and behind it if possible. 
The eye is placed so as to see its own reflection in the 
mirror in the direction of the pointer ; in this case the 
line of sight must be perpendicular to the scale. 



§26.J ESTIMATION OF TENTHS. 617 

§ 26. Estimation of Tenths. — One may readily dis- 
tinguish in most cases whether the pointer apparently 
coincides with a certain mark on a scale, or with the 
space between two marks; but this is by no means 
the limit of the eye's accuracy. If the pointer falls 
between two marks, it is generally possible to decide 
whether it is half-way between them, or nearer to one 
than to the other. In other words, the eye is accurate 
to fourths. It is, in fact, possible to imagine the space 
between two marks in an ordinary scale divided into 
it least ten parts, and to decide correctly in the ma- 
jority of cases in which of these parts the pointer lies. 



•o 

r 



It t ^l I t t it 

Fig. 1. 



The ten diagrams in Fig. 1 show the relative positions 
of a pointer dividing the space between two marks 
into various proportions, the figures indicating the 
. number of tenths to the left of the pointer in each 
case. A close study of such diagrams will in a short 
time justify the division of spaces into tenths by the 
eye. It is assumed henceforth that in the case of any 
index and scale under favorable conditions, the read- 
ing is expressed in tenths of the smallest divisions. 
The estimation of tenths is not confined to the eye. 
It will be found that the ear is equally reliable. Thus 
the time between two ticks of a clock can be divided 
into tenths, so that the occurrence of a sound can be 
determined with practice to a tenth of a second. 



618 OBSERVATION AND ERROR. [§27. 

§ 27. Mechanical Devices. — When a space or line is 
too small to be seen we generally resort to a lens or 
microscope, as in Experiment 19 ; but there are vari- 
ous other devices to measure small distances. One of 
the most delicate tests of the adjustment of the four 
points of a spherometer to the same plane is the noise 
made by rocking thfe instrument from side to side, 
(see Experiment 20), and an electrical contact is sen- 
sitive to a change of distance which the eye fails to 
see (see Experiment 65). The motion of the top of a 
vacuum chamber in an aneroid barometer is magnified 
by a system of levers, and finally by a chain passing 
round a small axle so as to render the smallest motion 
perceptible. When a motion is too rapid to be seen 
by the naked eye, we may still often observe it through 
some optical device. An instantaneous view, for in- 
stance, will show the body as if at rest, and in the case 
of periodic motion a series of instantaneous views may 
give it an apparent motion so slow that it is easily ob- 
served (see Experiment 51). Again motion may be 
made to record itself by marking on a moving surface. 
The vertical motion of a barometer is thus recorded 
by means of a pen on a piece of paper moving by 
clockwork horizontally beneath it. This method is 
called graphical. Any instrument which moves uni- 
formly so that time can be accurately recorded in this 
way is called a chronograph, literally a time-writer 
(from 'xp6vo<;^ time^ and r^pd<f>fa^ to write). A chrono- 
graph can be used to record the vibrations of a tun- 
ing-fork, even one which emits the highest or fastest 
audible note. 



§29.1 PERSONAL KQUATION. 619 

Similar results can be obtained wheu the pen is not 
moved directly by the tuning-fork or moving body, 
(see Trowbridge, New Physics, Experiment 155), but 
indirectly through the aid of electricity, and various 
electrical devices may be employed to magnify the 
effects of small intervals of time, and thus detect the 
smallest variation from coincidence (see ^ 147). 
Optical, Graphical, and Electrical Devices include 
the principal methods of aiding observation. 

§ 28. Use of Two 8611868. — When we wish to ob- 
serve two things in different places at the same time 
we often resort to the use of two senses. The Eye 
and Ear method ^ consists, for instance, in the use of 
the eye to watch one moving body while the ear 
listens for the occurrence of a sound defining the mo- 
tion of another. 

This is the method by which one ordinarily com- 
pares his watch with a striking clock or with a noon • 
gong. The sense of touch is used by the engineer to 
help him count correctly the revolutions of a wheel 
without looking off his watch, and a variety of meth- 
ods can be devised by which two or more senses bring 
together from different sources a knowledge of what 
is taking place at different places at a given time. 
The use of two senses often obviates the necessity of 
employing complicated mechanical devices. 

§ 29. Personal Equation. — It is generally found that 

the eye is quicker than the ear to report what is 

taking place, but the difference is greater in some 

persons than in others. Thus if two persons were 

1 See Pickering's Physical Manipulation, § 16. 



620 OBSERVATION AND ERROR. [§ 30. 

to estimate at what time the report of a cannon is 
heard, one would tend always to return figures greater 
than the other, let us say by several hundredths of 
a second. Such a difference, however small it may 
seem, might seriously affect a determination like that 
of the velocity of sound, and is a perpetual source 
of annoyance in astronomy. The allowance which 
each person must make to produce results eqtual to 
the true or average result is called his personal equa- 
tion. It is not specially considered in this course of 
measurement, being eliminated together with what is 
called " zero error," as explained in § 32. 

§ 30. Effects of Anticipation. — One of the most dan- 
gerous sources of error in observation lies in the habit 
of anticipating results. Experience shows that under 
the influence of a strong expectation, the eye is not 
only incapable of estimating fractions correctly, but 
that it becomes blinded to gross errors, — pronounces 
weights, for instance, equal when the balance-beam is 
not free to move ; reads sixty-odd centimetres instead 
of seventy-odd, several times in succession. It is 
sometimes necessary to prepare one's self by calcu- 
lating beforehand — particularly in astronomy — the 
values which one expects to observe; but indepen- 
dence of observation is obtainable only in ignorance 
of the meaning of the indications which one records, 
and particularly in ignorance of the fact whether the 
values obtained are likely to be too great or too small.^ 

^ The teacher may amuse himself at the expense of his class by 
determining the effects of " gravitation " towards various values which 
he may choose to suggest. 



§32.] ZERO ERROR. 621 

For these reasons the following rule will be found use- 
ful : Take your observations first ; second, give a copy 
to some one else ; third, reduce them ; fourth, report 
the result ; and fifth, inquire what values others have 
found.^ 

§ 31. Instrumental Errors. — Without any fault on 
the part of the observer, errors often arise through 
the imperfections of the instruments which he em- 
ploys. These may be divided into two classes : first, 
errors of adjustment, as when two parts are not ex- 
actly parallel or perpendicular ; and second, scale 
errors, for instance, irregularities in a graduated rod 
or in a set of weights. 

The various tests which have been devised to cor- 
rect errors of adjustment will be described in connec- 
tion with the several instruments to which they 
belong. Scale errors may arise either from a change 
in, or from the original misplacement of, certain fixed 
points ; like the " freezing " and " boiling " points of 
a thermometer, or from inaccurate calibration. They 
are avoided in general as explained in § 36. The 
commonest error of this sort is a misplacement of the 
zero of a scale. 

§ 32. Zero Error. — When the greatest care has 
been taken to read one end of a scale correctly, an 
error often arises because the other end is out of 
adjustment. The graduation of a tape measure sel- 
dom begins at the ring, and yet it is common to see 

^ The examination of substances whose composition is known only 
to the teacher — or to the apothecary — will afford a sufficient oppor- 
tunity to test the application of this rule 



622 OBSERVATION AND ERROR. [§33. 

distances measured by professional mechanics as if 
this were the case. It is always well, even when 
no error of this sort is suspected, to confirm an obser- 
vation by taking two others, the difference between 
which should agree with a previous result. Thus 
the length of a pencil might be found by laying it 
along the middle portion of a metre-rod instead of 
making one end of it even with the rod, and in this 
manner, even if the end of the rod were worn away 
or broken off, the true length of the pencil would be 
discovered. This is called the method of difference. 

The error due to the inaccuracy of the beginning or 
zero of a scale is called zero error, and it is necessary 
to guard against such errors in general. It should be 
borne in mind that every measurement, like that of 
length, depends upon at least two observations^ or their 
equivalent ; and that the accuracy of one is just as 
important as that of the other. However evident it 
may seem to be that if the quantity which is being 
measured were taken away, the index would point to 
zero, it is continually necessary to test the truth of 
this fact. The balance when both pans are empty, 
from a slight dislocation of one of the knife-edges, 
often tends to one side ; springs do not always return 
to their original length after stretching, owing to a 
permanent set; galvanometer-needles do not always 
point north and south when the current is cut off, 
— a bunch of keys may perhaps account for the 
variation. 

§ 33. Errors of Inference. — One must distinguish 
carefully between what he sees and what he infers. 



§33.] ERRORS OF INFERENCE. 623 

It would be impossible to state any general princfple 
by which errors of inference may be avoided ; but in 
order to correct them, it is often necessaiy to refer to 
the original observations from which the inferences 
have been drawn. Hence the necessity of preserving 
the records, however rough in form, made at the in- 
stant when a given phenomenon occurs. The turning- 
points of an index should for instance be recorded, 
and not simply the position where it is inferred that 
the pointer will come to rest ; or, if at rest, its actual 
position should be noted, not the weight which one 
inferB would produce an exact adjustment. Again, 
the reading of a standard English barometer should 
be written down first in inches, and afterwards re- 
duced to centimetres. 

In addition to the observations necessary to a given 
measurement, every circumstance should be noted 
which may have a possible influence on the result. 
The appearance of air-bubbleS, in hydrostatics, may, 
for instance, determine the relative accuracy of dif- 
ferent weighings. The time of an experiment enables 
us to supply the barometric pressure, roughly, at a 
later date, by consulting a weather report. An exact 
description of place may furnish a subsequent clue to 
the magnetic deviation. We must also be able to 
identify the instruments which we have used, if we 
would confirm the inferences drawn from their indica- 
tions. In fact, the severest test of a laboratory note- 
book must occasionally be applied, namely, one's 
ability to repeat with it a measurement from begin- 
ning to end. 



624 OBSERVATION AND ERROR. [§84. 

It is important to the clearness of one's notes to 
enter actual observations in one place and calculations 
in another. Errors in reasoning are almost always 
due to confusion in regard to the nature of the quan- 
tities dealt with. The student should learn from the 
first to write opposite each number what that number 
represents. Every figure necessary to the calculation 
of a result should be preserved for future reference, — 
even those which enter, for instance, into ordinary 
multiplication or division. In calculation, as in ob- 
servation, corrections are most easily made in those 
records which are most complete. 

§ 34. Logical Analysis. — The use of logical an- 
alysis for the purpose of discovering unknown sources 
of error is seldom dwelt upon by writers on physical 
measurement. It is, however, obvious that the re- 
duction of results may be thrown into the form of 
a demonstration ; and after errors of observation 
have been allowed for, if the reasoning is coiTect, 
unknown errors must lie in the assumptions. It is, 
therefore, important to determine what these assump- 
tions are. 

Thus in the case of a Nicholson's hydrometer 
we reason that since the weight required to sink 
it to a given mark is, let us say, 30 grams at 10 
o'clock without a load, and 10 grams at 11 o'clock 
with a load, assuming that a given weight always pro- 
duces a given result^ the apparent weight of the load 
must have been equivalent to that of 20 grams, accord- 
ing to the set of weights. 

Both theory and experiment show that the assump- 



§34.] LOGICAL ANALYSIS. 626 

tion is true only when the temperature of the water 
is constant and when various other conditions are 
fulfilled. Changes in quantities which we uncon- 
sciously assume to be constant are a frequent source 
of error in physical measurement. 



CHAPTER III. 

GENERAL METHODS. 

§ 35. Methods of Trial aud Approzimatiou. — The 
ordinary method used in the arts for testing the diam- 
eter of a wire is to fit it into a series of slits, each nar- 
rower than the one before it, until one is found which 
the wire cannot be made to enter. A series of trials, 
systematically arranged^ leads very quickly to the de- 
sired result. The trials are of course limited in prac- 
tice to a set of slits of about the same width as the 
wire. The first trial should be made with one near 
the middle of such a set ; for if this slit be two small, 
little time is lost, while, if it be too great, only half 
of the set remains to be tried. In any case, we find 
out which half contains the slit fitting the wire. The 
second trial should be made about the middle of this 
half. A quarter of the original set then remains to 
be tried. A third trial is made near the middle of 
this quarter, &c.^ By thus continually halving the 
limits between which an unknown quantity has been 
found to lie, its precise value may be determined with 
the smallest possible number of trials. 

In certain cases, we have no clew whatever to the 

magnitude of the quantity which we desire to meas- 

1 10 halvings reduce a quantity in the proportion 1024 : 1 ; 20 halv- 
ings reduce it in the proportion 1,048,576 to 1. 



§ 35.] METHODS OF TRIAL AND APPROXIMATION. 627 

uie.^ A bad electrical connectioa may, for instance, 
amount to a small fraction of an ohm (§ 20), or to 
several million ohms. We begin, therefore, by com- 
paring it with a standard which comes in the order of 
its magnitude, as expressed in the decimal system, 
about half-way between the extreme limits within 
which measurement is possible. With an apparatus 
capable of measuring resistances from 1 to 1,000,000 
ohms, we should first try, for instance, 1000 ohms. 
If 1,000 were too great, we should next try 10 ohms ; 
and if this were too small, 100 ohms. Very few tri- 
als are usually required to determine the order of mag- 
nitude to which any measurable quantity belongs. 

When the result of a given trial can be anticipated, 
this trial is needless, and should be omitted from the 
series which would otherwise be made. We begin, 
for instance, by compaiing an unknown weight with 
a standard as nearly equal to it as possible. Then a 
second standard or combination of standards is tried. 
A good practical rule is to try weights in their order 
of magnitude? each weight in a set being generally 
about half or twice as great as the one next above or 
below it. If the first estimate be reasonably close, 
the result of following this rule will be probably to 
turn the balance. It is evidently useless to make 

1 If there is any doubt whether the apparatus which we employ 
is capable of measuring the unknown quantity, it is well to compare 
this quantity at the start (1) with the smallest and (2) with the largest 
available standard. A reversal of the indication of an instrument 
obtained in this way is valuable, because it shows that the instrument 
is in working order and that a measurement can probably be made. 

2 See Pickering's Physical Manipulation, vol. i„ page 48. 



628 GENERAL METHODS. [§35. 

changes in weight which are certain to turn the scales. 
If, accordingly, two weights appear by any chance to 
be nearly balanced, a much smaller change should be 
made. 

The method of trial employed in weighing is es- 
sentially the same as that used in finding the diame- 
ter of a wire. When an unknown weight has been 
found to lie between two limits, in the absence of 
any indication which limit is the nearer, we tiy a 
weight as nearly half-way between these limits as 
convenience will allow. To avoid, however, compli- 
cated combinations of a set of weights, we follow 
this rule only in so far as may be possible by the ad- 
dition of one weight at one time or by the substitu- 
tion of one weight for another (see Exp. 1, ^ 2). A 
similar method is employed with a set of electrical 
resistances (Exp. 86). 

A great many physical instruments show only 
which of two quantities is the greater, without indi- 
cating how great the difference is between them. 
The best results are obtained with such instruments 
by the methods of trial described above. When, 
however, it is possible to calculate approximately the 
magnitude of an unknown quantity from the results 
of one or more trials, this method may be greatly 
shortened. Thus, by observing how much the tem- 
perature of a mixture is lowered by cooling one of 
the ingredients a certain number of degrees, we may 
calculate roughly how many degrees this ingredient 
must be warmed or cooled to bring about any desired 
temperature in the mixture (see ^ 99, 1.) A series 



§ 36] METHODS OF GRADUATION, ETG 629 

of trials may be arranged in this way so that each is 
much closer than the one before it. This is called the 
" method of trial and error," or the "method of suc- 
cessive approximations " (Pickering, Physical Manip- 
lation, vol. i., page 10). 

§ 86. Methods of Gradaation and Calibration. — (1) 
Production of a set of standards. The purposes 
of physical measurement frequently require the pro- 
duction of a set of standards, each of which must be an 
accurate multiple of a given unit. Let us first sup- 
pose that a suitable standard unit can be obtained. 
The first step is to make an accurate copy of this 
unit. This requires the aid of some instrument ca- 
pable of detecting the slightest difference \)etween 
two quantities (§ 42). With such an instrument, the 
copy is made as nearly as possible like the original 
by the method of trial and error (§ 35). Let us call 
the original -4, and the copy B. The two are then 
combined, and by the aid of the same instrument two 
standards, C and i>, are prepared, each equal to the 
sum of the standards A and jB, — that is, 2 -4, nearly. 
There are then two ways of producing a standard E 
equal to 5 A. We may combine C, i>, and A ; or C\ 
i>, and jB. The former is preferred because, in em- 
ploying the original standard -4., instead of a copy of 
it, there is one less chance of error j see (4). By 
combining -4, (7, i>, and E^ two standards, F and 6r, 
may be produced, each equal to 10 -4, nearly. There 
are, then, two ways of making a standard, J?, equal to 
20 A. One way is to combine F and 6?, the other is 
to combine one of these — J\ for instance — with -4, 



630 GENERAL METHODS. [§36. 

C, i>, and E, The latter is preferred because it makes 
use of the sum of the standards (A, (7, i>, and -ff) 
instead of a copy of this sum ; see (4). In a similar 
manner, we may prepare standards of the magnitudes 
50 A, 100 A, &o. 

Let us now suppose that a suitable standard unit can- 
not be obtained, and that the only available standard 
is some multiple of this unit, as for instance 1000 A. 
We then assume a provisional unit of any magnitude, 
a:, and construct a series of provisional standards, of 
the magnitudes 2 x, 6 a:, 10 re, &c., until we reach a 
value as great as the given standard. Then by the 
method of trial (§ 35) we find how many provisional 
units are equal to this standard. The values in the 
provisional series are now known; and by making 
and copying the proper combinations of this series, we 
may construct a series of standards which are more 
or less accurate multiples of the standard unit which 
we desire to represent. 

It would be out of place to consider here the me- 
chanical operations by which graduated scales and 
circles are produced. Standards must in general be 
subjected to a series of tests, as will be explained in 
(2) and (3). 

(2) Testing a set of standards. The con- 
struction of a set of standards may be considered as 
a first step toward the accuracy of results ; but no 
matter how carefully such a set may be prepared, it 
is almost always possible to detect a difference be- 
tween any two combinations of nominally the same 
value. It is generally easier to measure and allow 



§ 36] METHODS OF GRADUATION, ETC. 631 

for such differences than it is to avoid them. A set 
of standards may accordingly be tested by a series 
of comparisons involving essentially the same com- 
binations as those employed in processes of construc- 
tion; see (1). Instead, however, of comparing II 
with A-^-C-^-D-^-E-^F^WQ should in practice 
compare it with F -{- 0-^ since the latter combination 
(^F -\- (?), being more frequently employed, — see (4), 
— needs to be known with greater precision. We 
prefer, in fact, tests involving the use of the smallest 
possible number of standards. 

In addition to a series of comparisons by which we 
may determine the relative values of different stand- 
ards in a set (see Exp. 7), either the sum of the set 
or one or more of the larger standards which it con- 
tains should be compared with some standard of 
known value. 

(3). Calibration. Variations in the bore or "cal- 
ibre " of a tube may evidently give rise to errors in 
the estimation of its contents by means of a scale at- 
tached to the tube. Any process by which such errors 
may be eliminated is properly called '' calibration '* 
(see tt 68 and 71, Exps. 25 and 26). This term has, 
however, been extended to the correction of a scale 
of any sort. 

To obtain accurate results with an ordinary scale 
of length, it is obviously necessary that all the inter- 
vals of a given nominal value should be equal, or at 
least that they should not differ from one another by 
a perceptible amount. A simple way to test the ac- 
curacy of a scale is to lay beside it another scale 



632 GENERAL METHODS. [§36. 

graduated in exactly the same manner. Let a, 6, c, 
&c., represent the spaces on one scale, and a\ h\ &^ 
&c.,those on the other scale, and let us suppose that 
the division lines between these spaces are opposite 
one another. Then a = a', 6 = 6', c = c\ &c. The 
first scale is then to be moved along so that a may 
come opposite to V. If the division lines again come 
opposite, a = 6', 6 = (j', &c. Since in the firat case 
V = b, and in the second case 6' = a, it follows that 
a = 6, and in the same way all the intervals, a, 6, c, 
&c., must be equal. 

To test, accordingly, the uniformity of the milli- 
metre divisions on a metre rod, we place two such 
rods side by side, then we move one of them along 
1 mm. The equality of the centimetre spaces may be 
similarly established by moving one of the rods 1 <?m., 
and the decimetres may be tested by moving the rod 
10 cm. It must not be imagined, because there is no 
perceptible irregularity in the millimetre divisions, 
that there can be none in the centimetre or in the 
decimetre divisions. If for instance, the first 100 mm. 
spaces on each rod were longer than the next 100 
mm, spaces by j J^ mm, in each case, we should hardly 
notice the difference between them ; but the first 
decimetre would be longer than the second by a 
whole millimetre. For a similar reason it is impor- 
tant to compare the two halves of a scale, — see (4), 
— the two quarters into which each half may be di- 
vided, &c. (see Exp. 24). 

The relations between the magnitudes compared in 
testing a graduated scale or circle are, to a certain 



§ 36] METHODS OF GRADUATION, ETC. 683 

extent, the same as in the case of a set of standards; 
see (2). 

When there is no other way of testing the relative 
values of different scale indications, we do so by 
measuring with the scale different quantities bearing 
known ratios to one another (Exp. 96) ; the scale may 
then be used for relative indications. Every scale 
which is to be depended upon for absolute results 
must be compared in one case at least with a standard 
of known absolute value. 

(4) Direct and indirect processes. The cor- 
rection of a scale or of a set of standards usually de- 
pends, as we have seen, upon a series of comparisons, 
each of which must introduce a certain chance for 
error in the result. Standards should evidently be 
compared directly with the originals which they are 
intended to represent, whenever it is possible to do 
so, rather than with copies of these originals. Again, 
the two halves of a scale should be compared directly 
with one another, not indirectly, by means of the 
spaces into which they are subdivided; see (3). Short 
and direct methods of comparison are always prefer- 
able, other things being equal, to long and indirect 
processes. 

It will be seen from (1) that in certain cases the 
sum of several weights is more reliable than a single 
weight of the same nominal value. In general, how- 
ever, each weight in a set is subject to a certain error, 
especially when the set has been copied from another 
set, or when the weights are worn or corroded. In 
such cases the chances for error in weighing increase 

4 



634 GENERAL METHODS. [§37. 

ifi proportion to the number of weights which we 
employ. For this reason, as well as for convetuence 
in manipulation, we make it a general rule to use as 
few weights as possible. Further illustrations of the 
principles which underlie this rule will be found in 
§38. 

§ 37. Methods of Bubdivision.^ — The subdivision 
of a scale or of a set of standards may be carried 
theoretically to almost any extent by ordinary meth- 
ods of graduation (§ 36) ; but there is always a prac- 
tical limit to the process. The smallest quantity 
actually indicated by a given instrument is called the 
"least count*' of that instrument. Errors due to 
" least count " may easily arise. Their influence on 
a result may be lesBened by methods of multiplica- 
tion or repetition (§ 39) or by methods of "least 
error" in general (§ 38). It is, nevertheless, desirable 
that the " least count " of an instrument should be 
reduced to the smallest practicable amount. 

Even with the best analytical balances, weights 
smaller than 1 milligram are seldom employed. The 
fractions of a milligram are usually estimated by 
means of a "rider" or small weight sliding along a 
graduated scale on the beam of a balance. It has 
been found similarly impracticable to make use of 
standards of electrical resistance less than one tenth 
of an ohm. Fractions of the smallest available stan- 
dards are estimated in general by methods of inter- 
polation (§ 41). 

1 Keferences in this edition to the Method of Maltiplication or 
Bepetition shoald read § 89, not § 87. 



S 37.] METHODS OF SUBDIVISION. . 636 

In the measurement of length, there are certain 
methods of subdivision by which the least count of a 
scale may be greatly diminished without a proportion- 
ate increase in the number of divisions. Thus a 
centimetre scale 1 metre long, requires for its pro- 
duction 100 lines besides the zero ; but if the first 
centimeti*e be divided into 100 parts, we may with 
200 lines measure any length less than a metre to a 
tenth of a millimetre. 

When thi$ method of subdivision is employed, the 
application of corrections for errors in graduation 
(§ 36) is comparatively simple, since a given measure- 
ment can be made in only one way. We lose, how- 
ever, the advantage which is sometimes gained by 
making measurements in different parts of the scale, 
and averaging the results (see § 46). For this reason 
there would be an obvious advantage in using a short 
movable scale very finely divided, in connection with 
a scale of centimetres. This principle has been ap- 
plied in the construction of various sliding-scales or 
gauges. It is found, however, impracticable to read 
any scale with the naked eye unless the divisions are 
at least ^ of a millimetre apart. The use of sliding 
scales was therefore very limited until Vernier showed 
how, by a slight modification in these scales, compar- 
atively accurate results could be obtained. The divi- 
sions of a Vernier scale are made nearly but not 
exactly equal to one or more main-scale divisions. 

A common form of Vernier gauge consists of a 
fixed scale in millimetres and a sliding piece with ten 
or eleven marks, each nine tenths of a millimetre 



636 GENERAL METHODS. (§37. 

from the next (see Fig. 2). The first of these, num- 
bered 0, points out the reading of the instrument, in 
millimetres, upon the main scale, just as if there were 
no " vernier." It comes opposite a millimetre mark 
only when the reading is a whole number of milli- 
metres. In this case the next mark on the vernier 
(No. 1), being ^^^ mm, further on, falls ^ mm. short 
of the nearest main-scale division ; No. 2 falls ^ 
mm. short, and so on. Hence if the sliding scale be 
moved along ^^ wrw., the mark No. 1 will come oppo- 
site a mark on the main scale (not the one nearest 
the zero of the wire), and if the vernier is moved ^ 

mm. along, mark No. 2 will be exactly opposite still 

to 



/o 

Fig. 2. 

another mark on the main scale. In the same way 
Nos. 3, 4, 5, &c., will come opposite various marks in 
the main scale, when the vernier is respectively ^^j, ^, 
rJ^, &c., mm. beyond the original position. Obviously 
we have only to find the number of the vernier line 
which is opposite a line on the main scale (no matter 
which) to determine the number of tenths of a milli- 
metre between the zero of the vernier and the line 
just below it on the main scale. 

The same principle holds in the case of any ver- 
nier. By a series of steps, easily counted, the spaces 
on the vernier gain or lose one space with respect to 
the main scale. The reading of the main scale is 



§37.] METHODS OF SUBDIVISION. 637 

thus practically divided into as many parts as there 
are steps in the gain or loss of one space. 

It often happens that in comparing the vernier and 
the main scale, no two lines are found to be exactly 
opposite, so as to form a single continuous line ; in- 
stead, two lines are found, which, though nearly con- 
tinuous, show, when closely examined, more or less 
dislocation. We then estimate by the eye the relative 
amount of dislocation in each case, and reduce the 
result as accurately as possible to decimals. Thus if 
in a vernier the third and fourth lines are equally 
dislocated, the reading is .35 ; if the third line is only 



.CO 


•01 ' -ox • 


1. .03 


'OJi 








l«. 


n' ■"! 


H 


1 or 



Fio. 3. 

one fourth as much dislocated as the fourth, then the 
reading is .32. By reference to the diagrams in Fig. 
3, it will generally be possible to express the reading 
of the gauge to hundredths of a millimetre, and with 
almoBt as much accuracy as if the vernier contained 
a hundred lines. 

The use of a vernier for the subdivision of a scale 
is closely related to the method of coincidences 
(§ 40), and may be considered also as one of the 
various methods of interpolation (§ 41) by which 
fractions of the smallest available standards are 
customarily estimated. 



638 GENERAL METHODS. [§37. 

§ 38. Methods of Leaat Error. — It is desirable 
ill physical measurement that observations should 
be accurate ; it is equally desirable that the .condi- 
tions under which they are made should be favor- 
able for the exact determination of results. There 
are certain general principles by which experiments 
are, when possible, arranged so that a given error in 
the observations may cause the least possible error in 
the result. Any method in which these principles 
are applied may be called a method of least error. 

The advantages of direct methods of comparison 
have been already pointed out (§ 36). We prefer, 
in general, determinations which depend upon the 
fewest data, assume the fewest laws, and make use of 
the fewest and best-known physical constants. The 
present section is devoted especially to the relations 
which should exist between physical instruments and 
the quantities which they are used to measure. 

The delicacy of most instruments is somewhat di- 
minished by an increase in the magnitude of the 
quantities measured, but not in proportion to this in- 
crease. The best results are accordingly obtained 
with quantities nearly as great as the capacity of the 
instrument will admit. We employ, for instance, 
large quantities of a substance in determinations of 
specific gravity by means of a balance. On the other 
hand, it would be impracticable to measure accurately 
the weight of copper deposited (Exp. 81) on an elec- 
trode weighing several thousand times as much 
as the deposit in question; for a balance capable 
of weighing the electrode would not be sensitive 



§38.] METHODS OF LEAST ERROR. 639 

enough for the deposit. While, therefore, it is desir- 
able to increase the deposit of copper, the weight of 
the electrode should obviously be diminished. We 
avoid, in general, determinations of the difference be- 
tween two nearly equal quantities depending upon 
observations of the quantities themselves. Such dif- 
ferences should be measured directly if possible 
(§§ 41, 42). 

Some instruments are particularly adapted to meas- 
uring quantities of a given magnitude. A tangent 
galvanometer, for instance, gives the best results with 
electrical currents which deflect it 45**. Let us sup- 
pose that when three turns of wire are used, the 
needle points to 26° ; with six turns, to 45° ; with 12 
turns, to 63°. An error of observation equal to + 1° 
would give 27° instead of 26°, 46° instead of 45°, 
and 64° instead of 68°. Now the results depend 
upon the tangents of the observed angles (see Exp. 
78). The tangents of 26° and 27° differ (see Table 
5) by about 4.4 <^, and the tangents of 63° and 64° 
differ in the same proportion; but the tangents of 
45° and 46° agree within 3.6 %. We should ob- 
viously employ 6 turns of wire in preference to 3 
or 12. 

In making selections or modifications of the instru- 
ments which v^e employ, we must consider, in general, 
the nature of the formulae by which the results are to 
be reduced. It will be found, for instance, that a 
1 % error in a quantity causes an error of about 2 % 
in estimating the square of that quantity but only 
about J of 1 5^ in the estimation of its square root 



640 GENERAL METHODS. [§ 39. 

(see § 57). We prefer, accordingly, determinations 
depending on roots rather than on powers of the 
quantities directly observed. The relative value of 
different determinations must be judged, not by the ac- 
curacy of the observations, but by that of the results. 

The principles of ''least error" may require, under 
certain circumstances, the use of the method of mul- 
tiplication or repetition (see § 39), the method of co- 
incidences (see § 40), or the method of reversal or 
interchange (see § 44). 

§ 39. MethodB of Multiplication and Repetition.^ — 
It would be impossible to weigh a single drop of 
water very accurately on a coarse balance ; but if we 
knew under what circumstances the drop was formed 
it might be possible to produce a thousand drops of 
almost exactly the same size, and by finding their 
combined weight to arrive at that of a single drop. 

The error in measuring 1000 drops may not be per- 
ceptibly greater than in the case of a single drop, and 
since in the process of reduction this error is divided 
by 1000, we may obtain at least a comparatively ac- 
curate result. The use of any means for increasing 
the magnitude of a quantity in a given proportion 
for the purpose of finding a more accurate measure 
of that quantity constitutes in general a " method of 
multiplication." The value of suoh methods evi- 
dently depends on the accuracy with which a quan- 
4iity may be reproduced as compared with the accuracy 
of a direct measurement. 

1 References in this edition to the Method of Gradoation or Cali- 
bration should read § 87, not § 39 



§ 39.] METHODS OF MULTIPLICATION. 641 

We may find, for instance, the weight of mercury 
required to fill a capillary tube by emptying the con- 
tents of the tube sevei-al times in succession into a 
vessel, in which the mercury is collected and weighed. 
The same method could not, however, be employed 
with water, on account of the considerable portion 
which sometimes adheres to the tube. 

The method of multiplication is often used in the 
determination of times of vibration ; for it may be 
proved mathematically (see § 111) that successive vi- 
brations executed under certain conditions do not 
differ by a perceptible amount. The rate of a pendu- 
lum should accordingly be determined by a long 
series of observations. Such a series may be ex- 
tended, by a system of mechanical counting, for days 
or even for months. There must evidently be no 
break in the series. The method of multiplication is 
applicable only to consectUive intervals in the meas- 
urement of time. 

The method of multiplication is sometimes used 
for the estimation or detection of a series of small 
impulses given to a pendulum or to a vibrating needle 
at the middle point of a swing, so that the effects 
may be added together. A large allowance must 
sometimes be made for the effects of friction, or other 
causes tending to destroy the motion. For the 
" method of multiplication and recoil " see Kohl- 
rausch. Physical Measurement, Art. 76. 

The method of multiplication is applied in the 
construction and use of an ordinary galvanometer or 
" multiplier,*' the object of which is to increase the 



642 GENERAL METHODS. [§39. 

effect of an electrical current in a known or measur- 
able proportion. Methods of multiplication are also 
applied in the measurement of length. 

There are various mechanical devices by which a 
body may be moved in a straight line through suc- 
cessive distances, each equal (or nearly equal) to its 
own length. We have an example in the ordinary 
method of measuring distances with a rod or chain. 
This is, however, more or less inaccurate on account 
of the uncertainty of the marks which show where 
the ends of the measure are placed. One method by 
which greater precision may be obtained is to place 
a block end to end in front of a measuring rod, then 
to remove the rod, to place a second block behind the 
first, just touching it, then to remove the first block 
and to put the rod in front of the second block. 
This process is then repeated over and over until the 
length of the rod has been multiplied, or, as we say 
technically, *' repeated," a sufficient number of times. 
By this means very long distances may be quite 
accurately measured even with a short millimetre 
scale. This and similar methods are properly called 
" methods of repetition.'* 

Methods of repetition are frequently used in the 
measurement of angles. Let us suppose that a given 
angle, cut out of thin metal, reaches from the zero of 
a circle, graduated in degrees, to a point between 40** 
and 41® ; and that by some method of repetition simi- 
lar to that just described, the angle is found to i*each 
from the last point (between 40° and 41°) to one be- 
tween 80° and 81°, &c. We should obtain in this 



§40.] METHOD OF COINCIDENCEa 648 

way a series of observations like the following : — 
0\ 40^ +, 80" +, 120^ +, 160" +, 200" +, 240" +, 
280" +, 320" +, 360" +, 401" — 441° — , &c. We see 
from any two successive observations that the angle 
must lie between 40" and 41", but we have no means 
of estimating the fraction of a degree over 40. If 
however, we consider the first and last observations, 
we see that the angle must be less then ^ of 441 ^ 
which gives 40^\ as the superior limit of the angle. 
In other words, the angle becomes known within ^ 
of a degree. By considering two observations which 
differ by 360" (or any multiple of 360") we escape 
from a great variety of errors by which the results 
obtained with graduated circles are apt to be affected. 
A method by which we may utilize, not simply the 
first and last, but nearly all of a series of consecutive 
observations will be considered in § 61. 

§ 40. Method of Coincidences. — We have seen 
(§ 39) that some lines on a vernier come almost ex- 
actly opposite the lines nearest them on the main 
scale, while others do not. In the same way, when 
any two scales are compared together, cases of more 
or less approximate " coincidence " usually occur. 
Every fifth inch on an English scale coincides, for 
instance, as nearly as the eye can judge, with every 
127th division on a millimetre scale. We should evi- 
dently prefer to calculate the length of the inch in 
millimetres from a case of perfect coincidence than 
from one where a given number of inches was found 
to be greater or less than a given number of milli- 
metre^ by a fraction which could only be estimated 
by the. eye. 



644 GENERAL METHODS. [§40. 

The method of coincidences may be nsed with ad- 
vantage to avoid errors due to " least count " (§ 37) 
in the comparison of any two sets of standards of the 
same sort, no matter what kind of physical quantity 
they represent. 11 Troy ounces happen, for instance, 
to balance 342 grams within a few milligrams. With 
two ordinary sets of weights, the smallest of which 
are 1 ounce and 1 gram respectively, it is possible 
accordingly, to find the value of the Troy ounce in 
grams within a small fraction of a milligram. 

The most important application of the method of 
coincidences is, however, in the comparison of inter- 
vals of time. Let us suppose that two pendula differ 
slightly in their rates of oscillation, so that one gains 
slowly upon the other, and that they start together 
at a given point of time. After a certain number of 
oscillations have been executed by one of the pen- 
dula, the two will be swinging in opposite ways, and 
again after a given number of oscillations, they will 
be swinging the same way. The relative rate of 
oscillation may be accurately determined by count- 
ing the number of oscillations in question. If, for 
instance, the faster pendulum makes n vibrations be- 
tween two successive coincidences, the slower pendu- 
lum must make n — 1 ; hence the relative rate is w h- 
n — 1. Let us suppose that through an error in ob- 
servation w 4" 1 oscillations were counted instead 
of n ; the relative rate would tlien be estimated as 
n + 1 -^ w. The error committed would therefore be, 

n-|-l n n^ — 1 rfi — 1 

n n — 1 V? — n rfi — n n^ — n 



§ 41-1 METHODS OF INTERPOLATION. 645 

If n is moderately large such an error would be 
inappreciable. 

§ 41. Methods of Interpolation. — We have seen 
that errors due to the " least count " of an instru- 
ment may be almost indefinitely reduced by the 
methods of multiplication, repetition, and coinci- 
dences (§§ 39, 40). Such methods cannot, however, 
always be applied. The value of an observed quan- 
tity, J, is usually found to lie between two limits, 
one il, the other -4 + a, where a represents the 
" least count " or smallest change which can be pro- 
duced in a set of standards. That is, we have — 

A + a>q>A. 

If more precise results are required, we seek some 
instrument or indicator by which we may estimate, 
relatively at least, the differences between the quan- 
tity q and the two nearest values of the standards, A 
and A'\'a^ with which we are able to compare it. 

The sensitiveness of any instrument used as an 
indicator may be defined as the number of scale divi- 
sions by which its reading changes when the smallest 
possible change {a) is made in the standards. We 
will first suppose the sensitiveness to be known. Let 
the quantity q be compared with the combination of 
standards {A) just below it in magnitude, and let the 
indicator show a motion of x scale divisions. Then 
since % divisions correspond to the quantity a, we 
may infer that x divisions must correspond to x «*^ of 
a, hence the true magnitude of y is — 

q = A+—. 



646 GENERAL METHODS. [§41 

In the same way, if the indicator shows a motion of 
y scale divisions when the quantity q is compared 
with the combination of standards {A + a) just above 
it, we have — 

SB 

By comparing this equation with the last, we see that 
X must be equal to 9 — y, or — 

x + y = 9. 

The last equation enables us to calculate the sensi< 
tiveness of any indicator from two deflections, ob- 
tained as stated above. The value of 9 may vary 
according to circumstances. The special value here 
determined is the sensitiveness of the indicator to a 
change of the magnitude a in the quantity q. The 
process of estimating a quantity (5^) from the relative 
differences (x and y) separating it from two magni- 
tudes {A and A •\' a) between which it lies is called 
"interpolation" ("putting in between"). 

We have instances of the method of interpola- 
tion when, in the use of a Nicholson's Hydrometer 
(Exps. 2, 3, 4), the distances of a certain mark above 
or below the surface of the water are used to esti- 
mate fractions of a centigram, or when in the use of 
a vernier (§ 38), the relative dislocations of two lines 
are used to estimate hundredths of a millimetre. The 
vernier itself may be considered as one means of in- 
terpolation. The use of a " rider " (^ 259) enables 
us to determine weights exactly by interpolation 
even if the weight of the rider be unknown. The 



§42.] NULL METHODS. 647 

indications of the pointer of a balance afford an- 
other means of interpolation in weighing (see ^ 20). 
The deflections of a galvanometer are similarly used 
(see Exp. 98) to estimate small differences between 
two opposing electromotive forces which we seek to 
bring into equilibrium. 

§ 42. NtUl Methods. — Most physical quantities 
cannot, like scales of length, be directly compared 
with one another, but are measurable only through 
the effects which they produce upon some instru- 
ment. Electrical currents, for instance, are usually 
determined by their action upon the needle of a gal- 
vanometer. When two effects lie in the same direc- 
tion, they are generally compared by the method of 
substitution (§ 43). It is, however, frequently de- 
sirable to oppose two effects, especially when they are 
nearly equal, in order that the difference between 
them may be directly measured (see § 38). In 
weighing with a balance, the effects of two nearly 
equal weights upon the instrument are thus opposed. 
Any method by which two effects may be made to 
neutralize or annul each other may be called a null 
method. 

In electrical measurements, the term " null method " 
is usually applied to cases where two equal electro- 
motive forces are opposed to one another so as to 
produce no current through a delicate galvanometer. 
Null methods are characterized by the fact that the 
conditions of perfect adjustment between the different 
parts of an apparatus is shown by the absence of any 
indication on the part of some delicate instrument. 



648 GENERAL METHODS. [§42. 

Null methods do not require the use of instru- 
ments which indicate the magnitude of the difference 
between two nearly equal quantities, although it is 
often convenient to employ such instruments for pur- 
poses of interpolation (see § 41). It is only necessary 
that an instrument should show whether two quanti- 
ties are equal or unequal. Being used solely to de- 
tect differences, such instruments are sometimes 
called "detectors." They take the place of sight, 
touch, or bearing (§ 23) with quantities which do 
not affect these senses. 

There are two principal precautions to be observed 
in the use of null methods. One is to make sure 
that the instrument employed responds to the slight- 
est variation in either of the two quantities which 
are compared; the other is to test the zero of the 
instrument (§ 32). Errors may occur, for instance, 
from a break or from a cross-connection in the circuit 
of a galvanometer ; for in this case there will be no 
perceptible deflection, no matter how great may be 
the difference between the electromotive forces which 
are compared together. Again, if the needle of a 
galvanometer does not naturally point to zero, it may 
require a current to make it do so (see Exps. 89, 90). 
We should infer wrongly in such a case that the cur- 
rent had been reduced to zero. 

Null methods usually depend upon the use of very 
sensitive instruments ; but the conclusions which we 
draw from them, being founded upon purely negative 
indications, must be examined with great care. Null 
methods are considered highly desirable on account 



% 4.3.] METHOD OF SUBSTITUTION. 649 

of their precision, but they need in general some 
kind of confirmation. 

§ 43. Method of Substitution. — The " method of 
substitution " is the fundamental method for testing 
any result the accuracy of which is questioned. It 
is so called because a known quantity is BubstittUed 
for an unknown. Thus if the resistance of a wire 
has been found by means of any electrical combina- 
tion sensitive to variations in resistance (Exps. 86, 87) 
to be equivalent to 10 ohms, we have only to substi- 
tute for it a resistance knotvn to be 10 ohms to fiud 
whether there is or is not any error in our work. 

The scale of a densimeter (Exp. 15) may be tested 
by substituting a liquid of known, for one of unknown 
density, or the indications of a volt-meter (Exp. 96) 
bj substituting known for unknown electromotive 
forces. The method of substitution is often used 
where no other is possible, as in Experiments 2, 8, 
and 4. It depends upon the principle that two quan- 
tities must be equal if they can be substituted one 
for the other without affecting a combination sen- 
sitive to variations in the magnitude of the quanti- 
ties in question. Evidently the known and unknown 
quantities thus compared should be as nearly equal 
as possible. 

In the method of substitution, as in null methods 
(§ 42), we must make sure that the instrument which 
we employ is free to move, since otherwise very un- 
equal quantities might apparently produce the same 
effect upon it. The *' zero-error *' of an instrument 
(§ 32), and instrumental errors in general (§ 31), are 

5 



650 GENERAL METHODS. I§44. 

usually eliminated by the method of substitution. 
Borda's method of weighing is to counterpoise accu- 
rately an unknown weight in one pan of a balance 
with material of any sort in the opposite pan, then to 
substitute known weights for the unknown until an 
exact balance is again established. In a similar 
manner, when, in electrical measurements, null 
methods (§ 42) are employed, it is well to test the 
accuracy of the results by substituting known for 
unknown quantities. The use of the method of 
substitution in combination with null methods is 
the most general way of obtaining both accuracy 
and precision in physical measurement. 

§ 44. Methods of Interchange and Reversal. — In 
the ordinary method of double weighing (see Exp. 
8) an unknown weight is first placed in the left-hand 
pan of a balance, and a known weight in the right- 
hand pan. Let us suppose that the former is greater 
than the latter by a small amount, which is sufficient 
to send the pointer of the balance x divisions to the 
right of its natural resting-point. The unknown 
weight is next placed in the right-hand pan, and the 
known weight in the left-hand pan. The pointer 
will evidently move about x scale divisions to the 
left of its natural resting-point. The total move- 
ment produced by interchanging the weights will 
therefore be about 2 x scale-divisions. If, how- 
ever, the unknown weight were exactly counter- 
poised, the substitution of the known weight for it 
would cause a motion of the pointer through only x 
scale divisions. It is easier, accordingly, to detect 



§45] CHECK METHODS. 651 

a diflference between two weights by the method of 
interchange than by the method of substitution 
(§ 48). 

The method of interchange is generally used in 
connection with null methods of comparison (§ 42) 
when reversible instruments are employed. Whatever 
may be the difference between the two nearly equal 
quantities thus compared, its effect upon a reversible 
instrument is doubled by interchanging these quan- 
tities. For this reason the method of interchange, 
when applicable, is always preferred to the method 
of substitution. 

A similar method is employed in case of reversible 
instruments in general. Thus an electrical current 
which deflects a galvanometer needle 2^ to the east 
of north, should if reversed deflect it x^ to the west 
of north. The needle is thus moved, by a reversal 
of the current, through 2 x°. Since an angle of 2 ic® 
can be measured as accurately as an angle of a;°, the 
method of reversal has to a certain extent the ad- 
vantage of a method of multiplication (§ 89). In the 
inethods of interchange and reversal "zero-errors" 
are eliminated (§ 82), for the increase of one reading 
due to an error in the zero will be nearly offset by a 
decrease in the reversed reading. Methods of rever- 
sal are always, when practicable, employed. 

§ 45. Check Methods. The methods of substitu- 
tion and of reversal are instances of check methods. 
In physical measurement, as in arithmetic, an indefi- 
nite number of such methods may be devised. The 
use of check methods is not, however, limited to such 



652 GENERAL METHODS. [§45. 

as yield accurate measurements. We often find an 
advantage in checking results which we believe to be 
precise, with others obtained by different methods, 
which we consider comparatively unreliable. It is 
in this way, principally, that gross mistakes are dis- 
covered, such as are otherwise likely to be repeated 
over and over. But the use of check methods is 
also important in the detection of smaller errors. 
Even if a method is uncertain, there is probably 
some limit to its inaccuracy, and if the results fail 
to agree with those of a different method by an 
amount greater than this limit, we are led immedi- 
ately to suspect an unknown source of error in one 
of these methods. The densimeter, for instance 
(Exp. 15), though not nearly so exact as the specific 
gravity bottle (Exp. 14) should be accurate at least 
within 1% ; hence if the results differ by more than 
1% we at once repeat the determination with the 
specific gravity bottle. On the other hand an agree- 
ment of the two results within 1 % indicates the ab- 
sence of gross mistakes in either determination. 

Whenever the results of check methods, however 
rough, agree with previous results as closely as may 
be expected, there is always a certain degree of mu- 
tual confirmation. It should be remembered, how- 
ever, that a check method is such only in so far as 
it makes use of different data, different constants, 
different instruments, and different laws or principles 
from those already employed. Accuracy in physical 
measurement is generally obtained only when every 
possible variation has been made in the conditions of 



S 47.| ALLOWANCE FOR ERRORS. 663 

an experiment, the results compared, and the differ- 
enees between them explained. 

§ 46. Method of Averages. — When finally all pos- 
sible care has been taken to avoid sources of constant 
error, and to increase the accuracy of determinations, 
there remains one general method of escaping from 
what are known as accidental errora (§ 24), or those 
which tend sometimes to increase, and at other times 
to diminish, the result. Tliis method is simply to 
take a great number of measurements, and to find 
the average. It is not likely, for instance, that in ten 
observations all should by accident be greater, or all 
less, than in the long run ; in fact, the chances are 
more than one thousand to one against it. It is much 
more likely that three or four should be affected one 
way, and the rest the other way. In fact, we must 
expect that the errors due to chance shall to a cer- 
tain extent offset one another. The consequence 
is that the average of several observations is more 
reliable than any one alone. For a discussion of 
the advantages gained by taking the average of 
several observations, see § 61. 

§ 47. AUowance for xsrrors. — We have considered, 
so far, the principal methods by which errors may be 
eliminated from physical measurement. There are, 
however, certain errors which cannot thus be avoided. 
The effect of some of these may be submitted to cal- 
culation. The buoyancy of air, for instance, is com- 
puted and allowed for in all accurate weighings (§ 67). 
There is another class of errors which cannot be cal- 
culated in this way from data already in our posses- 



654 GENERAL METHODS. [§47, 

sion. The causes from which such errors arise may 
require separate investigation. Thus the heat lost 
in transferring a hot body from one place to another 
can be estimated only by comparing results of dif- 
ferent experiments (see Part I. ^^ 93, 94). 

No single observer can expect to discover all the 
sources of error which are likely to arise in measure- 
ments. Our knowledge of the corrections which 
are to be applied in the determination of a given 
physical quantity is one of slow historical growth. 
It is necessary to refer continually to examples which 
have stood the test of long criticism. At the same 
time, each observer must be on the alert against new 
sources of error. The slightest alteration in the 
conditions of an experiment may entirely change the 
nature of the corrections to be applied. 

Errors of greater or less magnitude are sure to 
creep into our work notwithstanding every possible 
effort to avoid them. The student is advised not to 
pay too close attention to fine corrections, lest in so 
doing he may overlook others of much greater im- 
portance. It is a well-known fact that the accuracy 
of results is apt to be grossly overestimated (see In- 
troduction). Sufficient allowance for errors is seldom 
if ever made. 

The application of corrections to the results of 
physical measurement must be considered separately 
in connection with each experiment or class of ex- 
periments. The discussion of errors and corrections 
belongs perhaps to the " Reduction of Results " 
(Chap. IV), rather than to *' General Methods" of 



§48.] STANDARD OF ACCURACY. 666 

measurement. The student must not, however, for- 
get that a just allowance for errors constitutes one of 
the most important parts of an accurate physical 
measurement. 

§ 48. Standard of Accuracy. — The distinction be- 
tween accuracy and precision has been pointed out 
in the Introduction. One generally knows by ex- 
perience, roughly at least, what degree of accu- 
racy is attainable with a given instrument. Thus 
a weighing with ordinary prescription scales will 
doubtless be accurate to centigrams, but not to 
milligrams ; temperatures taken with a common 
laboratory thermometer are reliable to degrees, but 
not generally to tenths of degrees; lengths may 
be true to hundredths, but not perhaps to thou- 
sandths of a centimetre. From such data we may 
generally estimate roughly the degree of accuracy 
attainable in the final result. All parts of a meas- 
urement should be made with a corresponding de- 
gree of accuracy. 

Let us suppose, for instance, that it is desired to 
determine the density of alcohol at a given tempera- 
ture (^. g, 20**) within a few hundredths of 1 % by 
means of a specific gravity bottle (see Exp. 14) of 
about 100 cu. cm. capacity. To do this, the weight 
of water and the weight of alcohol required to fill 
the bottle must be determined within a few centi- 
grams ; the temperature of the water must be known 
within about V (see Table 25), and that of the 
alcohol within a few tenths of V (see Table 27). 
The real difficulty in this experiment consists ac 



656 GENERAL METHODS. [§49. 

cordingly in the accurate determination of the tem- 
perature of the alcohol^ — a point to which the stu- 
dent's attention needs generally to be directed. An 
accurate reading of the barometer would be wholly 
out of place in such a determination, since an 
error of several centimetres (see Table 22) would 
scarcely affect the last significant figure (§ 55) in 
the result. 

§ 49. Distribution of Time. — Time is often mis- 
spent in the exact determination of quantities which 
have comparatively little influence in the result. 
Thus the correction for atmospheric pressure seldom 
affects the decigrams in a weighing, and ordinary va- 
riations make only a few milligrams' difference in the 
result. It is therefore unnecessary, in many experi- 
ments, to read a mercurial barometer closer than to 
millimetres, much less to correct it for variations of 
temperature, for capillarity, or for the tension of mer- 
curial vapor. A double weighing, with a rough al- 
lowance for the buoyancy of air, takes about the 
same time as a single weighing with the exact cor- 
rection, and is, with rough balances, decidedly to 
be preferred. 

When a measurement depends on several deter^ 
minations of about the same degree of precision, we 
generally devote an equal amount of time to each ^ 
but if we can see that the result will be affected by 
the errors in one case more than in another, the num- 
ber of observations is increased in proportion. Thus 
in the determination of the volume of a cylinder from 
its length and diameter we take twice as many ob- 



§49.] 



DISTRIBUTION OF TIME. 



607 



servations of the latter as of the former, because the 
diameter occurs twice as a factor, while the length 
occurs only once in the calculation of the result. A 
fuller discussion of this principle will be found in 
Part IV. 



CHAPTER IV. 

REDUCTION OF RESULTS. 

§ 50. Probable Error. — When several observations 
of a given quantity have been made, their " probable 
error " may be found roughly by the following rule : 
throw out alternately the highest and lowest values 
until only a majority remains; take half the range 
of that majority as the probable error of a single 
observation. 

Thus from the ten following observations of the 
boiling-point of alcohol — 

780.79 780.33 78O02 780.93 780.46 
780.67 78P.00 78o.81 780.43 780.66 

we have, throwing out 78°.93, 78°.00, 78^81 and 
78°.02, a majority of six, ranging from 78^.33 to 
78^.79, that is, through 0°.46. The probable error of 
a single observation is therefore about 0°.23. 

In saying that the probable error is 0°.23, we do not 
mean that this error is more probable than any other, 
0°.20 for instance. We mean simply that in the long 
run more than half the errors will probably be less 
than 0^.23 (see Table 7), and hence, as some errors 
are positive and others negative, that a majority of 
the observations will be scattered through a range not 



}53.] PROBABLE ERROR. 659 

exceeding 0''.46. This is evidentlj the case if the 
observations above are a fair sample of those which 
would be obtained in an extended series. 

§ 61. Probable Bnror of an Average. — To find the 
probable error of the average of several observations, 
we divide that of a single observation by the square 
root of the number of observations. 

Thus if the probable error of a single observation of 
temperature is, as in the last section, 0°.28, that of the 
mean of ten observations is 0^.23 -f* ylo* or less than 
0^08. 

The relation between the probable error of an aver- 
age and that of a single observation is established by 
the theory of the combination of errors as explained 
in Part IV. 

§ 52. Probable Brror of a Raaolt. — The probable 
error of a result can be calculated if we know that of 
each datum upon which it depends, as will be ex- 
plained in Part IV. It is often, however, less labo- 
rious to work out several independent results, the 
probable error of which can be found by inspection, as 
shown at the beginning of this chapter. Thus instead 
of calculating the density of a block (in Experiment 1) 
from its average weight, length, breadth, and thick- 
ness, we may use each measurement of length, breadth, 
and thickness for a separate calculation, and average 
the results. In all such cases the probable error 
should be determined. 

§ 53. Representation of Probable Error. — The aver- 
age of the ten observations of the boiling-point of 
alcohol mentioned in § 50 is 78''.50 ; the probable 



660 REDUCTION OF RESULTS. [§ 55. 

error of this average as found in § 51 is 0°.08. We 
say, accordingly, that alcohol boils (probably) at 
78^50 ± 0°.08. 

In the same way the probable error of any result is 
often written after it with the " plus - or - minus " 
sign. 

§ 54. Notation. — It is convenient for many reasons 
to express results in units of such magnitude that 
the probable error may lie below the decimal point. 
When no such units exist, we introduce as a factor 
10 raised to the necessary power. Thus the mechani- 
cal equivalent of the unit of heat is not written 
41,660,000 ergs, but 41.66 megergs, or 4.166 X 10^ 
ergs. 

In this notation we escape any possible confusion 
between ciphers which are the result of actual meas- 
urement and those which we are obliged to use from 
the necessity of the case. 

Ciphers are used in physical measurement at the 
end of a decimal as freely as any other figure. Thus 
the average of ten observations in the last section was 
written 78°.50. The cipher informs us that the aver- 
age was between 78°.495 and 78°.505, Without the 
cipher we should infer simply that the average was 
between 78^.45 and 78°.55. The existence of a cipher 
in the last decimal place has therefore as much signifi- 
cance as that of any other figure. The question how 
many figures it is advisable to retain is discussed in 
the next section. 

§ 55. Sisnificant Figures. — In arithmetic any num- 
ber of figures may be significant. In physical meas- 



§ 56.] BIGNIFICAirr FIGUBB8. 661 

urement those figures only are significant to the left 
of which the probable error does not extend. 

Thus, in the observations at the beginning of this 
chapter, the degrees and tenths are significant, but 
the hundredths are not, because the probable error is 
0**.23. In the average of the ten observations, the 
hundredths, also, are significant, since the probable 
error is O^'.OS, One figure is generally enough to de- 
scribe the probable error. The place which this fig- 
ure occupies is the same as that of the last significant 
figure. 

It is customary to retain only significant figures 
either in an observation or in a result. Some author- 
ities use two or more places affected by probable error. 
When the probable error is stated, there is no objec- 
tion to this practice. Otherwise it is equivalent to a 
false pretension to accuracy.^ 

§ 56. Use of Significant Figurea. — Labor is saved in 
physical reductions by using only significant figures. 
The rejection of subsequent figures is not found in 
practice to impair the accuracy of the result. In de- 
ciding how many places to retain, the following ap- 
proximate rules may be of assistance : — 

1st. In addition or subtraction, retain the same 
number of decimal places throughout, — as many as 
are significant in the least accurate of all the terms. 

2d. In multiplication or division, retain the same 

number of figures throughout, — as many as are sig- 

^ The student is cautioned in particular against cases where the 
result of some mathematical process is to generate an indefinite num- 
ber of figures. It is true that a metre is about 3^ feet ; but it would 
be misleading to state that it is about 8.33333, etc., feet. 



REDUCTION OF RESULTS. l§ 57. 

nificant in the least accurate of the factors, — not 
counting, of course, initial ciphers. 

8d. In logarithmic work, use as many decimal 
places as there are significant figures in the least ac- 
curate of the arguments. 

Thus in weighings with a balance accurate only to 
a fraction of a centigram, we carry out corrections 
only as far as the milligrams. Again, in calorimetry, 
where results are often proportional to differences of 
temperature less than lO*' and accurate only to tenths, 
these results seldom contain more than three signifi- 
cant figures, and corrections not affecting the third 
figure may be disregarded. 

§ 67. Rules for Approximation. — A great deal of 
time is often saved by applying rules which give ap- 
proximate but not rigorously accurate results. Thus 
to add 1% or 2% to anj'- quantity corresponds nearly 
to adding twice that per cent to the square of that 
quantity, three times that per cent to its cube, half 
that per cent to its square root, or to subtracting the 
original per cent from its reciprocal. The truth of these 
assertions will be seen by reference to Table 2. 

It is obviously the same thing to add a certain per 
cent to a quantity as to add it to a product in which 
that quantity occurs as a factor ; and nearly the same 
thing, if the per cent is small, as to subtract it from a 
quotient obtained with the quantity as a divisor. 

One of the most valuable rules for approximation is 
that used in finding the product of several quantities, 
each nearly equal to unity. Instead of multiplying, 
we (utd them together. The resulting decimal is ap- 



§58.] USE OF TABLES. 668 

proximately the same. Since the product cannot be 
far from unity, the figure in the unit's place is easily 
supplied. 

Thus if the ratio of the arms of a balance is 0.99996, 
the correction for the use of brass weights in air 
0.99984, for the buoyancy of air on water 1.00122, and 
the space occupied by 1 gram of water is 1.00175, the 
volume of water is found by multiplying its apparent 
weight by the factors 0.99996x0.99984xl.00122x 
1.00176. The product found by the ordinary laborious 
process is 1.0027715+, or, to five places of decimals, 
1.00277. The same decimal is found by adding the 
four numbers together. 

The arithmetic mean (or half-sum) of two quanti- 
ties difiering by less than 2% may usually be substi- 
tuted for their geometric mean (or square root of 
their product) which is harder to calculate. 

It will be noticed in Table 3, i, e^ d, and «, that the 
sine, tangent, arc, and chord of small angles are ap- 
proximately equal. It is frequently useful to substi- 
tute one for the other. It is also seen that the cosine 
of a small angle is nearly equal to unity, so that the 
difierence may often be disregarded. 

The above rules for approximation may be applied 
without injury to all results which are not expected 
to contain more than four significant figures, pro- 
vided that the corrections do not exceed 2^ nor the 
angles 2^. 

§58. Use of Tables. — The reductions in physical 
measurement are often facilitated by the use of tables. 
There are two kinds of these : one in which the quan- 



664 REDUCTION OF RESULTS. [§ 58. 

tity sought is given in terms of a single argument; the 
other where it is given in terms of two arguments. 
The first kind is readily understood by any one who 
has used logarithms. In one column, generally at the 
left of the page, we find the argument ; in the next 
column, the corresponding values of the quantity 
sought. Generally, however, there are ten such col- 
umns on the same page. The argument is not printed 
at the left of each column, but, to save space, the last 
figure of it is at the head of the column and the rest 
at its left in the first column on the page. The num- 
bers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 at the head of different col- 
umns usually indicate a table of the fii-st kind. 

When the argument lies between two values in the 
table, we cannot directly find the quantity which we 
seek. We have to make use of interpolation, the 
rules for which need hardly be explained. 

Interpolation depends upon the principle that slight 
differences in any quantity are nearly proportional to 
the corresponding differences in its argument, and 
upon the application of the rules of simple proportion 
to the differences in question. 

The second kind of table is similar to the fii*st, only 
that at the head of the different columns is contained 
a second and independent argument upon which the 
quantities in the body of the table also depend. 

Thus the density of air at different pressures and 
temperatures is contained in Table 19. We follow 
the line corresponding to a given pressure until we 
reach the column corresponding to the given temper- 
ature, and there find the density in question. 



159] 



GRAPHICAL METHOD. 



665 



Interpolation in such a table is more difficult than 
in one of the first kind, because the variation due to 
both arguments must be taken into account, as ex- 
plained in ^ 168. Interpolation is, however, unne- 
cessary when the quantities are, as in Table 20, close 
enough together, or where only a rough value is 
required. 

§ 69. Graphical Method. — Co-ordinate paper (that 
is, paper ruled, in small squares) is useful in many ex- 
periments, both for representing results so that any 
gross error is visible to the eye, and for purposes of 
interpolation. At the left of the paper there is usu- 






ili!JJH!in!!HT^'^n'''l!TT 









n:B:iH »;;:: ::=:»:;i;^ 







Iff 



ill III 


III - "1 




iHFiiiiin 


lllll 


Pms ^ ^ *.i 


nil II 


m 



Fig. 4. 



ally constructed a vertical scale, like the scale of 
degrees in the diagram. At the top there is a hori- 
zontal scale, like that in the diagram representing the 

6 



666 REDUCTION OF RESULTS. [§60. 

weights floated by a Nicholson's hydrometer. The 
correspondence of two values is represented by a point 
opposite the two values in question. Thus in Fig. 4, 
A represents that at 30® the hydrometer floats 30.1 
grams ; -B, that at 20® it floats 30.4 grams ; (7, that 
at 10® it floats 30.6 grams. The dotted line ABC 
drawn with a bent ruler thus supplies an indefinite 
number of approximate values. . To find the weight 
floated at 15®, we find a point P opposite 15®, and 
then a point Q opposite P. The answer is 30.62 
grams. In the same way the relation between any 
two quantities can be represented by points, and in- 
termediate values found. 

§ 60. Use of Rough Methods. — It is always prudent 
to revise any reduction involving much numerical work, 
applying the various tests which arithmetics contain. 
It is, however, easier to reason clearly about small 
quantities than about large ones, since the former only 
can be carried in the head. Mistakes in reasoning can 
often be discovered by rough mental processes when 
no error can be detected in the figuring. 

Thus, if the buoyancy of air relieves water of a little 
more than a thousandth part of its weight, 60 grams 
will lose a little over 6 centigrams. If we find that we 
have introduced a correction of 6 decigrams or 6 milli- 
grams, we at once detect the mistake. 

The use even of rough tables, when they can be 
found, is a very convenient check upon numerical 
work. When a multiplication runs into the millions, 
logarithms will be useful, — not always, however, five 
places. Gross errors are most easily detected by loga- 



§ 61.] REDUCTION OF CONSECUTIVE OBSERVATIONS. 667 

ritbms carried out only to a single place of decimals, 
the whole attention being placed upon the character- 
istic. It is thought advisable in physics to use negative 
characteristics in preference to subtracting from 10. 
The student may be reminded that a most serious and 
at the same time a most common mistake in calcula- 
tion is the misplacement of the decimal point. 

§ 61. Reduction of Consecativa Obsarvatiozui. — In 
§ 88 we obtained the following series of angles : 0°, 
40°+, 80^+, 120°+, 160°+, 200^+, 240°+, 280°+, 
320°+, 360°+, 401°—, and 441°— ; the first and last 
give us a difference of 441° — , indicating less than 
40^^° for the angle ; the second and next to the last 
give less than 40^°, but the 3d and 3d from the last as 
well as the 4th and 4th from the last give each 40°. 
The average of these four results is 40^°, or 40°.05 
nearly. 

Again, the 1st and 9th, the 2d and 10th, the 3d and 
11th, and the 4th and 12th give respectively 40°+, 
40°, 40^°—, and 40^^—; the average of these four 
values is 40^^^°, or 40°.06 nearly. Either of these 
methods of reduction is accurate enough for the 
measurements in question. In each case the 5th, 
6th, 7th, and 8th observations were omitted. By 
using them we could have obtained two more pairs 
of observations; but the shortness of the interval 
between them takes off from their value. The prob- 
able error of the result would actually be increased 
by treating them as we have the others. It is gene- 
rally advisable to omit in this way the middle third 
of a series of consecutive observations. 



668 REDUCTION OF RESULTS. [§ 61. 

There is a third way of reducing consecutive inter- 
vals against which the student must be cautioned. 
The differences between the 1st and 2d, the 2d and 3d, 
etc., are in 10 cases 40°, in one 41°. There is a com- 
mon fallacy to the effect that the average of these, 
40^^**' Drakes use of all the observations. It is easy, 
however, to see that in taking the average we must 
first add the intervals together, and that we shall ob- 
tain as a result the interval between the 1st and 12th 
observations, since the whole is equal to the sura of all 
its parts. We subsequently divide by 11, but the 
result depends solely upon the 1st and 12th, and not 
in any way upon the intermediate observations, the 
value of which is therefore completely lost. 

This method of averaging consecutive intervals 
should be accounted a serious error, not simply be- 
cause it is unnecessarily laborious, but because of the 
self-deception which it involves. 



CHAPTER V. 

HYDROSTATICS. 

§ 62. Pascal's Principle. — From experiments in 
weighing liquids we might infer that their weight 
exerted simply a downward action. By immersing 
a pressure-gauge^ in any liquid we find, however, 
that at a given depth the liquid exerts an equal force 
upon it in all directions, whether horizontal, vertical, 
or oblique, whether up or down. The same in- 
strument shows that when a fluid is at rest the 
pressure is the same at all points on the same level. 
If this were not so, a perfect fluid would evidently 
be unable to remain at rest. Conversely, all points 
in a stationary liquid which are subject to a given 
pressure are found on a given level.* 

§ 63. Hydrostatic Pressure. — If we have a column 
of liquid in a tube with vertical sides which it cannot 
cling to, the whole weight of the column must rest 
upon the bottom of the tube. Let the tube be 1 sq. 
em. in section ; then the weight of the whole column 

1 For the construction of such a gauge see Descriptive list of Ex- 
periments in Elementary Physics, 1889, Exercise 6. This experiment 
is due to Professor Hall. 

^ When (see Fig. 60, page 127) the air-pressure is greater on one 
part of a liquid surface (c) than on another (6), the liquid stands at 
unequal heights in two parts of the apparatus, but if the air-pressure is 
the same it stands at the same level in both places (Fig. 61). That part 
of a liquid in a U-tube which lies below a given level transmits or com- 
municates pressure along this level without increasing or diminishing it. 



670 HYDROSTATICS. [§ 64 

rests upon a surface 1 sq. em. in area, and the pressure 
in dynes per sq, cm, is numerically equal to this weight 
reduced to dynes. The weight of the column is evi- 
dently the product of its volume in cu, cm.<, the 
density (or weight of 1 cu, cm. in grams), and the 
intensity of gravity (or weight of 1 gram in dynes) ; 
and as the tube has a unit cross section, the volume 
is numerically equal to its height. The hydrostatic 
pressure (that is, the pressure of the liquid per unit 
of area) at the bottom of a tube is therefore the 
product of the depth and density of the fluid and the 
intensity of the earth's gravitation. It is clear that 
the size of the tube makes no difference, for in a tube 
of twice the cross-section wc should have twice the 
weight distributed over twice the area, and the pres- 
sure per 8q. cm, would be the same. Since pressure 
#/is the same in all directions, we may therefore state 
Tyis a general principle that pressure increases with the 
(/depth. 

§ 64. Principle of Archimedes. — Suppose we sus- 
pend a solid in a fluid. The pressure on the solid 
will of course be greater the more we lower it into 
the fluid, but the pressure on the bottom of the solid 
will always be greater than on the top ; hence the 
fluid will buoy up the solid more or less. One can 
calculate the amount of this buoyancy by the princi- 
ples which have already been stated if the shape of 
the solid is not too complex, but there is a much 
simpler way of arriving at the result. Imagine the 
solid out of the fluid, and its place filled by a separate 
portion of that fluid, having the same shape and 



! 65 ) BUOYANCY OF AIR. 671 

bounding surfaces as the solid. The pressures on 
this new portion of the fluid must be the same as on 
the actual solid, because the surfaces and their depths 
are the same ; but the forces produced result simply 
in holding the fluid in place, hence their resultant is 
equal and opposite to the weight of a portion of the 
fluid equal to the solid in bulk. This principle is 
known by the name of its discoverer, Archimedes, 
(287 to 212 B. c.)» and may be thus stated : a solid 
immersed in a fluid is buoyed up by a force equal to 
the weight of the fluid displaced. The difference 
between the weight of a body and the buoyant force 
of a fluid in which it is submerged may be called the 
effective weight of the body in that fluid. 

§ 65. Buoyancy of Air. — According to the princi- 
ple of Archimedes just explained, a body loses weight 
in air just as it would in any other fluid. Seven 
grams of brass displace, for instance, about five-sixths 
of a cubic centimetre of air ; that is, about one milli- 
gram, or one 7000th of their nominal value. Bodies 
weighed against them also lose in weight according to 
the amount of air displaced. Ordinary weighing con- 
sists, therefore, in a comparison of effective weights. 
The number of grams which balance a body in air is 
called its apparent weight in air. If, however, the 
body is in water (the weights being as before in air), 
we find what is called the apparent weight in water. 
The effective weights in air or in water can always 
be found roughly from the corresponding apparent 
weights by subtracting, for reasons above explained, 
one part in 7000 from the nominal values of the brass 



/ 



672 HYDROSTATICS. [§ 67. 

weights. The exact correction is given in § 67. Only 
apparent weights are obtained by Nicholson's hydrom- 
eter, by the hydrostatic balance, or by the specific- 
gravity bottle. 

§66. Apparent Specific Gravities. — It is obvious 
that in weighing a body first in air, then in water, as 
in Experiments 2, 3, and 4, or 8 and 9, we find first 
the apparent difference between the weight of the 
body and that of an equal bulk of air, and second, 
the apparent difference between the weight of the 
body and that of an equal bulk of water. Subtract- 
ing the latter from the former we have the apparent 
difference of weight between the water and air dis- 
placed, or what is the same thing,^ the apparent 
weight in air of an equal bulk of water. The ratio 
between the apparent weight of a body (in air) and 
that of an equal bulk of water (in air) is called the 
apparent specific gravity of the body. Without cor- 
rections for the buoyancy of air, we can obviously find 
only apparent specific gravities. 

§ 67. Correction of Apparent Weights. — Given the 
apparent weight of a body in air and in water, we 
usually proceed as follows : First calculate by sub- 
traction the weight of an equal bulk of water, as 
explained in § 66. Multiply this by the space ap- 
parently occupied by 1 gram (see Table 22) to find 
the volume in question. This is obviously equal to 
the number of eu. cm. of air displaced by the sub- 
stance. Multiply it, therefore, by the weight of 

^ This holds strictly for effective weights from the principle of 
Archimedes; hence also for apparent weights, to which the former 
are proportional. See § 06. 



§68] APPARENT SPECIFIC GRAVITIES. 678 

1 cu. cm. of air (see Tables 19 and 20) to find the 
weight of air displaced. Next multiply the weight 
in grams of the body in air by the weight of air dis- 
placed by 1 gram of brass (Table 20, A) to find the 
weight of air displaced by the brass weights. Sub- 
tract the latter from the apparent weight of the body 
in air to find its effective weight in air (§§ 64, 66). 
Add to this the weight of air displaced by the body 
to find its true weight in vacuo. 

When the density of a substance is approximately 
known, either by reference to Tables 8-11, or from 
an actual determination of its apparent specific grav- 
ity, we may at once reduce its apparent weight to 
vacuo by applying the appropriate coefficient from 
Table 21. 

The apparent weight of a liquid, obtained either 
by methods of displacement or by the specific gravity 
bottle, must be reduced to vacuo ^ like any other ap- 
parent weight, starting with either (1) the volume, or 
(2) the density of the liquid, or (3) with the weight 
of an equal bulk of water. The apparent weight 
of a body in a liquid needs, however, to be corrected 
only, as has been explained above, for the buoyancy 
of air on the brass weights by which the body is 
counterpoised. 

§ 68. Correction of apparent Specific Gravities. — 
To find the density of a body, we first find, as ex- 
plained in § 67, the volume of the body from the ap- 
parent weight of water displaced, and second the 
weight of the body in vacuo. The weight in vacuo 
is then simply divided by the volume to find the true 



674 HYDROSTATICS. [§ 69. 

density of the substance at the given temperature 
and pressure. 

In case we have given, as in Experiment 13, not the 
apparent weight of water displaced by a solid, but 
that of some other fluid of known density, we may 
divide the corrected weight of the fluid, in vacuo^ ob- 
tained as above, by the density of the fluid, to find 
the space occupied; or we may divide its apparent 
weight by its apparent specific gravity, if we know it, 
to find the apparent weight of an equivalent bulk of 
water, and work out the result as before. 

We notice that, in reducing apparent specific grav- 
ity to density, we apply to the numerator of a fraction 
a factor from one table, and to the denominator a 
factor from another table. The same result, essen- 
tially,^ may be obtained (see § 57), by a single process. 
Subtract the factor in Table 22 from that in Table 21, 
multiply the apparent specific gravity by the algebraic 
difference, and apply the correction thus found. The 
difference between density and specific gravity is 
usually less than one per cent. 

§ 69. Density and Specific Gravity distinguished. — 
Specific gravity is defined as relative density. Hence 
density bears to specific gravity (referred to water) the 
same ratio that the density of water bears to unity. 
(See Table 26.) By the specific gravity of a sub- 
stance at a given temperature, we understand, in 
the absence of any statement to the contrary, the pro- 
portion between its weight and that of an equal bulk 

1 Results thus reduced show a slight error, usually confined to the 
sixth place of decimals. 



§ 70.] CALCULATION OF DIFFERENCE OF DENSITY. 675 

of water at the same temperature. It is understood 
also, unless otherwise stated, that both bodies are under 
atmospheric pressure (76 cm.). Specific gravities of 
gases, however, are often stated with respect to hy- 
drogen or air at the same temperature and pressure. 
Specific gravities are also referred to water at its tem- 
perature of maximum density. Having accepted the 
value 1.000013 for the maximum density of water, we 
see that such specific gravities are less than densities 
by an amount (13 parts in a million) which is small 
compared with the probable error of observation. 

§ 70. Calculation of Difference of Density. — Since 
density, D, is the quotient of mass, M^ by volume, 
r, or 

-!• 

two bodies having the same volume, F, densities 
l>i, 1>2, and masses 2Mi, iM^, have a difference of den- 
sity equal to the difference in their masses divided by 
the volume, that is, 

n - n ^' M,_ M,^M, 
^^ ^ - F F ■" F • 
Hence we may find the difference in density between 
two liquids or two gases (as in Experiment 18) from 
the difference in weight of a flask of known capacity 
filled first with one, then with the other. It is obvious 
that in weighing a flask filled first with air, then with 
a liquid (as in Experiments 11 and 14), we might deter- 
mine in this way the difference of density between 
the liquid and air, and that by adding to this result 
the density of air, B^ (from Tables 19 and 20), we 



676 HYDROSTATICS [§ 71. 

should find the density, 2>a, of the liquid in question ; 
that is, 

When the substance weighed is (as in Experiment 
18) lighter than air, the difference of density may be 
considered negative, and must be subtracted numeri- 
cally from the density of air as indicated by the 
formula identical with the above, 

V 

§ 71. Accuracy of Meteorological Instruments. The 
density of the atmosphere is found to affect all delicate 
weighings. For many purposes it is suiBBciently accu- 
rate to assume a mean density of 1.2 mgr. to the cubic 
centimetre ; ^ but for the most accurate determinations 
we need to correct it for temperature, pressure, and 
humidity. The corrections are so slight that a rough 
estimate is suflBcient for this course of measurements, 
and hence we may accept provisionally the indications 
of such weather instruments as may be found in the 
laboratory. We shall learn, later on, the means of 
detecting errors in these indications, and shall expect 
to prove that these errors have not perceptibly affected 
our results. 

In place of the ordinary weather instruments, we 
may employ a sensitive baroscope, or harbdeik^ con- 
sisting of a hollow cylinder which has been counter- 
poised in vacuo against a weight occupying say 1000 

^ The probable error under this assumption may be estimated as 
between 1 part in 10,000 and 1 part in 100,000. 



§73.] THE DENSITY OF WATER 677 

eu. cm. less space than itself. The apparent difference 
of weight between the hollow cylinder and its coun- 
terpoise indicates at once the actual density of the 
atmosphere. 

§ 72. Accuracy of Oram- Weights. We must choose 
between accepting such copies of the gram as are at- 
tainable, and determining independently the weight of 
a cubic centimetre of water. Experience shows that 
weights can be copied (and that they generally are 
copied) with a very great degree of precision, while it 
is comparatively diflBcult to copy standards of length, 
and still more difBcult to reproduce them.^ There is 
also more or less uncertainty as to the temperature at 
which a cubic centimetre of water may be assumed to 
weigh one gram (see § 6 and Table 25), and it is by no 
means easy to find the weight of a cubic centimetre of 
water with any degree of precision. It is, moreover, 
important to express our results in conventional units. 
For these reasons we prefer to accept a set of gram- 
weights, provided, however, that we are not able to 
detect any gross error in them by such means as are in 
our power. 

§ 73. The Density of Water. — On account of the 
inaccuracy of our standards of length we are unable 
to determine the volume of a body very accurately 
from its length, breadth, and thickness ; and hence we 
cannot find its density absolutely, as in Experiment 1, 
with any degree of precision. The same inaccuracy 
affects the volume of water which such a body dis- 

^ The error in the original determinations was nearly a tenth of one 
per cent. (See § 5.) 



678 HYDROSTATICS. [§ 73 

places, and hence also the density of water, which is 
found by comparing the weight and volume displaced. 
We prefer, therefore, to accept the results of a great 
number of determinations (see Table 25) rather than 
any rough measurements of our own, and we make 
use of this table of density for testing or correcting 
our standards of length, and not of our standards of 
length for the determination of a new table of densi- 
ties. It is thought that measurements of length cor- 
rected in this way will be nearer the conventional 
standard than those depending directly on such rough 
copies as are found in the market. The approximate 
agreement of our actual standards of length and mass 
is the first of a series of tests to which these standards 
must be subjected, and through which, finally, any 
gross error in either is sure of detection. 



CHAPTER VI. 

HEAT. 

§ 74. Temperature. — Temperature is believed to 
depend upon the vibration of the molecules of which 
a body is composed, and hence be akin to what we 
call heat. Temperature is not, however, heat, but the 
state of saturation with heat which determines, under 
certain conditions, whether heat will be imparted or 
absorbed. Bodies which can communicate heat to 
others are said to have a higher temperature. Two 
bodies in contact are said to have the same tempera- 
ture when no heat flows from one to the other. It 
is found that two bodies at the same temperature as 
a third are themselves in thermal equilibrium. Heat 
corresponds in a certain sense to quantity, temperature 
to intensity of vibration (see § 84). The temperature 
of a gas is seen from its nature to be intimately 
connected with pressure ; for pressure is explained as 
the effect of the perpetual bombardment of the mole- 
cules against the sides of a vessel which contains 
them. 

§ 76. Absolute Zero. — We must distinguish the 
absolute zero of temperature from that which we have 
provisionally adopted. At the absolute zero, the par- 



680 HEAT. [§ 76. 

tides of a body are supposed to be at rest. Gases 
therefore exert no pressure at this temperature, and 
occupy no space, save that which their molecules take 
up when closely packed together.^ The absolute zero 
must be the same for all bodies, since when their heat 
is wholly taken away they cannot communicate any 
from one to another, and hence have, by definition, the 
same temperature. There is reason to believe that the 
absolute zero of temperature is, on our provisional 
scale, about 273° centigrade below the freezing-point 
of water. 

§ 76. Absolute Temperatures. — We have seen that 
the temperature and pressure of gases are intimately 
connected. The absolute scale of temperature is 
founded upon this fact. By definition, absolute tem- 
perature u proportional to the pressure of a perfect gas 
confined to a constant volume. All permanent gases 
are found to be essentially perfect in this sense. 

To compare absolute temperatures, we may seal up 
a mercurial barometer in a tube, or an aneroid barom- 
eter in a preserving jar. The corrected indication of 
the pressure of the air enclosed will be proportional 
to the absolute temperature. 

We are still at liberty to adopt any length of degree 
which we please, and for convenience we will choose 
that of the centigrade scale. Let us suppose that the 
barometer rises ten inches when we heat the air from 
the freezing to the boiling point of water. Then a 
tenth of an inch will represent a degree. The abso- 

^ The molecules are thought to occupy at least one half as much 
space as the liquid formed hy the condensation of a gas. 



§ 77.] VELOCITY OF MOLECULES. 681 

lute temperature of freezing or boiling can now be 
found from the corresponding pressure of the barome- 
ter in tenths of an inch. We discover in this way 
that water freezes at 273°, and boils at 873° on this 
absolute scale. 

Whatsoever means we adopt for estimating the pres^ 
sure of a confined gas, the same result is obtained, 
since the pressure at boiling is to that at freezing as 
373 is to 273. 

It is found that all temperatures on the mercurial 
thermometer may be converted approximately to the 
absolute scale by adding 273°. 

§ 77. Velocity of MolectdeB. — From the definition 
of force (§ 12) depending on mass, time, and change 
of velocity, it is clear that the pressure of a gas must 
depend both upon the number and upon the velocity 
of the molecules which strike a given surface in a 
given time. If we double the velocity of the mole- 
cules without changing the distance they must travel 
before hitting the sides of the vessel, the blows will 
be twice as frequent and twice as strong; hence the 
pressure will be quadrupled, — also, by definition, the 
absolute temperature, as the volume remains the same. 
So, in general, temperature may be shown to vary as 
the square of the molecular velocity. 

We do not know the mass of a single molecule, 
except within wide limits; but we can find the 
weight of a cubic centimetre of a gas, and thus 
independently of the number of molecules in the 
given space, we can calculate the average velocity 
which will account for a given pressure. Molecu- 



682 HEAT. 1§ 79 

lar velocity is not therefore a matter simply of 
conjecture.^ 

§ 78. Pressure and Density of Gases. — The density 
of a gas is evidently proportional, other things being 
equal, to the number of molecules in a given space. 
In the case of exceedingly rarefied gases, the mole- 
cules are so far apart as not practically to interfere 
with one another; hence each will hit the sides of 
the vessel as often as if the others were not pres- 
ent.2 It follows from the principles explained in the 
last section that in such a case pressure and density 
are proportional when the average velocity, or tem- 
perature, remains the same. Hence at a constant 
temperature, the pressure of a perfect gas varies with 
the density. Experiment confirms this assumption in 
the case of exceedingly rarefied gases. 

As a gas becomes more and more condensed, there 
is less and less space between the molecules free for 
vibration, and cohesion may come into play, partic- 
ularly in the case of a vapor near its point of con- 
densation. In such cases the law connecting density 
and pressure cannot be applied. Even the most 
permanent gases are more or less compressible than 
theory would indicate (see Table 12), though in most 
experiments the variation is barely perceptible. 

§ 79. Law of Boyle and Mariotte. — As the volume 
of a gas increases, the density obviously diminishes, 

1 The average velocity of a hydrogen molecule at 0® is found to be 
not far from a mile per second ; that of oxygen is one fourth as great. 
For a farther discussion of this subject, see Maxwell's Theory of Heat, 
chapter 22. 

> See Daniell's Principles of Physics, page 224. 



I 81] REDUCTION OF DENSITY. 683 

and the pressure, as we have seen, diminishes in pro- 
portion. Hence the volume of a perfect gas at a given 
temperature varies inversely as its pressure, 

§ 80. Law of Charles. — As the volume of a gas in- 
creases, the pressure diminishes ; but as the absolute 
temperature increases, the pressure increases. It fol- 
lows that if both the volume and the absolute tem- 
perature increase in the same proportion, the pressure 
will remain the same. Hence the volume of a perfect 
gas at a constant pressure is proportional to its absolute 
temperature. 

By this principle absolute temperature can be esti- 
mated from the Volume of a gas at a constant pressure 
as in Experiment 26, as well as from the pressure of 
a gas at a constant volume, as in Experiment 27 (see 
§76). 

§ 81. Reduction of Density to Standard Temperature 
and Pressure. — If D is the density of a gas, P its 
pressure, and T its absolute temperature, then the 
pressure, Pi, at the standard temperature, T^^ will 
be given by the proportion, P^x P i i T^i T^ or P^ 
z=i PTq-^ T; the density, jDqi at the standard pres- 
sure, Po9 is given by the proportion, D^: D : : PqI Pi; 
whence J), = D P,^P^ = D P^-^CP T,-^ T) = 
DP.T^P T,. 

If the pressure, J9, is expressed in centimetres of 
mercury, and the temperature, <, is on the ordinary 
centigrade scale, we have 

76 273 + ^ 



684 _ HEAT. [§84. 

§ 82. Expansion of Solids and Liquids* — <- In tbe case 
of solids and liquids, the effects of temperature iu 
causing expansion are slight in comparison with those 
in the case of gases. It is probable that the cohesive 
forces which bind their particles together leave very 
little available space for their vibration, and it is quite 
possible that this available space obeys the same laws 
in general as in the case of gases. We have, however, 
several cases where bodies contract with heat, the 
most notable of which is water below 4°. Such cases 
may be explained as the result of the gradual re- 
arrangement of the particles consequent on a rise of 
temperature, — that is, to the same caiuse which makes 
water occupy about ten per cent less space than 
the same weight of ice* 

§ 83. Linear and Cubical Co-efficients of Expansion. — 
A co-efficient of expansion is a number which always 
occura as a factor or co-efficient in calculating expan- 
sion produced by heat. The increase of the volume 
of one cubic centimetre caused by a riseof I'' in tem- 
perature is called the cubical co-efficient of expan- 
sion of a substance. The increase of the length of 1 
cm, is called the linear co-efficient of expansion. Un- 
less otherwise stated, the co-efficient of expansion of 
gases and liquids is assumed to be cubical ; that of 
solids, linear, affecting length, breadth, and thickness 
alike, and hence only one-third as great as the corre- 
sponding cubical co-efficient. 

§ 84. Relation between Heat and Temperature. — 
The relation which temperature bears to heat is an- 
alogous to that which hydrostatic pressure bears to 



§86.] SPECIFIC HEAT. 685 

water. Heat flows from high temperature to low 
temperature, water from high level to low level. 
When we pour water into a vessel, the level rises; 
so heat increases the temperature of a body. It takes 
more water to fill a large jar to a given depth than 
a small one, more heat to warm a heavy body to a 
given temperature than a light one. Heat, like water, 
is indestructible, though it can be transformed into 
many shapes. We usually estimate quantities of heat 
relatively to a certain unit, which has been defined 
(§ 16), or, in the absolute system, by the quantity of 
work to which it is equivalent. 

§ 85. Thermal Capacity. — The thermal capacity of a 
substance may be defined as the total amount of heat 
necessary to raise its temperature one degree. It cor- 
responds to the cross-section of a vessel. A common 
measuring-glass, flaring a little at the top, requires 
more and more water to raise the level by a given 
amount. So most substances require more heat to 
raise their temperature one degree as the tempera- 
ture increases. The variation is, however, frequently 
imperceptible. 

§ 86. Specific Heat. — If we put pebbles into a 
vessel it will take less water to fill it than before; 
still less if the spaces between the pebbles are filled 
with sand. 

Specific heat corresponds to the material which a 
vessel contains before water is added. It is some- 
thing irrespective of the weight or bulk of a body 
which gives it a greater or less capacity for heat. 
From experiments in mechanics we infer that the 



HEAT. [§ 87. 

fineness of subdivision of the particles of a body is 
what fits them to be set in vibration, that is, to 
absorb heat. Specific heats accordingly increase as 
what we call the "molecular" weight diminishes. In 
the case of elementary substances this can almost be 
called a law.^ 

§ 87. Latent Heat. — If a small vessel is put inside 
a large one, and water poured into the space between, 
the level rises up to the edge of the small vessel, then 
is constant until the small vessel is filled, after which 
it rises again. So when ice is heated it rises in tem- 
perature until it begins to melt, then the temperature 
is constant until the ice is all converted into water, 
then it rises again. 

A certain quantity of heat disappears in melting the 
ice, without raising the temperature, just as a certain 
quantity of water disappears in filling the inner vessel. 
The quantity which is thus absorbed in melting a 
gram of a substance is called its latent heat of lique- 
faction. In the same way heat disappears when a 
liquid is changed into a vapor. The amount of heat 
necessary to convert a gram of a liquid into a vapor 
is called its latent heat of vaporization. 

Thus it takes about 80 units of heat (or 3,300 meg- 
ergs) to change a gram of ice at 0° into a gram of 
water at 0°. The water is not any warmer than the 
ice, because water and ice may remain indefinitely in 
contact and yet perfectly distinct. In the same way 

1 The products of the atomic weights and the corresponding specific 
heats (see Table 8, a) will be found in most cases to be nearly equal 
to the number 6. 



§ 89.] LAW OF COOLING. 687 

it takes about 536 units ^ of heat (or 22,000 megergs) 
to change a gram of water at 100° into- a gram of 
steam at 100° when the atmospheric pressure has to 
be overcome. 

§ 88. Explanation of Latent Heat. — When the par- 
ticles of a body are separated in such a way as to 
overcome certain forces called cohesive, because they 
tend to hold particles together, it is clear that work 
must be done. If a particle of ether escaping from a 
drop of that fluid is held back by the attraction of 
that drop, it will evidently lose a part of its velocity ; 
and as only the swiftest particles can escape at all, the 
slowest must remain, and the drop will grow cooler 
and cooler. The work done in evaporation is at the 
expense of temperature. When finally the liquid has 
been all converted into vapor, heat must be com- 
municated to the latter to restore to it the same tem- 
perature that it bad in the liquid state. The boiling 
of a liquid depends upon the continuous communica- 
tion of heat necessary to maintain a constant tempera- 
ture. This heat is said to be latent, because it does 
not affect the thermometer. It can, however, be 
recovered; for the heat absorbed in vaporization is 
given back in the act of condensation. The process 
is in fact reversed. A particle of vapor is acceler- 
ated by the attraction of the liquid mass into which 
it falls, and gains in velocity what before it lost. 

§ 89. Law of Cooling. — There are three ways in 
which heat is likely to escape from a calorimeter: 

^ Of this, about 40 units are consumed in overcoming the pressure 
of the atmosphere. 



688 HEAT. [§ 90. 

iSrst by conduction, or passing from one particle to an- 
other ; second by convection, or being carried bodily 
by currents of air ; and third by radiation, or directly 
passing from one place to another as the sun's heat 
does in waves or rays. When all these causes have 
been guarded against, there is apt to be a very slight 
loss of heat, which has to be allowed for. In all three 
ways in which heat can escape the amount is found 
to be proportional, nearly, to the difference of tempera- 
ture between the contents of the calorimeter and the 
surrounding air. Hence we have Newton's law of 
cooling : L099 of heat per unit of time is proportional 
to difference of temperature. 

If, for instance, the temperature within the calo- 
rimeter is 40"^ and that outside of it 20° and the rate 
of cooling 1° in 5 minutes, we should infer that if the 
calorimeter were at 80° the temperature would fall 
only about 1° in 10 minutes. We are thus able to 
estimate the temperature at a point of time when ob- 
servation would be impracticable. (See Experiment 
31.) 

§ 90. Principle of Calorimetry. — When substances 
at different temperatures are mechanically mixed in 
a calorimeter so that no chemical or physical reaction 
takes place, with the exception of a small quantity of 
heat which escapes as has just been explained, the 
total amount remains constant. What is lost by one 
body is therefore taken up by another. 

If m^is the mass of one body, 8;^ its specific heat, ^1 its 
temperature before mixture, and t its temperature 
after mixture, then the number of units it has ab- 



§ 91.1 HEAT DEVELOPED IN A CALORIMETER. 689 

sorbed is^i^x (^-^i). If it has lost heat instead of 
gaining it, the expression will be negative. Denoting 
by subscripts 1, 2, 3, &c. in the same way the proper- 
ties of the several substances contained in the calor- 
imeter, we have 

mi «i (t-ti) + Wa «a (^-^2) + Wj «j {t-Q -\-etc. = 0. 

The temperature of the mixture, t, is the same 
for all. The products Wi «i, m^ «„ wfj 85, etc.^ are evi- 
dently the thermal capacities of the bodies in ques- 
tion. For if s is the heat required to raise 1 gram 1% 
m 8 will be that required to raise m grams 1°. 

To calculate the thermal capacity of a calorimeter, 
we multiply the weight of the inner vessel in grams 
by the specific heat (from Table 8, a) of the material, 
usually brass, of which it is composed. The thermal 
capacity of a stirrer attached to the bulb of a ther- 
mometer is calculated in the same way. The thermal 
capacity of a thermometer is about one-half of the 
number of cubic centimetres immersed, whether of 
mercury or of glass, — more exactly, ^ in the case of 
mercury. The various methods of calculating specific 
heat by the above principles will be explained in 
Experiments 32, 83 and 34. 

§ 91. Heat Developed in a Calorimeter. — When a 
substance contained in a calorimeter undergoes a 
change of state, whether physical or chemical, heat is 
usually developed or absorbed. The fact is recog- 
nized by the departure of the temperature of the 
mixture from that which it would be expected to 
have if the mixture were purely mechanical. The 



690 HEAT. f§ 91. 

heat developed or absorbed when a gram of a solid is 
dissolved is called the (latent) heat of solution ; when 
it unites chemically with another substance, it is the 
heat of combination ; or if it bums in the process, the 
heat of combustion. The calculation of these heats is 
explained in Experiments 36-38. 



CHAPTER VII. 

SOUND AND LIGHT. 

§ 92. Wave Motion. — When a row of marbles is 
set in a crack of the floor, and one at the end of the 
row is hit, it strikes the one next to it and comes to 
rest after giving up nearly all its motion, the second 
marble gives up its motion to the third, and so on, 
until finally the last marble is set in motion. In the 
same way a string can transmit a pulse. The string 
however, has generally a lateral motion and each por- 
tion pulls the next one side instead of pushing it for- 
ward. A wave of sound in air is transmitted like a 
pulse through a row of marbles, a wave of light like 
a pulse through a string. In both cases, however, the 
pulse, if not obstructed, is carried from the origin 
not simply in one direction but in all. The different 
paths by which light spreads out are illustrated by 
a system of strings radiating in all directions from a 
given point. These strings represent also what are 
called rays of light. To explain the distribution of 
sound we may imagine a space filled with solid 
bodies having springs of some sort between them so 
as to keep them apart and yet allow any one to 
transmit a blow to its neighbor, as in the case of 
the marbles. 



692 SOUND AND LIGHT. [§94. 

§ 93. The Air and the Ether. — A pulse of SOUnd iu 
air is in reality transmitted by the impact of the mole- 
cules of air, which are perfectly elastic, whereas mar- 
bles are not. The velocity of sound in air is a little 
over 33 thousand cm. per sec. While sound is inter- 
cepted by what we call a vacuum (there being no mole- 
cules to transmit it), light passes more easily through 
a vacuum than through air. What carries light we do 
not know* We call it the ether. The ether, like air, is 
perfectly elastic; but it has no weight, and no per- 
ceptible resistance to motion through it ; it seems to 
pass between the particles of the densest solids "as 
freely as the wind passes through a grove of trees." ^ 
And yet it transmits, as we have seen, transverse 
vibrations, after the manner of a string. 

In some respects the ether reminds us of magnet- 
ism, which, though perfectly immaterial, can hold a 
piece of iron firmly through a piece of glass. Elec- 
tricity, however, affords the only true analogy to light. 
It is well known that telephone messages are carried 
from one wire to another, either through a vacuum or 
through almost any medium which we can interpose. 
The fact is certainly significant that electrical vibra- 
tions may pass in this way with the velocity of light 
(30 thousand million cm. per sec.)^ and the belief 
is gaining ground that light is carried by what is 
called electromagnetic induction from one particle to 
another. 

§ 94. Law of Inverse Squares. — Since both sound 
and light spread out equally in every direction, a pulse 

1 Lloyd's Undttlatory Theory of Light, § 21. 



§ 95.] RELATION OF WAVE-FRONT AND RAYS. 693 

naturally takes the form of a hollow shell, perfectly 
spherical, and growing larger as the wave passes farther 
from the source. The area of such a shell is propor- 
tional to the square of its radius ; hence the intensity 
of sound or light per square centimetre varies inversely 
as the square of the distance, — the same amount of 
energy being distributed over a greater amount of sur- 
face. The transmission of sound and light without 
any perceptible loss affords another illustration of the 
principle of the conservation of energy. 

§ 95. Relation of Wave-Pront and Rays. — The sur- 
face of a shell such as is formed by a pulse spreading 
out in all directions, or any portion of such a surface, 
is called a wave front. It is clear that a wave-front 
is perpendicular at every point to the ray of light 
passing through that point, as the radius of a sphere 
is perpendicular to its surface. When a portion of a 
wave passes through an orifice, the rest being inter- 
rupted, most of it still continues to advance very much 
as if the whole wave were present. It is found, in- 
deed, that waves tend to move in straight lines, and 
in all cases in a direction at right angles to their front. 
It follows that any cause which can change the direc- 
tion of the wave-front wDl also cause a bending of the 
rays. In the absence of any such cause, the general 
direction will remain constant. 

This tendency of waves to move in straight lines is 
much mote marked when a great number of pulses are 
sent one behind the other, as is always, practically, 
the case. The wave-fronts then find it impossible to 
bend much without interfering with one another. 



694 SOUND AND LIGHT. [§ 96. 

A series of wave- fronts issuing from an orifice con- 
stitutes in the case of light what is called a beam. 
The middle part of a beam is perfectly straight ; the 
bending is confined to an almost imperceptible portion 
at the edges. Sound shows also a tendency to move 
in straight lines ; but, owing to the great distance be- 
tween the pulses, not nearly to the same extent. 

§ 96. Frequency of Vibration. — When a toothed 
wheel, by striking on a card, gives a regular series of 
pulses .to the air, a musical note is often produced. 
The pitch of the note depends on the number of pulses 
per second. There are three classes of notes, one in 
which the pulses are too infrequent to produce a con- 
tinuous effect upon the ear, the second audible (say 
from 30 to 30,000 pulses per second), and the third 
too rapid to be heard. In the same way there are 
three classes of vibration in light; one too slow to 
affect our organs of sight, a second visible (from 400 
to 800 millions of millions per second), and a third 
more rapid still and in consequence invisible. 

When sound is intercepted, it is usually changed 
into heat. All kinds of light when absorbed by aa 
opaque body are generally transformed into heat. In 
all such cases the heat is equivalent, erg for erg, to the 
energy spent in producing the vibrations in question. 
All kinds of light act on a photographic plate, but 
principally those of the third class alluded to, often 
called actinic. In sunlight the principal source ot 
energy is from invisible vibrations of the first class, 
often called calorific for this reason.^ 

^ See Tyndall's Fragments of Science, pages 182-184 



§ 99.] RESONANCE. 695 

§ 97. Reflection. — All waves are reflected from a 
surface as an elastic ball is from the floor. That part 
of the motion which is perpendicular to the surface is 
reversed, and that parallel to it preserved ; hence the 
path of the ball makes the same angle with the sur- 
face before and after reflection. One can see, without 
a special examination of the motion of separate par- 
ticles, that a reversal of one component accounts for a 
similar change of direction in a wave. 

§ 98. 'Wave-length. When sound is reflected back 
and forth between two walls, an echo is heard at in- 
tervals corresponding to the time it takes sound to 
traverse the distance back and forth between the walls. 
When the walls are only a few feet apart, the echo 
may become so frequent as to produce a musical note. 
Thus a tube closed at both ends exhibits this phenome- 
non. The distance which sound travels between two 
successive pulses is called in general a wave-length, 
and is clearly equal in this case to twice the length of 
the tube. When a particular color is produced in the 
same way by the reflection of light back and forth be- 
tween two pieces of glass very close together, its wave- 
length is twice the thickness of the space between the 
glasses. 

§ 99 Resonance. — The vibration of a tube closed 
at both ends may be described as a periodic rush of 
air from one half to the other and back again. When 
such a tube is cut in two in the middle, each half has 
the power of vibrating essentially as before. The at- 
mosphere receives the rush of air out of the tube and 
supplies air to fill the vacuum thus caused, taking in 



696 SOUND AND LIGHT. [§ 100. 

fact to each half the same place as the other half of 
the tube. Since the whole tube was equal to half 
a wave in length, the halves will be nearly quarter- 
wave-lengths ; but as the vibration extends a little be- 
yond the open ends,^ a tube closed at one end only is 
not quite a quarter of the length of the wave to which 
it responds. 

When a tuning-fork emitting the corresponding 
note is held near the mouth of the tube, the sound is 
greatly increased. The downward pulses from the 
fork are reflected from the bottom of the tube so as 
to reach it in the middle of its upward motion, which 
is therefore reinforced in its eflFect upon the air. The 
slightest variation in the length of the tube causes the 
phenomenon to disappear; but if the tube is made 
just one half a wave-length longer, or any number 
of half-wave-lengths, the reflected pulses, traversing 
the distance twice^ are retarded a whole wave-length 
or several whole wave-lengths, meet the fork as before, 
and resonance reappears. 

A tube open at one end therefore responds to a 
given note when its depth is equal to J, |, |, etc., 
wave-lengths or thereabouts. The first quarter-wave- 
length is approximate ; the other lengths are greater 
than the first by exactly J, 1, l^^^ etc., wave-lengths 
respectively. 

§ 100. Interference. — When two series of pulses 
arrive at the same place at the same time the efl^ect 

^ It has been estimated that the vibration yirtually extends beyond 
the open end of a tube to a distance equal to a fourth or a fifth part of 
its diameter. 



§ 101] DIFFRACTION-GRATING. 697 

is greatly increased ; but if they arrive at different 
times, each tends to fill up the gaps in the other, and 
thus often to diminish the eff^ect. Hence if a musical 
sound enters a room by two windows, a person stand- 
ing between the windows on the opposite side might 
receive the pulses from each at the same time, while 
one by his side, being nearer one window than the 
other, would receive the pulses at different times. 

Again, a person still further to one side would re- 
ceive pulse No. 1 from the further window at the same 
time as pulse No. 2 from the nearer window, and the 
sound would be reinforced. Evidently the difference 
of his distances from the two windows must be the 
same as that between two pulses, or in other words, 
a wave-length. There will be reinforcement again 
when one window is 2, 3, 4, etc., wave-lengths further 
off than the other ; but whenever there is a fraction of 
a wave-length involved there will be more or less in- 
terference. The same holds for a series of windows, or 
when sound arrives by any two channels whatsoever. 
We can always find the wave-length of a given note if 
we know the smallest difference 
in the length of different chan- 
nels producing reinforcement or 
interference. 

§ 101. Diffiraction-Gratiiig. — 
Precisely the same method is ap- 
plied to light. An ordinary 

diffraction-grating (see illustra- 

,. ^ . ^ /. . /. ,. Difpraction-Gratino. 

tion) consists of a series of lines 

with slits between them, through which light passes. 

8 




698 SOUND AND LIGHT. [§ 102. 

We find the difference in length of the paths followed 
by the light arriving at a given point by two succes- 
sive slits, and this is the wave-length of the light 
which is reinforced at that point by the grating. 

There is an obvious advantage in employing a 
grating with a large number of lines, let us say 
a thousand. If each line is exactly one wave-length 
further off than the next, a thousand pulses will 
arrive simultaneously at the eye ; but if there is the 
least error in adjustment, let us say a thousandth of 
a wave-length, the pulses will all arrive at different 
times, and thus produce complete interference. 

It is to be observed that waves of light and sound 
tend to move in straight lines only when the breadth 
of the waves is considerably greater than the distance 
between them ; hence the phenomena of bending or 
diffraction in passing through narrow orifices. Sound- 
waves, being on the average a million times farther 
apart than waves of light, bend much more readily, 
and require a screen proportionally broad to produce 
a distinct " sound-shadow." The longest light-waves 
are, however, comparable with the shortest waves of 
sound. All waves bend round a small obstacle very 
much like the waves of the sea. 

§ 102. Refraction. — If a line of soldiers should 
march obliquely into a swamp, those who met it first 
would be most retarded, and their front would change 
its direction. In the same way a wave changes its 
direction in entering a medium in which it moves 
more slowly. Let AB (Fig. 5) be the wave-front 
in vacuo advancing in the direction A Q,t right 



§ 102.] LAW OF LENSES. 699 

angles to AB; and let CD be the wave-front ad- 
vancing in the direction 5i> at right angles to <72>, 
after passing through the surface £ (7 of a refracting 
medium. Since the time in passing from J. to (7 is 
the same as from -B to 2>, -4. (7 is to J3 2> as the veloc- 
ity in vacuo is to the velocity in the refracting 




Fio. 5. 

A C 
medium ; but ■^— ;, is the sine of AB C^ which may 

B T) 

be called the angle of incidence (i), and ■^— ^ is the 

sine of B (7i>, the angle of refraction (r) ; hence 

sint A C ^ BD A C 

sinr''BG'^ BC " B D' 

The ratio of A to BB, or the velocity in vacuo to 

the velocity in a given medium, is called the index of 

refraction of that medium, /*, and hence is calculated 

by the formula 

sin i 

fi= -, — • 
sm r 

The index of refraction of glass, for instance, is given 
as 1.5, nearly. This means that light travels half as 
fast again in vacuo as in glass. 



700 SOUND AND LIGHT. [§ 103. 

§ 103. Law of Lenses. — When waves of light di- 
verging from a point B (Fig. 6) pass through a lens 
A J, and converge to a point H^ the central portions 
are clearly retarded by a constant amount J)F in- 




^r- im-~ --i^ 



Fig. 6. 

eluded between two spherical surfaces AFIaniADl 
with B and S respectively as centres. J) F may be 
divided by a plane AEI into two portions, DE and 
EF^ which, by geometry, are inversely as the dis- 
tances BE and fl'J? (nearly), called conjugate focal 
lengths. As DF must be constant, DE '\- E F must 
be constant, — hence also the sum of the reciprocals 
of the conjugate focal lengths. 

When rays emanate from a distant point, like a star, 
so as to be nearly parallel, they are focussed at the 
shortest possible distance by a given lens. This dis- 
tance is called the principal focal length. As its con- 
jugate is very large, the reciprocal of this conjugate 
may be neglected. Hence the law of lenses : Tlie re- 
ciprocal of the principal focal length {F^ i% equal to the 
sum of the reciprocals of any two conjugate focal 
lengths {Fi and F^^ or 

Jl - 1 i_ 



104] 



IMAGES. 



701 



The calculation of the index of refraction of a lens 
will be explained in Part X. 

§ 104. Images. — If waves of light emanate, not 
from a single point, as £ in Fig. 6, but from several 
such points, as B, -B', S' (Fig. 7), they will be focussed 
at several points, as IT, if', I?", so situated as to be 
in the straight lines BEH, B EH\ B"EH\ as the 
middle of a lens, having two parallel surfaces, does 
not bend the rays. 




::w% 



VP 



***• 



Fio. 7. 



Since every point B is represented, we find at JST a 
perfect image of an object at J?, but completely in- 
verted ; and the separation between any two points is 
clearly proportional to the relative distance of the 
image and object from the lens. 

We distinguish between real and virtual images. 
IT, iT, H'^ is a real image of 5, B\ B'\ because 
the rays of light from -B, B\ B" actually meet at 
H H\ jy, respectively, and again diverge from these 
points as from a real object. A photograph requires 
a real image for its production. On the other hand, 
an image in a looking-glass is virtual, because rays do 
not really meet in it or diverge from it. 



702 SOUND AND LIGHT. [§ 104. 

A virtual image may be located as in experiment 43. 
When for instance an object is too near a convex lens 
to have a real image on the opposite side, we may still 
find a virtual image behind the object. That is, rays 
diverging from the object may, after passing through 
the lens, seem to diverge from a more distant point 
on the same side of the lens as the object.^ Concave 
mirrors furnish similar examples of real and virtual 
images. Convex mirrors and concave lenses do not 
tend to bring rays to a focus, and give therefore only 
virtual images. 

^ By a construction similar to Fig. 6 it may be shown that in such 
cases the reciprocal of tlie principal focal length is equal to the 
difference of the reciprocals of two conjugate focal lengths. 



CHAPTER VIII. 



FORCE AND WORK. 



§ 105. Components and Resultants. — When a body 
moves from AtoB (Fig. 8), then from B to C, it passes 
of course from A to O; the two motions A B and B C 
may also be thought of as relative motions taking place 
at the same time. Let the points A^ B^ and C all start 
at A; let B move with respect to J., through the 
distance and in the direction A By and at the same 
time let move with respect to B through the distance 
and in the direction B C; then clearly C has moved 
with respect to A through the distance and in the 
direction A C. 




Fig. 8. 



We express this fact by calling the motion A C the 
remltant of the two motions A B and B (7, and by 
calling A B and B C components of A (7, because when 



704 FORCE AND WORK. l§ 105. 

compounded together they produce A C. We shall 
have occasion to consider only components which are 
at right-angles. 

It AB and B are motions which take place in the 
unit of time, they represent velocities ; hence clearly 
the resultant of two velocities A B and B is A 0. 

Again A B and B C may represent component veloc- 
ities which a body acquires in the unit of time; in 
other words, component accelerations (§ 11) ; evi- 
dently the resultant of two accelerations A B and B 
must be an acceleration A 0. 

Finally, we may multiply the accelerations AB^BC^ 
and A C hy the mass of the body which they affect, 
without disturbing their relative values; but the 
products of mass and acceleration are forces (§ 12) ; 
hence two component forces, A B and B C, must give 
a resultant force A 0. 

In fact it is evident that all quantities involving 
distance and direction, whether motions, velocities, 
accelerations, or forces, must be compounded by the 
same rules as lines in geometry. 

Now since A B and B Csltb geometrically equivalent 
to A CyB Cmust be the geometrical difference between 
AB and AC. Hence a change of velocity from A Bio 
A means the acquisition of a new velocity, B C. We 
are thus able to represent the chjinge of velocity con- 
sequent on a change of direction as well as from a 
change in magnitude. 

Again, a motion A O carries a body as far away 
from the line AB slq the motion B (7, and a motion 
AC carries it as much nearer to -B (7 as a motion A B. 



§ 106.] ABSOLUTE MEASUREMENT OF FORCE. 705 

Hence if the components, A B and B C, are at right- 
angles, A B and B (J measure respectively the effects 
of a motion ACy m the general directions A B and 
BC} 

§ 106. Absolnte Measurement of Force. — If a body is 
free to move in every way, the force acting upon it is 
always said to have the same direction as the velocity 
which the body acquires, as explained in the last sec- 
tion. It is also said to have a magnitude such that 
the product of the force / and the time t it acts is 
equal to the product of the mass m acted upon and 
the velocity v acquired. This definition of force is 
expressed also by the formula 
ft = mv. 
Experience shows that force defined as above corre- 
sponds to that which we ordinarily measure with a 
spring-balance. 

The student should bear in mind that the funda- 
mental law of motion contained in the formula ap- 
plies only to bodies perfectly free to move, like masses 
in astronomy. It is a common fallacy to suppose that 
force is necessary to maintain motion. Our formula 

1 The relation between the components and resultants of forces 
may be illustrated by the strains which they produce. Let A be the 
head of a nail bent by one force from A to B^ and by another force 
from Bio C, As a result, it is bent from A to C. Now by Hooke's 
Law, as explained in § 114 below, forces are proportional (with certain 
limitations) to the strains produced; hence two forces ^^ and BC 
must have a resultant A C when estimated in this way. 

Again, a nail bent from ^ to (7 is bent in the general .direction A B 
by the same amount as if bent from Aio B; and in the general direc- 
tion B C the same as if bent from B to C Hence A B and BC an 
the components of ^ C in their respective directions. 



706 FORCE AND WORK, [§ 108. 

expresses the fact that, in the absence of friction or 
other interference, motion maintains itself ; for if 
/= 0, t; = 0, — ^that is, in the absence of force there 
is no change of velocity either in magnitude or in 
direction. This is essentially Newton's first law of 
motion. The force which one body exerts upon 
another is found to be equal and opposite to that 
with which the second body reacts upon the first. 
It is necessary, therefore, to measure only one of 
these forces. 

§ 107. Average Velocity. — If we take any series of 
consecutive numbers beginning at 0, we shall find the 
average value to be half the last value. Thus the 
average of 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10 is 5. So if 
we begin with a body at rest, and increase its velocity 
uniformly up to a given point, the average velocity 
will be half the final velocity. 

The average velocity is also found if we divide 
the distance traversed by the time ; or the distance a 
body moves is the product of the average velocity and 
the time. 

§ 108. Laws of Falling Bodies. — The force in dynes 
which gravity exerts upon a body is the product of 
the mass m in grams and the intensity of gravity ^, 
in dynes per gram. Substituting mff for / in the 
general formula of § 106, we have 
mgt = mvy or 
gt = v. 

The velocity acquired by a falling body is therefore 
proportional to the intensity of gravity and to the time 
it acts. 



§ 109]. BALLISTIC PENDULUM. 707 

The final velocity is, by the last section, equal to ^v ; 
and the distance d traversed, being the product of the 
average velocity and the time, is 
e? = i vt. 

Substituting the value of v above we have 
d=igtxt = igt\ 

In other words the distance a body falls is propor- 
tional to the intensity of gravity and to the square of 
the time. 

Again, we find the value of ^, 

'=1' 

and substituting this in the last formula, we have 

S^^ 9 
The square of the velocity which a body acquires is 
therefore proportional to the distance fallen. 

The same formulae express the relation between the 
velocity lost by a body projected vertically upwards, 
the time it takes it to reach its highest point, and the 
distance it rises in so doing. 

§ 109. BalUstic Pendulum. — When a body A sus- 
pended by a vertical cord -4.(7 (Fig. 9) is given a hori- 
zontal velocity v along the arc AB^ it continues until 
it reaches a point 5 at a vertical height AD above A 
the same as if it had been projected vertically upwards. 
The reason of this will be seen later on, when we have 
considered problems in the conservation of energy. We 
have from the last section 

AD^\—. 
^9 



708 



FORCE AND WORK. 



[§109. 



Drawing the diameter A U, and the chords A B and 
BE, we have in the similar triangles ABE and ADB, 



\ /./" 

B^' ..... 

Fig. 9. 

J15:AJS::AB:AE,ov AI> = AB'-^JlE. Hence, 
substituting, 

ZTB" -^ A^= «* -^ 2 ^ ; 
and as AlS= 2 AOj 

v^ = ZTS' xg-^ AC, 

v = AB\/^. 

^ AG 

The velocity of a pendulum at its middle point is 
therefore proportional to the distance AB oi the point 
where it turns, measured in a straight line ; that is the 
velocity is proportional to the chord of the arc A B. 
This is the principle used in comparing velocities by 
the ballistic pendulum. 

We shall see that a suspended magnet differs from 
a pendulum chiefly in the nature of the force which 
causes it to return-to its normal position. When a 



{ 111.] ISOCHRONISM. 709 

needle, previously at rest, is given a sudden angular 
velocity, the arc through which it swings is called the 
throw of the needle. The velocity is therefore propor- 
tional to the chord of the throw. 

§ 110. Laws of Vibration. — The square of the veloc- 
ity of a pendulum at the middle point of its swing 
resulting from a given displacement is seen from the 
last section to vary as the intensity of gmvity, and 
inversely as the length of the pendulum. We may 
infer that the length of a pendulum is proportional 
to the square of the time occupied by a single swing ; 
and the force acting upon it is proportional to the 
square of its rapidity of oscillation. 

The same principle applies to a magnetic needle, 
and is frequently used in comparing the strength oi 
the forces which are exerted upon it. See Experi- 
ments 75 and 82. 

§ 111. laochroniBxn. — It is well known that a pendu- 
lum vibrating in a very small arc keeps almost exactly 
the same time as in a comparatively large one. This 
shows that the average velocity of the pendulum (§ 107) 
must be proportional to the arc. The explanation is 
simply this, that the force urging the pendulum towards 
its middle point becomes greater as the arc increases. 
This force is proportional to AF (Fig. 10), perpendic- 
ular to B 0^ drawn as in § 109 and hence approxi- 
mately equal to the distance A B which the pendulum 
must travel. We have already seen that the velocity 
acquired in reaching the middle point is proportional 
to the chord A B and hence approximately to the 
arc. 



710 FORCE AND WORK. I§ 113. 

From the fact, however, that the lines A F and A B 
are not quite equal to the arc -4.J5, we infer that a 
common pendulum is not perfectly isochronous. The 
effect of different arcs on the rate of vibration will be 






JZ 



Fig. 10. 

found in Table 3, column g. In all experiments with 
a pendulum or with a vibrating needle, we must limit 
the arc of oscillation according to the degree of accu- 
racy required. 

§ 112. Point of Application of a System of Forces. — 
It may be observed that the weight of a body acts as 
if a single force were applied to a certain point called 
the centre of gravity, and that it must be sustained by 
a single force, or its equivalent, applied in the same 
vertical line with the centre of gravity, equal and op- 
posite to the weight of the body in question, in order 
that the body may remain at rest. In the case of a 
magnet the forces which it exerts act for most pur- 
poses as if they came from two points, represented in 
Fig. 13, § 126. We say therefore that the point of 
application of the forces exerted by gravity is at the 
centre of gravity, while the centres of magnetic forces 
are at two points called poles. 

§ 113. Couples. — A pair of forces equal in magni- 



§ 114.] HOOKE'S LAW. 711 

tude bat opposite in direction are said to constitute a 
couple. The perpendicular distance between the lines 
in which they act is called the arm of the couple ; the 
product of the magnitude of either force and the arm 
of the couple is called the magnitude of the couple. 

'D< <mc 



A^j> ^ 



Fig. 11. 

Thus AB and CD (Fig. 11) constitute a couple with 
an arm AQ and magnitude AB'AC. The effect of a 
couple in a given plane (-4.5 (7i>) does not depend upon 
the location or direction of the arm A with respect 
to the (rigid) body acted upon, and it is indifferent at 
what points in the lines AB and CD the corresponding 
forces are applied. A left-handed couple (AB-AC) 
can be balanced only by an equal and opposite right- 
handed couple (^AB'-AC) such that 

ABiA'B'iiA'C'iAC. 

§ 114. Hooke's Law. — The effect of a force applied 
at the end of a rod is either to stretch or to bend it ; 
the effect of a couple is to twist a rod. These effects 
are found to be proportional to the magnitude of the 
forces or couples in question. Hooke's law " ut tensio 
sic vis " may be translated, strains are proportional to 
stresses. (See § 22.) The ratio of a stress to a strain 
constitutes what is called a modulus of elasticity. 



712 FORCE AND WORK. [§ 116. 

§ 115. Laws of Flexure. — The force required to 
bend a beam is evidently proportional to its breadth, 
but the thickness must be taken three times into 
account, first, because a greater strain or distortion 
necessarily accompanies a given amount of bending ; 
second, because (as in the case of breadth) there is 
more material to be bent, and third, because the force 
has less purchase upon the material. 

The force required is in fact proportional to the 
cube of the thickness. It can be shown in a similar 
way to be inversely as the cube of the length, for 
less force will be required, first, because it has a 
greater purchase ; second, because the longer the beam 
is, the less sharply need it be bent to deflect it through 
a given angle ; and third, because it takes a smaller 
angle to produce a given deflection. 

§ 116. Laws of Torsion. — The couple required to 
twist a rod of a given shape increases with its breadth 
or thickness, first, because the average strain or dis- 
tortion is greater — at the edges, for instance ; sec- 
ond, because the purchase of the forces is less ; third, 
because the material acted upon is proportional to the 
breadth ; and fourth, because the material is also pro- 
portional to the thickness. In the case of a square or 
round rod the couple is therefore ^ proportional to the 
fourth power of the diameter. It is also inversely as 
the length, because the strain is less in proportion to 
the length of the rod for a given. amount of twisting. 

^ It may be remarked that if there are N independent reasons why 
ooe quantity should increase in proportion to another quantity, the 
former always yaries, other things being equal, as the JV^ power of the 
latter. 



§ 118] WORK OF WATER UNDER PRESSURE. 713 

§ 117. Measurement of Work. — Work is measured 
by multiplying together the distance through which 
a point has moved and the force which has been 
overcome. Thus the work transmitted through a belt 
can be found if we know the difference of tension 
between the two portions moving respectively to and 
from the driving-wheel, and the total distance trav- 
ersed. If the belt is prevented from moving, as in 
Experiment 69, we can find the work done by the 
wheel in rubbing against the belt. We multiply to- 
gether in this case the difference of tension in the belt 
and the distance traversed by the rim of the wheel. 
The work in question is transformed by friction into 
heat, but it could easily be utilized by allowing the 
belt to turn machinery. The measurement of work 
transmitted through a belt while in motion is more or 
less complicated. 

§ 118. "Work of "Water under Pressure. — The work 
represented by a flow of water under pressure is easily 
calculated. Suppose the orifice to be 1 sq. cm. in 
section ; then the force behind the stream is numeri- 
cally equal to the pressure (see § 63). Let the stream 
advance 1 cm, ; then the work done, being the force 
times the distance, or in this case the pressure times 
the distance, is also numerically equal to the pressure. 
The volume of water which escapes from the orifice is 
clearly 1 cu. cm. Hence the work done on 1 eu. cm. is 
numerically equal to the pressure. The same is also 
true, no matter what the size of the orifice may be ; for 
with a given pressure per %q. cm, the force must vary 
with the cross section of the stream, and hence also 

9 



714 FORCE AND WORK. [§120. 

the work represented by an advance of 1 cm. ; but the 
volume in cu. cm. delivered also increases in the same 
proportion, and therefore the work per cu, cm. remains 
the same. 

Since pressure in dynes per square centimetre is 
numerically the same as work in ergs per cubic cen- 
timetre, we have the following rule : To find the 
work in ergs represented by a flow of water under 
pressure, multiply together the flow in cubic centime- 
tres and the pressure in dynes per square centimetre. 

§ 119. "Work done by ObUque Forces. — When the 
direction of the force and the motion is not the same, 
we consider only the effect or component of the force 
in the direction of the motion (see § 105) ; or we 
may, on the other hand, take the component of the 
motion in the direction of the force, and multiply by 
the whole force in question; because in taking the 
component of either the force or the motion we re- 
duce it in a given proportion determined by the angle 
between the two directions in question (see § 105). 
Evidently it makes no difference which of the two 
terms in a product is thus reduced. 

§ 120. Conservation of TVork. — It follows from the 
principle set down in the last section that moving 
from A to B (Fig. 12), then from B to (7, against a 
force acting in any fixed direction, F F' ^ requires the 
same amount of work as in moving directly from A 
to (7. For if we drop perpendiculars A A\ BBf^ C C^ 
upon the line FF' representing the direction of the 
force, the components of the motions are A^ J9', B^ 6^, 
and A' (7 respectively, and since these are in the same 



§121.] ENERGY OF A MOVING BODY. 715 

straight line, A! B -\' B Cf = A^ Cf. That is, the sum 
of the component motions is the same by a direct or 
by an indirect path, and hence also the work required, 




Fig. 12. 

or the product of these components by the whole 
force in question. The fact that no work is gained 
or lost by choosing different paths is an illustration 
of the more general principle of the conservation of 
energy. 

§ 121. Energy of a Moving Body. — A question 
which often arises is, how much work is stored up 
in a moving body, as for instance in a gram of matter 
with a velocity one cm. per sec. Suppose a dyne to 
act on a gram at rest, we know that it would give it, 
by definition (§ 12), in one second a velocity of one 
cm, per sec. We know (by § 107) that the average 
velocity for this second will be, iialf a centimetre per 
second, or that the gram will have moved J cm. The 
work done upon it is therefore J dyne-cm. = \ erg. 

To give a gram twice the velocity in the same time 
would require twice the force and double the average 
velocity ; the distance would also be doubled. This 
would mean four times the work. In the same way 
three times the velocity would mean nine times the 



716 FORCE AND WORK. [§ 122. 

work, or in general the work done upon a moving 
body is proportional to the square of its velocity. It 
is obviously also proportional to the mass ; and as 
1 gram with a velocity of 1 cm, per sec. has been 
found to contain J erg, we' have the following rule : 
Multiply the mass in grams by the square of the 
velocity in centimetres per sec, and divide by 2 to find 
the work in ergs which a moving body contains. 

It is easily found by calculation that a moving body 
in coming to rest can do the same amount of work as 
was required to set it in motion. A gram, for in- 
stance, with a velocity of 1 cm. per sec. will be 
brought to rest by a force of 1 dyne in 1 second. 
The average velocity is therefore ^ cm, per sec. ; the 
distance traversed J cm ; the work done against 1 
dyne through a distance of J cm. is J erg, — the same 
that was required to start it in motion. 

§ 122. Conservation of Energy in Mechanics. — Work 
stored in a body is often called energy. Energy is 
again defined as the power of doing work. We dis- 
tinguish between the energy of motion of a body 
(kinetic energy) and its energy of position (potential 
energy), due to the level, for instance, to which it has 
been raised. All kinds of energy are measured in 
ergs. 

We have seen that it takes the same amount of 
work to raise a body from one level to another, no 
matter by what path it may be raised (§ 120). When 
it returns to the original level the work is given back. 
The energy spent in setting a body in motion is 
also restored when the body comes to rest (§ 121). 



§ 122.] CONSERVATION OF ENERGY IN MECHANICS. 717 

Energy of position may be changed into energy of 
motion and the reverse, as is particularly evident in 
the case of falling bodies or bodies projected into the 
air ; but in mechanics no energy is ever lost. This 
statement is an illustration of a more general principle 
known as the " Conservation of Energy " (§ 149). 



CHAPTER IX. 

ELECTRICITY AND MAGNETISM. 

§ 123. Nature of Electricity and Magnetism. — We do 
not know what electricity and magnetism are; that 
is, we are ignorant of their fundamental relations to 
matter and motion. Electricity circulating around 
the particles of steel is believed by many to be the 
sole cause of its magnetism. This hypothesis accounts 
for all the observed effects. It has been suggested 
by leading scientific men that the rapidity with which 
light is transmitted may be due to electrical action 
(see § 93), and it is suspected that chemical affinity 
is closely related to electricity. (See §§ 142-144.) 
We speak of electricity as if it were a fluid ; but there 
are three reasons why neither electricity nor magne- 
tism can be regarded as a fluid in the ordinary sense : 
first, they have no inertia (or resistance to being set 
in motion) ; second, they have no weight (or attrac- 
tion for ordinary matter under the law of universal 
gravitation) ; and third, they repel, instead of attract- 
ing their own kind. 

In the first two respects electricity and magnetism 
resemble heat more than a fluid. It has been sug- 
gested that they may be forms of energy ; but there 
are more objections to this view than to the other. 



§ 125] ELECTRICAL ATTRACTIONS, ETC. 719 

and comparatively little help is to be derived from it. 
Even if electricity were proved to be a kind of mo- 
tion, we should still think of it as a fluid, as we do of 
heat when it is said to flow from one point to another 
(§ 74). 

§ 124. Positive and Negative Electricity. — As com- 
pressed air can be distinguished from rarefied air, so 
positive may be distinguished from negative electricity. 
When mixed together they neutralize one another; 
and in this neutral condition, electricity, like the at- 
mosphere, seems to be everywhere present. Positive 
electricity can be separated from negative by vari- 
ous means ; but we produce in all cases equal quanti- 
ties of both. For instance, glass rubbed with a piece 
of silk receives a positive charge ; an equal charge of 
negative electricity is found in the silk. Some writers 
maintain that there are really two distinct kinds of 
electricity which unite, somewhat as an acid does with 
a base to form a neutral compound ; and mathemati- 
cians are apt to take this view, finding it convenient 
to treat electricity as incompressible. Positive elec- 
tricity may, however, be thought of as under greater 
pressure than negative, whether it yields to that pres- 
sure or not. We imagine that it is this pressure 
which causes electricity to flow from one place to 
another. We consider only the flow of positive elec- 
tricity ; though it is maintained by some that half the 
effect is due to the flow of an equal quantity of neg- 
ative electricity in the opposite direction. 

§ 125. Electrical Attractions and Repulsions. — Two 
bodies charged with positive electricity repel each 



720 ELECTRICITY AND MAGNETISM. 1§ 126. 

other, or two charged with negative electricity repel 
each other ; but a body charged with positive electri- 
city attracts one with a negative charge. The force 
exerted is proportional to the charge, or quantity of 
electricity in each body. It is, in fact, equal to the 
product of the two charges, divided by the square of 
the distance between them. There is also a mutual 
repulsion between different portions of the same 
charge, which tend therefore to fly as far apart as 
possible. Hence electricity collects in the surfaces of 
bodies which conduct it, and (except while flowing 
through them) is never found at any appreciable 
depth. 

§ 126. Nature of a Magnet. — In a similar way posi- 
tive and negative charges of magnetism may be sepa- 













Fig. 13. 



rated, but only in a few substances like steel. With 
magnetism, as with electricity, a positive charge im- 
plies an equal negative charge; but in the case of 
magnetism both charges are always found in the same 



§ 128.1 FIELD OF FORCE. 721 

body. Such a body constitutes a magnet, and is said 
to have two poles, corresponding to the centres of 
positive and negative magnetism. The position of 
the poles N and S (Fig. 13) is shown by sprinkling 
iron-filings on a piece of glass over the magnet. The 
iron-filings arrange themselves in lines as in the dia- 
gram, radiating from the two poles N and S. One 
of these poles, N^ is called north because, when the 
magnet is freely suspended, it tends to point approxi- 
mately in that direction ; ^ the other is called the south 
pole. The direction in which a magnet is said to 
point is alwaj'S determined by its north pole. 

§ 127., liineB of Force. — The iron-filings arrange 
themselves along what are called "lines of force." 
A small compass-needle placed close to the glass 
always points parallel to the lines of iron-filings, 
and gives the direction of the lines of force, as indi- 
cated by arrows in the diagram. The lines accord- 
ingly are said to come fi'om the north pole, and go to 
the south pole. It is found that where the lines are 
closest, the magnetism is strongest. A strong horse- 
shoe magnet can hold a solid mass of iron-filings 
between its poles. 

§ 128. Field of Force. — The space around or be- 
tween the poles of a magnet, wherever its action is 
felt, is called the field of force, or simply the field of 
that magnet. By the intensity of this field we mean 
the force exerted by the magnet on a unit quantity 
of magnetism (§ 17) placed at any point of the field. 

^ At Cambridge, Massachusetts, a magnet points very nearly north 
by west. 



722 ELECTRICITY AND MAGNETISM. § 129. 

The intensity varies in different parts of the field. 
At a given point the intensity of the field due to a 
single magnetic pole is equal to the strength of the 
pole divided by the square of its distance from the 
point in question. Both poles of a magnet must, 
however, be taken into account in calculating the 
intensity of a field. The resultant (§ 105) of the 
forces upon a unit of positive or north* magnetism 
determines, by its direction and magnitude, both the 
direction of the lines of force, and the intensity of the 
field. 

The earth, for example, is a great, though weak 
magnet. The intensity of its field at Cambridge, 
Massachusetts, is about 1 dyne per unit of magnet- 
ism ; or more exactly, | dyne. The lines of force are, 
however, more nearly vertical than horizontal, and 
only their horizontal component, or about one quarter 
of the whole effect, is felt by a compass. The angle 
between the lines of force and a horizontal plane 
(70°-80°) is called the magnetic dip. 

The field of a dynamo machine may be several 
thousand times stronger than that of the earth. 

§ 129. Magnetic Attractions and Repulsions. — Two 
north poles, or two south poles, repel each other ; a 
north and a south attract; the force exerted is pro- 
portional to what we call the strength of each pole — in 
the case of two poles, it is equal to the product of their 

^ By '' north magnetism " we mean the kind of magnetism contained 
in that end of a magnet which points north. This is evidently the 
opposite kind to that which we find in the north polar regions of the 
earth, since only dissimilars attract. The "magnetic north pole" of 
the earth is therefore technically a negative or south pole. 



§ 129.] MAGNETIC ATTRACTIONS, ETC. 723 

strengths divided by the square of the distance be- 
tween them. Comparing this statement with that in 
§ 128, we see that the force acting on a magnetic pole 
is equal to the product of its strength, and that of the 
field of force in which it is placed. The strengths 
of the north and south poles of a given magnet are 
always alike. 



'N^ 



Fig. 14. 

When two magnets with poles, -Z\r, S^ N\ S', of 
nearly equal strengths, ±«, and ±«', are placed par- 
allel and opposite to one another, as in Fig. 14, if the 
distance between them is d, there is a perpendicular 
repulsion between N and N' equal to ««' -^ d^. and 
one between S and aS', of the same amount. There is 
furthermore an oblique attraction between iVand S\ 
also between N^ and S; but if the distances NS' and 
N' S are great in comparison with NN\ or rf, the 
oblique forces may be disregarded.^ The resultant is 
therefore approximately equal to 28 s' -~ d^. 

By supposing one of the magnets reversed, we find 
in the same way a resultant attraction nearly equal to 
2 88' -r- d\ Counting attractive forces as negative, the 

1 The effective components of the oblique forces bear to the perpen- 
dicular forces a ratio equal to (N N^ -r- NS')^. If N S' is 6 times as 
great sls N N\ the error committed by disregarding the oblique forces 
will be less than 1 per cent. The chief source of error in the applica- 
tion of the principles contained in this section lies in the fact that mag- 
netic forces are only approximately centred in poles. 



724 ELECTRICITY AND MAGNETISM. l§ 130. 

algebraic difference,^ A, between the repulsion and the 
attraction will be 

A = 4 — , nearly. 

We measure A by an ordinary balance in experi- 
ment 72, with a small distance, d, between two nearly 
equal magnets, and thus determine roughly the mean 
strength of the poles in question. 

§ 130. Action of Currents on Magnets. — When an 
electric current flows through a wire, it affects all 
magnetic bodies in its vicinity. It creates, in fact, 
a magnetic field. When only a short portion of the 
wire is considered, the intensity of the field due to 
this portion is proportional to its length and to the 
strength of the current passing through it ; the inten- 
sity also varies inversely as the square of the distance 
from the wire. The lines of force are perpendicular 
to the wire at every point. They are in fact circles 
with the wire at their centre, as 
^'■'^i^ shown by the arrangement of iron- 
^(ilvTjs^S^^^^;"^ filings about a vertical current, in- 
,^^i;: i;^ > Fig- 15. Hence, a magnet tends 
^■^£^^^^^^^' •/ ^^ point at right angles to an elec- 
*(»'"^'' i'' ^^^^ current, and to the line join- 

ing the two. To remember which 
way the magnet points, place the 
thumb across the forefinger of the right hand ; if the 

1 Charges of magnetism which each magnet " induces" upon the 
other increase the mutual attraction of the magnets, but decrease their 
mutual repulsion by a nearly equal amount. The algebraic difference 
remams essentially the same. 



§ 132] MAGNETIC CURRENT MEASURE. 725 

finger represents the direction of the current, the thumb 
shows how the north pole of a magnet points. 

§ 131. Action of Magneto on Corrento. — Conversely, 
an electric cuiTcnt is acted upon by magnetic bodies 
in its neighborhood. It is, in short, affected by a mag- 
netic field. The effect is equal, under the most favoi^ 
able circumstances, to the product of the length of 
wire, the strength of the current, and the intensity 
of the field. In general, however, we consider only 
that portion or component of a current which is per- 
pendicular to the lines of force. The direction in 
which a field acts upon a current is at right angles 
to the lines of force and to the current. To remem- 
ber which way the field acts on the current, let the 
thumb represent a north pole as before, and the fore- 
finger a current; then the thumb will point in the 
direction in which the pole is urged ; hence as action 
and reaction are equal and opposite, the current must 
be urged towards the base of the thumb. 

The lines of force due to the current are, as we 
have seen, parallel to the thumb ; but those due to 
the pole are perpendicular both to the thumb and to 
the forefinger. They issue in fact from the north 
pole (see § 127) and follow, accordingly, the line of 
pressure between the thumb and forefinger. It is 
these lines alone which affect the current. Neither 
the pole nor the current is influenced by the field of 
force which it itself creates. 

§ 132. Magnetic Current Meaanre. — From our defini- 
tion of the unit of current (see § 18) and the laws 
stated in the last section, it is clear that the field of 



726 ELECTRICITY AND MAGNETISM. [§ 134. 

force due to a current C flowing through a length of 
wire X at a distance D is equal to C L -f- 2>^, and 
that the action of a field of force F on the same 
current, if they are at right angles, is CLF. These 
expressions enable us to measure a current through 
its magnetic action, as will be explained further in 
§§ 133-135. 

§ 133. Constant of a Coil. — The constant of a coil of 
wire is defined as the field at its centre due to a unit 
of current passing through the wire. If the radius of 
a circular coil is r, the number of turns of wire w, the 
length of wire is 2 tt r X w, every portion of which 
acts in the same direction on a magnet at the centre 
(see § 130) ; hence the constant is 

^__X _2 IT rn _2 irn 

§ 134. Magnetic Area. — A rectangular coil, ahcd^ 

of wire in the plane of this paper, would be acted upon 

differently in different parts by a field of 

force in the same plane. Suppose that the 

current C circulates with the hands of a 

watch; and that a field acts from left to 

right. Then (by § 131) the sides a h and 

c d (Fig. 16) will not be affected ; a d will 

be depressed with a force (7 X ad X F^ and 

Em. 16. b cwiW be raised with the same force; the 

two forces then constitute a couple, with 

an arm a h and magnitude C F X ah X ad. The 

couple acting on a rectangle, a 6 c i, is therefore equal 

to the product of the current and field of force multi- 



135] 



ELECTRO-DYNAMOMETER. 



727 



plied by the area of that rectangle. The same clearly 
holds for any number of rectangles or for their sum. 
A rectangular coil of wire consists essentially of a 
series of rectangles, abed, each carrying the current, 
C» The total area, A, enclosed by these rectangles is 
called the magnetic area of the coil, and determines 
the couple, FA, acting upon the coil in a magnetic 
field, F, in its own plane. 

§ 135. Electro-Dynamometer. — A common form of 
electro-dynamometer consists (see illustration) in a 




Electro-Dynamometer. 

coU of wire a, with a smaller coil i, at right angles to 
it near its centre. The larger coil is usually circular ; 



728 ELECTRICITY AND MAGNETISM. [§ 135. 

the smaller may be rectangular. If K is the constant 
of the large coil, a current C, circulating through this 
coil, will cause a field of force {F=C IT) to act on 
the small coil ; if the magnetic area of this is Ay and 
the same current, (7, passes through the small coil, the 
couple acting on the latter will be C FA= C^KA. 

When the constant K and magnetic area A are 
known it is only necessary to measure the couple in 
order to determine the current. A current is thus 
primarily measured by the force with which it acts on 
itself. We shall not need to consider currents through 
long conductors, except where, as in § 133 or in § 134, 
every portion is similarly situated with respect to the 
forces in question. 



CHAPTER X. 

ELECTROMOTIVE FORCE AND RESISTANCE. 

§ 136. Heating by Electricity. — When a current of 
electricity passes through a wire, heat is developed in 
proportion to the square of the current and also to 
what we call the electrical resistance of the conductor. 
This is known as Joules's Law. When the power, or 
the rate at which heat is generated, reduced to watts 
(see § 15) is P, when the current in amperes (§ 19) is 
C, and when the resistance in ohms (§ 20) is B^ we have 

The resistance iZ of a conductor is thus easily found 
if we know the amount of heat developed in it by a 
given current in a given time. (See ^ 172.) 

§ 137. Electrical Power. — The work spent in one 
second in maintaining a current is obviously the same 
thing as the power, F ; and the quantity of electricity 
flowing in one second is by definition equal to the 
current O ; the ratio of the power to the current is 
therefore the same thing as the work spent per unit 
of electrical quantity, and is defined as electromotive 
force, U. Electromotive force corresponds therefore 
to hydrostatic pressure (see § 118), or rather, to a 
difference of hydrostatic pressure. 

We have, therefore, 

E=P^ C or P=CE; 

10 



730 ELECTROMOTIVE FORCE AND RESISTANCE. [§ 139. 

that is, electrical power (in. watts) is equal to the pro- 
duct of the current (in amperes) by the electromotive 
force (in volts). 

§ 138. Ohm's Law. — Since in the last section we 
found ^= P -7- C, and in the section before, P= C^R; 
we have, substituting, 11= C^B-^ C=. OJt. In other 
words, the electromotive force (in volts) is equal to the 
product of the current (in amperes) and the resistance 
(in ohms). It follows that the current (in amperes) 
is equal to the electromotive force (in volts) divided 
by the resistance (in ohms), or 

B 

This is known as Ohm's Law. 

A similar law discovered by Poiseuille holds for the 
flow of liquids through capillary tubes. If R is the 
resistance of such a tube as defined in § 20, E the hy- 
drostatic pressure in dyne9 per %q, cm.^ and C the cur- 
rent in cu. cm. per «e<?., we have 

R 

§ 139. Electrical Potential. — Electrical potential is 
analogous to pressure, or head of water. As water 
flows through a horizontal tube from places of high 
pressure to places of low pressure, so electricity flows 
from points of high potential to points of low poten- 
tial. The electromotive force of a battery is the same 
thing as the difference in potential which it is capable 
of producing. Hence we may apply Ohm's Law as 
follows: the current (in amperes) through any con- 



§ 140.] RESISTANCE IN SERIES AND MULTIPLE ARC. 731 

ductor (containing no source of electricity) is equal 
to the difference in potential of its two extremes (in 
volts) divided by the resistance (in ohms) between 
them, no matter how the difference of potential is kept 
up; and the difference of potential at the two ex- 
tremes of such a conductor (in volts) is the product of 
the current (in amperes) and the resistance (in ohms). 
Denoting by c the current, by r the resistance, and 
by e, the difference of potential in any portion of the 
conductor, we have 

e = (?r. 

Clearly, when a given current of electricity, tf, travels 
along a wire it loses in potential by an amount, e, pro- 
portional at any point to the resistance, r, which has 
been overcome. 

§ 140. Resistance in Series and in Multiple Arc. — 
When a current passes first through one conductor 
then through another, as we say in series^ the total 
resistance is clearly the sum of the separate resist- 
ances ; but if the current has a choice of two paths, 
like a congregation dispersing through two doors, it is 
less retarded than if confined to one alone. 

l^Qt ABO and ^2>(7 (Fig. 17) be two such chan- 
nels as we say, in multiple arc ; 



Fig. 17. 

if the resistance of vl JB (7 is J?i, and that of J. 2) (7, Bj, 
and the difference of potential between A and C is -ff. 



732 ELECTROMOTIVE JFORCE AND RESISTANCE. [§ 141. 

then the current Cy^ through AB C\& Ci = II-^ R^; 
that through AD C is C^z=zE-^B^; the total current 
is (7=(7,+ (7j = -ff-5-^i + J?-^^a. But if the com- 
bined resistance is J?, we have C = E-^ B. Equat- 
ing the two values of (7, and cancelling E^ we have 

111 

— = — + — ; or 

B B^ B^ 

the reciprocal of the combined resistance of two (or 
more) conductors in multiple arc is equal to the sum 
of the reciprocals of the separate resistances. 

We notice also that the current through each chan- 
nel is inversely as its resistance, or Ci : C^ : : B^ : B^^ 
from which (7^ : Oi: B^: B^ +^2» ^^c- 

§ 141. Wheatotone'B Bridge. — We have seen (§ 139) 
that loss of potential is proportional by a given path 
to the resistance overcome. Since in Fig. 17, § 140, 
in passing by either path from A to (7, the total loss 
of potential must be the same, the loss in reaching B 
will be the same as in reaching D if the resistances of 
AB and AD bear the same proportion to the total re- 
sistances of ABO and A D (7 respectively. In this case 
no current will flow through a wire joining B and D 
(Fig. 18), since these points will have the same po- 




FiG. 18. 



tential. A cross connection 52), between two par- 
allel circuits A (7, is called a Wheatstone's Bridge ; and 
the absence of any current through it shows that the 



§ 143] ELECTRO-CHEMICAL EQUIVALENTS. 733 

four resistances AB^ BC^AD^ and B C are in propor- 
tion; that is, 

AB:BCi:AB:DC. 

§ 142. ElectrolyBia. — When a current of electricity 
passes into and out of a fluid by means of two conduc- 
tors, often called electrodes, the liquid is almost al- 
ways decomposed, and its constituents liberated. The 
metallic elements are generally carried with the cur- 
rent, the acid constituents against it until they reach 
the electrodes. There they are either deposited, as in 
electroplating, or set free in the gaseous form, as in 
the electrolysis of water, or made to combine with the 
material of one of the electrodes, as the acid does with 
the zinc of an ordinary battery. 

§ 143. Electro-chemical Equivalents. — As concerns 
the quantity of a given substance acted upon in elec- 
trolysis, neither the surface of the electrode nor the 
chemical nature of the reaction seems to have any 
effect. A given quantity of electricity always affects 
a given quantity of a given substance. Thus one am- 
pere in one second causes about one 3000th of a gram 
of zinc to be dissolved from a zinc plate forming one 
of the electrodes, or deposits about three times as 
much mercury. The quantity of mercury is found 
to be the same, whether the nitrate or chloride is 
used ; and a similar uniformity is found, in the case 
of other elementary substances, in regard to the quan- 
tity set free from their various salts. The weight of 
a substance acted upon by the unit quantity of elec- 
tricity is called its electro-chemical equivalent. (See 
Tables 8 6, 11 and 12.) 



734 ELECTROMOTIVE FOliCE AND RESISTANCE. [§ 145. 

§ 144. La^v of Zilectro- chemical Equivalents. — Ob- 
servation shows that the electro-chemical equivalents 
of different substances are to each other as their chem- 
ical combining proportions. Thus 2 parts of hydrogen 
combine with 16 parts of oxygen to form water, or 
with 71 parts of chlorine to form muriatic acid ; again, 
71 parts of chlorine or 16 parts of oxygen unite with 
63 parts of copper or 65 parts of zinc ; one ampere 
in about 192 seconds sets free 2 mgr. of hydrogen, 16 
wgr. of oxygen, 71 mgr. of chlorine, dissolves 66 mgr. 
of zinc, and precipitates 63 mgr, of copper. There is 
evidently an intimate connection between electricity 
and the bonds which bind atoms chemically together ; 
though no one as yet has offered a satisfactory expla- 
nation of the law of electro-chemical equivalents. 

§ 145. Calculation of Electromotive Force. — Since 
we know the quantity of zinc dissolved by one ampere 
in a second {-^-q g), the amount of heat which a 
gram of zinc gives out in combining with nitric acid 
(about 1500 units), and the value of one unit of heat 
per second in watts (4.2 nearly), we can evidently find 
the power spent on one ampere by multiplying these 
three together, and this should be (§ 137) the electro- 
motive force developed by the action. Hence a bat- 
tery in which the only reaction is the dissolving of 
zinc in nitric acid should have an electromotive force 
of about ^Vir X ISOO X 4.2, or 2.1 volts. 

The electromotive force H generated by any chem- 
ical action is accordingly 4.2 times the product of the 
electro-chemical equivalent and heat of combination 
in question. In the Daniell cell we must offset 



§146] ARRANGEMENT OF BATTERIES. 735 

against the electromotive force due to the solution 
of zinc, that due to the precipitation of copper, which 
is about one-half of the former, because the copper 
which is separated from the acid with which it is 
combined has very nearly half as much aflBnity for 
it as the zinc which takes its place. The electro- 
motive force of a Daniell cell is therefore about 1 
volt. 

Experiment shows that electromotive forces can be 
calculated with more or less exactness in this way, as 
nearly all of the chemical energy is spent on the 
electric current. The actual electromotive force can 
never exceed its theoretical value. 

§ 146. Arrangement of Batteries. — When we join 
several batteries together in multiple arc (Fig. 19), the 




Fig. 19. 

zinc poles having all the same potential, and the cop- 
per (or carbon) poles all the same potential, we gain 
nothing in electromotive force, any more than we 
should gain in pressure by connecting two reservoirs 
on the same level. The current is, however, often 
increased, owing to the diminished resistance (see 

§ 140): 



/C^OOO. 



Fig. 20. 

When, however, we join batteries in series (Pig. 20), 
so that the current {^fi^es in all cases from zinc to 



736 ELECTROMOTIVE FORCE AND RESISTANCE. [§ U7. 

copper, a given amount of work is done on the same 
current by each cell, as explained in the last section, 
and hence the electromotive force is increased in pro- 
portion to the number of cells. Unfortunately, the 
resistance is also increased in the same proportion, 
(§140). 

In seeking to increase a current, it is as important 
to diminish resistance as to increase electromotive 
force (see Ohm's Law, § 138) ; and a practical rule 
often of service in the arrangement of a battery is 
to reduce the resistance of a battery by arrangement 
in multiple arc or to increase its electromotive force 
by arrangement in series until the internal resistance 
is equal as nearly as may be to the resistance of the 
outside circuit which is to be overcome.^ In this 
way the greatest possible current will be obtained 
from a given number of cells through a given outside 
resistance. Thus for a very long telegraph line we 
prefer an arrangement of batteries in series ; for a 
very short circuit an arrangement in multiple arc. 

§ 147. Induction of Electricity. — When a wire of 
length jL, carrying a current (7, at right angles to the 
lines of force of a magnetic field F^ is moved at right 
angles both to these lines and to itself with a velocity 
F", against the forces acting on it, evidently power is 
required of the magnitude CLFV Qvg^ per second; 
for the force overcome is C L F {^ 132) and the dis- 
tance traversed in one second is V. The power re- 
quired per unit of current to keep up the motion is 

1 A similar rule applies to the arranjrement of several electrical 
instruments, but from lack of space it cannot be dwelt upon here. 



§ 148 ] THERMO-ELECTRICITY. 787 

therefore CLF V-i- C, or LFV. Experiment shows 
that this power is not spent, as one might expect, in 
heating the wire, but, through some agency which we 
do not understand, it acts upon the current in the 
wire. It produces, in fact, an electromotive force, JS?, 
which we have seen (§ 137) is equal to the power per 
unit of current.^ That is E^= LFV. The current 
is given accordingly by Ohm's Law (§ 138), if the 
resistance of the circuit is known. We are thus 
able, given the phenomenon, to anticipate the law 
governing what is called the induction of electricity. 

We make use of induced currents, in Experiment 
76, to compare the intensity of two fields of force ; 
and in Experiment 77, to compare the intensity of the 
same field in two directions. In each case the motion 
of the wires is limited to a certain distance. If the 
distance is traveled rapidly we get a strong current 
for a short time ; if slowly, a small current for a long 
time ; the sudden throw of a galvanometer-needle 
(see § 109) is therefore dependent simply upon the 
strength of the magnetic field. 

§ 148. Thermo-Electaicity. — In regard to the elec- 
tric current generated by heating or cooling a junction 
of two dissimilar metals, we observe that the electro- 
motive force is approximately proportional to the tem- 
perature of the junction, within narrow limits. As 

1 The electromotive force in this formula is expressed in ergs per 
second per unit of current. Reducing the power to watts and the 
current to amperes, we find that the electromotive force in volts is 
equal to the jlroduct of the length of wire in centimetres, its velocity 
in centimetres per second, and the strength of the field in dynes per 
unit of magnetism divided by 100,000,000. 



738 ELECTROMOTIVE FORCE AND RESISTANCE. [§ U9. 

one janction in an electrical circuit implies another, 
it is the difference of temperature of these two junc- 
tions which we take into account. 

When the range of temperature is considerable, the 
thermo-electric force is rarely proportional to the dif- 
ference of temperature of the two junctions. Thus 
the current which flows ordinarily from copper to 
iron through a hot junction, increases up to 275°, 
then diminishes, and is reversed at a still higher 
temperature. 

§ 149. Conservation of Energy. — The principle of 
the conservation of energy explained at the end of 
chapter VIIL, applies to all transformations of energy, 
and forms the basis, as we have seen, of most impor- 
tant calculations. Whatever light, electricity, and 
magnetism may be, they return to us eventually in 
some form the energy spent in creating them. En- 
ergy, like matter, may be transformed or scattered, 
but cannot be destroyed. 



ADDENDA, 



AMBIGUOUS TERMS. 

§ 150. Gravity. — Ordinary matter has two char- 
acteristic properties : inertia (§ 151), and gravity. 
The continual changes which take place in the ve- 
locities of heavenly bodies, or in the directions of 
their motions, are attributed to gravity. To account 
for these changes, it is necessary to suppose an at- 
traction between different bodies which, other things 
being equal, varies inversely as the square of the 
distance between them. Tliis is known as Newton's 
Law of Universal Gravitation. It is not confined to 
heavenly bodies alone, but holds for any two bodies 
of matter, however small ; though the operation of 
the law may be concealed by other phenomena. 
That property in matter which makes it attract other 
matter is properly called its gravity. We say, for 
instance, that "gravity" draws all bodies toward the 
centre of the earth. In such exprest^ions as the 
"acceleration of gravity," the earth's gravity is 
usually referred to. A body cannot strictly be said 
to fall under the influence of its own gravity. Grav- 
itation is a mutual attraction, existing only between 
two different bodies of matter. We must distinguish 



740 ADDENDA. [§ 15a 

between forces of gravitation, which depend upon the 
distances between bodies, and their gravity proper, 
which is invariable so long as no change is made in 
the quantity of matter which they contain. An esti- 
mate of the quantity of matter, founded upon this 
invariable property is usually designated by the word 
ma««, notwithstanding the fact that '' mass " is strictly 
defined without any reference to gravity whatsoever 
(see § 152). It is also designated by the word 
" weight," though this has properly an entirely dif- 
ferent signification (see § 153). 

Either the word " mass " or the word " weight " 
may mean, accordingly, an estimate of the quantity 
of matter which a body contains ^founded upon gravi- 
tation. Thus the number of grams (§ 6) by which 
a body can be balanced determines its " weight in 
grams.'' The word " weight " should always be 
qualified in this way when it refers to a quantity of 
matter ; and when thus qualified it is preferable to 
the word "mass" as applied to measurements de- 
pending upon gravity. 

§ 151. Inertia. — Bodies do not move instantly 
from one place to another under the action of forces. 
More or less time is always required to set a body in 
motion, to turn it one side, or to bring it to rest. 
These facts are explained as the result of a universal 
property of matter called inertia. There is, how- 
ever, no agreement amongst scientific men as to the 
exact meaning of this term. Inertia is described by 
some writers (in accordance with the original mean- 
ing of the Latin word) as the "inability" of matter 



§ 152.] AMBIGUOUS TERMS. 741 

to move itself. According to Ganot,^ '* Inertia is a 
purely negative, though universal, property of mat- 
ter." Other writers associate with inertia a certain 
power or necessity. An old term, vis inertice (force 
of inertia), illustrates this view. Inertia has been 
defined as "that property of matter which makes 
the application of a force necessary for any change 
in the magnitude or direction of a body's motion." ^ 
*' The fundamental principle of physics,", says Des- 
chanel,^ '^ is the inertia of matter." 

We must distinguish between the so-called forces 
of inertia — that is, forces of greater or less magni- 
tude required under different conditions to produce 
changes in the motion of a body — and the inertia 
proper of a given body, which, like its gravity (§ 150), 
depends only upon the quantity of matter which it 
contains. An estimate of a quantity of matter, 
founded upon this invariable property, is designated 
by the word mass in its strict scientific signification 
(see § 152). 

§ 152. Mass. — The word mass is thought to have 
the same origin as the German maaS, and to denote, 
literally, a measure of the quantity of matter which 
a body contains. The mass of a body is strictly de- 
fined as the number of standard units of quantity 
(§6) to which a body is equivalent in respect to 
inertia (§ 151). This is what is always meant by 
the '' dynamical mass " of a body. There are various 
dynamical devices by which masses may be compared 

1 Ganot's Physics, § 19. * Hall's Elementary Ideas, page 6. 

» Deschanel's Natural Philosophy, 1878, § 6. 



742 ADDENDA. [§ 154. 

(Exps. 69-60) ; but none leading to very accurate 
results. It is, however, inferred from results obtained 
with pendula constructed of different materials (Exp. 
58), that there is no perceptible difference between 
the mass and the weight of a body when both are 
estimated in grams. The best comparisons of mass 
are made, accordingly, by means of an ordinary bal- 
ance. In practice the word " mass " means the num- 
ber of grams to which a body is equivalent in respect 
to weight. It is in other words (practically) the 
same thing as "weight in grams" (§ 160). 

§ 153. Weight. — Weight is, as we have seen 
(§ 160), sometimes used to denote the quantity of 
matter which a body contains. The proper use of 
the term is, however, in the sense of a force. The 
weight of a body is strictly defined as the force with 
which it is attracted by the earth's gravity. In this 
sense weights should be accordingly expressed in 
dynes (§ 12). To avoid confusion between the dif- 
ferent meanings of the word " weight," it is well to 
qualify it even when used in its strictest sense. 
To speak, for instance, of the " weight in dynes " of 
a body leaves no doubt that it is the idea of force 
which we wish to convey. 

It may be observed that the " weight in dynes " of 
a body varies with the intensity of the force of grav- 
ity exerted upon it ; but that the " weight in grams," 
being practically the same thing as the mass of the 
body, remains always the same. 

§ 154. Density. — The density of a body is strictly 
defined as the ratio of its mass to its volume (§ 9). 



§ 155 ] AMBIGUOUS TERMS. 748 

Since, however, we usually estimate masses by bal- 
ancing them with gram weights, and since volumes 
are measured in cubic centimetres (§ 9), density 
means in practice the quotient obtained when the 
weight in grams of a body is divided by its volume 
in cubic centimetres. The weight is supposed in all 
cases to be corrected for the buoyancy of air, or in 
other words, reduced to vacuo (§ 67) ; the volume 
is supposed to be measured at 0° or reduced to 
0°, unless the temperature of the experiment is 
stated. 

If V IS the volume of a body in cu. cm.^ il!f its 
mass (or practically its weight) in grams, and D its 
density, we have accordingly — 





-f. 


(1) 


whence 


M=DV, 


(2) 


and 




(3) 



It follows that the density of a substance is nu- 
merically equal to the number of grams contained 
in 1 cu. cm. Thus 1 eu. em. of lead weighs (see 
Table 8) from 11.3 to 11.4 grams; and 1 cu. cm. of 
dry air usually weighs (see Table 19) from .0011 to 
.0013 grams. 

§ 155. Specific Volnme. — The specific volume of 
a body is defined as the ratio of its volume to its 
mass. It is found in practice by dividing its volume 
in cubic centimetres by its weight in grams. The 



7'44 ADDENDA. [§ 156. 

specific volume (aS') of a substance is accordingly 
the reciprocal of its density ; that is (see § 154), 

S=^, (1) 

whence S= ^, (2) 

V=MS, (3) 

and M=^. (4) 

We must distinguish apparent specific volumes 
from true specific volumes. The true specific vol- 
ume of a substance is the space occupied by a quan- 
tity of that substance weighing 1 gram in vacuo. 
The apparent specific volume is the space occupied 
by a quantity weighing apparently 1 gram in air. 
Apparent specific volumes are accordingly affected 
by the density of air. The apparent specific vol- 
umes of water under different conditions are con- 
tained in Table 22, and are useful in calculations of 
volumes in hydrostatics. If w is the apparent weight 
of water, and « its apparent specific volume, the true 
volume V is given by the equation (see 3), 

V = W8. (5) 

§ 156. Correction and Error. — Mistakes sometimes 
arise from confusion between the terms " correction " 
and " error." If o is the observed magnitude of a quan- 
tity, y, the error of observation is o — q. A correc- 
tion is defined as a quantity which added algebraically 



§ 156] AMBIGUOUS TERMS. 745 

to an observed magnitude (o) will give the true mag- 
nitude (g'). It is equal, accordingly, to q — o. If 
the observed value is greater than the true value, it 
follows that the error is positive, the correction nega- 
tive ; but if the observed value is less than the true 
value, the error is negative and the correction posi- 
tive. In every case the correction and the error are 
equal and opposite. 

If e is the ''^probable error'*'' of observation (see 
§ 50), we have by definition, 

o + g>y>o — e, probably^ 

or in the conventional system of representation (§ 53), 
q^=o ± e. 

The student must not be led by this expression to 
imagine that the " probable error " of a result is to 
be added to it or subtracted from it. He should 
bear in mind that the so-called '* probable error " is 
not literally a probable error (see § 50), but simply a 
limit within which the error is probably confined. 
Even if we knew the magnitude of the error, it 
would still be impossible to correct for it, since the 
sign is unknown. No matter how great the probable 
error of our observations may be, results strictly cal- 
culated from these observations are generally le88 
improbable than those obtained by making allowances 
for errors which we do not know to exist. 



11 



NOTES 



ARRANGEMENT OF MATHEMATICAL AND 
PHYSICAL TABLES* 



METHODS OP CONDENSATION. 

The object of constructing mathematical or physi- 
cal tables is to condense into a small space a large 
number of results obtained either by calculation 
or by observation. There are various well-known 
methods by which condensation may be effected. 
Thus, instead of writing 

The square of the nninber 1 is 1. 
The square of the number 2 is 4. I. 

The square of the number 3 is 9. 
etc. etc. etc 

we may express these results more concisely as 
follows : — 

Numben. SquarMi Numbers. SqnarM. Nnmbers. Squares. 

113 9 6 26 

2 4 4 16 6 36 



II. 



or in a still more condensed form : — 



Numbers. 
Squares. 



3 4 5 
9 16 25 



7 8 
49 64 



9 
81 



in. 



METHODS OF CX)NDENSATION. 



747 



The fact that a certain column or line of figures 
contains numbers, another the squares of these 
numbers, is indicated by the words ^^ numbers " or 
'' squares ' at the beginning of the column or line. 
It is not, however, explicitly stated which number 
each square corresponds to; this is left to be inferred 
from the proximity of the printed figures by which 
the squares and the numbers are represented. Thus 
in either of the tables II. or III. above, the fact that 
25 is the square of 5 is indicated by printing the 5 
much nearer to the 25 than to any of the other 
squares contained iu the table. 

Sometimes a heavy or a double line is used, as be- 
tween the 9 and the 5 of the second table (II.), to 
indicate a wide separation. In this case an arrange- 
ment of figures similar to that in Table II., is to be 
interpreted in accordance with the fact that 25 (not 
9) is the square of 5, even if the 9 is closer than 
the 25 to the figure 5. 

It is occasionally desirable to print side by side 
on the same page the results of performing different 
operations upon a given number. "Reciprocals," 
*' square roots," " squares/* and " cubes " might thus 



be represented : 


— 






Numbers. 


Beciproetlf. 


Numben. 


Square BoqIb. 


1 


1 


X 


1 


2 


0.5 


2 


141 


&c. 


&c. 


&c. 


&c. 


Numben. 


Squares. 


Numben. 


Cubei. 


1 


1 


1 


1 


2 


4 


2 


8 


&c. 


&c. 


&c. 


&c. 



IV. 



748 ARBANGEMENT OF TABLES. 

It is obviously unnecessary in such cases to repeat 
the same numbers in each alternate column ; and by 
omitting to do so, as in V., considerable space is 
gained. 



Numbers. 


Beciprocali. 


Square Roots. 


Squares. 


Oubei 


1 


1 


1 


1 


1 


2 


05 


141 


4 


8 


&c. 


&c. 


&c. 


&c. 


&c. 



ARGUMENT, VARIABLE, AND FUNCTION DEFINED. 

Starting in such a table (see Table 2, page 
798), in the left-hand column, with any number 
between 1 and 100, we find in a line with it 
its reciprocal, square root, square, or cube. The 
number which one starts with is called the argu- 
ment. Different values of the "argument" are al- 
most always placed in the left-hand column of a 
table, and are printed in heavy type, so as to be dis- 
tinguished from the rest of the table. The " argu- 
ments " represent certain values of a quantity which 
may or may not vary between wide limits. This 
quantity is called in any case the " variable." It will 
be seen by reference to Table 2 (page 798) that when 
a number increases, its reciprocal diminishes; but 
that its square and its cube increase faster than the 
number itself. The reciprocal, square, cube, &c., of 
a variable are QdlleA, functions of that variable (Jungo^ 
to perform). Logarithms, sines, cosines, &c., are also 
called ''functions," and in general, whenever two 
variable quantities are connected together, either by 
mathematical or by physical laws, so that if the first 



ORDINARY MATHEMATICAL TABLES. 749 

is given the second may be found, the second is 
called a "function" of the first. The name of a 
table relates to the function which it represents. If 
several functions are given in the same table (see V.) 
the name of each is usually printed at the head of 
each column or at the beginning of each line contain- 
ing the function in question. 

ORDINARY MATHEMATICAL TABLES. 

When the argument and the function require each 
3 or 4 figures to represent it, the same page cannot 
conveniently contain more than 200 or 300 values of 
each. If, however, the argument increases regularly 
(as is generally the case), it is not necessary that it 
should be printed opposite each value of the func- 
tion. It is, in fact, sufficient that the argument 
should be given for every 10th value of the function, 
since the intermediate values of the argument can be 
easily supplied. This principle is utilized in the or- 
dinary arrangement of mathematical tables, and af- 
fords a considerable saving of space. 

Different values of the argument, corresponding in 
such tables to every 10th value of the function, are 
placed in a column at the left of the page. Opposite 
them, in a second column, the corresponding values 
of the function are given. 

Thus in the first two columns of Table 3, Gr (page 
810),relatingtothe areas of circles, we find 



VL 



10 


78.6 


11 


95.0 


12 


113.1 


etc. 


etc. 



750 ABRANGEMENT OF TABLES. 

The letters Diam, are printed over the first column 
to show that it relates to the diameters of circles. 
The words " Areas of Circles " apply to the second 
as well as to the succeeding columns. We see, there- 
fore, that a circle having a diameter equal to 10 
units of length, must have an area equal to 78.5 
units of area (as nearly as the result can be expressed 
by three figures). The use of the first two columns 
by themselves does not differ in any respect from 
cases which we have already examined. 

It has, however, been stated that the first two col- 
umns give only every 10th value of the argument and 
function. The functions of "round numbers" are in 
fact confined to the second column, which is accord- 
ingly headed o, Intermediate values of the function 
are contained in the succeeding columns, headed by 
the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9. The values are 
arranged so as to follow in regular succession when 
read from left to right like a page of ordinary print. 
This succession should be continued in passing from a 
number in the column headed 9, to the number in 
the next line in the column headed 0. The table 
for the areas of circles becomes accordingly : — 

Diam. .0 J. .2 ^ .4 .5 .6 .7 .8 .9 

10 78.5 80.1 81.7 88.3 84.9 80.6 88.2 89.9 91.6 98.8 

11 96.0 96.8 98.6100.3102.1103.9106.7107.5109.4111.2 VIL 

12 113.1 115 116.9 118.8 120.8 122.7 124.7 126.7 128.7 180,7 
&c. &c. &c. &c. &C. &c. &c. &c. &c &c. &c. 

The chief peculiarity of a table constructed in this 
way is that, instead of printing the argument at the 
left of each value of the function, as in IV., part of 



ORDINARY MATHEMATICAL TABLES. 751 

the argument is to be found at the left of the line 
containing the function, while the remainder of the 
argument — usually a single figure — is placed at the 
head of the column containing the function. The 
areas in the first line of the main body of the table 
(VII.) correspond, accordingly, to the diameters 10.0, 
10.1, 10.2, &c., those in the next line to 11.0, 11.1, 
11.2, &c., &c. 

The argument corresponding to any number in a 
given column and line may alwaj's be found by the 
following rule: Add the figure at the head of the 
column to the fi^gurei at the left of the line to find 
the argument in question. In making this addition, 
attention must of course be paid to decimal points^ 
which in cases of doubt are given both at the 
head of each column and at the left of each line. If 
the decimal point is omitted in either of these two 
places, it may be taken for granted that the figure at 
the head of the column is to be written after the fig- 
ures at the left of the line.^ Thus in Tables 47 and 
48, page 897, since the first column contains latitudes 
0% 10°, &c., while at the head of the columns we find 
0°, 1°, 2'', &c., we infer that the first line refers to lat- 
itudes 0°. I'', 2°, &c., while the second refers to 10**, 
11°, 12°, &c. In table 8, F (page 809), however, in 
the absence of any decimal point in the left-hand 
column, we infer that the figures in that column, 10, 



^ For an arrangement of tables (having certain advantages) in 
which the reverse is taken for granted, see Pickering's Physical Man- 
ipulation, Vol. IL 



752 ARRANGEMENT OF TABLES. 

11, &c., are simply to be prefixed to the figures 10, 
20, &c., in the top line. 

The first line of " circumferences " relates, accord- 
ingly, to circles with the following diameters : 1000, 
1010, 1020, &c. ; while the diameters corresponding 
to the second line of circumferences are 1100, 1110, 
1120, &c. 

EXTENSION OF TABLES. 

Most tables contain arguments reaching from 1, 10, 
or 100 to a value 10 times as great,^ so that it is pos- 
sible to find the value of a function corresponding, if 
not to a given argument, at least to some decimal 
multiple or submultiple of that argument. From 
this the desired result may often be obtained by point- 
ing off the proper number of decimal places. Thus 
to find the circumference of a circle 300 cm. in di- 
ameter, we observe that 300 cm. = 3000 mw., and 
that the corresponding circumference is (see Table 
3, jP, page 808) 9425 mm., or 942.5 cm. Again, in 
finding the area of this circle, we reduce the di- 
ameter (300 cm.') to decimetres; and starting with 
the result (30.0 decim.) as an argument, in Table 
3, G (page 810), we find the area to be 706.9 sq. 
decim. or 70690 sq. cm. (since 1 sq. decim. = 100 sq. 



1 Tables of reciprocals, squares, cubes, logarithms, &c., often 
reach from 1 to 11 instead of from 1 to 10. The extension of such 
tables from 10 to 11, though strictly involving a repetition, is of 
great convenience in physical problems in which factors just above 
unitj are of comparativelj frequent occurrence. 



OMISSION OF CIPHERS, ETC. 768 

cm,'). In finding the volume of a sphere with the 
same diameter (300 c?w.)» we should reduce this diam- 
eter to metres; then with the result (3.00 metres) 
as an argument, we should find the volume of the 
sphere to be 14.14 cubic metres^ according to Table 3, 
-^(page 812), or 14,140,000 cu. em. (since 1 cubic 
metre = 1,000,000 cu. cm,). For the extension of 
tiigonometric or logarithmic tables beyond their natu- 
ral limits, special rules must be observed (see explan- 
ation of the tables, page 761 et seq.). 

OMISSION OF CIPHEBS, ETC. 

It may be remarked that it is not customary to re- 
peat initial ciphers or decimal points throughout the 
whole of a table. These are given either at the head 
of each column, or at the beginning of each 5th line. 
In some books other omissions take place. It is well 
always to look through a new table carefully before 
deciding how it is to be read, and where the decimal 
point is to be placed. A negative sign placed before 
a number applies not only to the integral part of that 
number, but also to the decimal part which follows. 
A negative sign placed over a figure applies only to 
that figure. If the figure is an integer followed by a 
decimal, the integer is negative, the decimal positive. 
In logarithmic tables, decimal points are frequently 
omitted both in the argument and in the logarithm. 
In such cases they are always understood to exist 
after the first figure of the argument and before the 
first figure of the logarithm. 



754 ARRANGEMENT OF TABLES. 



COMPLEMENTARY ARGUMENTS. 

Some tables (for instance, Table 4, page 814) eon- 
tain two arguments. One of these is printed in the 
ordinary manner, partly at the left and partly at the 
top of the page, and is to be used in connection with 
the function mentioned at the top of the page. The 
other argument is printed partly at the right and 
partly at the bottom of the page, and is to be used in 
connection with the function named at the bottom of 
the page. The object of this arrangement is to make 
a double iise of the figures in the body of the table. 
An extra column of figures is usually added to avoid 
certain difliculties. No number is placed at the head 
of this column, and no attention is to be paid to it in 
dealing with the functions named at the top of the 
page. The argument corresponding to the function 
at the bottom of the page is found, in the case of a 
number in a given column and line, by adding the 
figure at the bottom of the column to the figures at 
the right of the line. The values of the argument at 
the right of a page increase upwards; those at the 
bottom of the page increase from right to left 

INDEPENDENT ARGUMENTS. 

The two arguments employed in the class of tables 
mentioned above are not independent, but represent 
quantities each of which is usually the "comple- 
ment " of the other. The use of two independent 
arffuments introduces an entirely different kind of 



INDEPENDENT ARGUMENTS. 766 

tables. The two arguments correspond in these 
tables to two independent variables upon which the 
value of the function depends. The first argument 
is arranged in a column, usually at the left of the 
table ; the second is arranged in a line, usually across 
the top of the table. To find the value of a function 
corresponding to given values of both arguments, we 
follow the line containing the given value of the first 
argument until we reach the column containing the 
given value of the second argument. Table 1 (which 
is a form of multiplication table, see page 797) is an 
example of the use of two independent arguments. 
The first argument is a series of factors from 1 to 100, 
arranged in column at the left of either half of the 
table. The second argument is an independent series 
of factors, .1, .2, .3 .4, .6, .6, .7, .8, .9, in the head- 
line of either half of the table. The body of the 
table consists in results obtained by multiplying these 
two sets of factors together. The number occupying 
a place in a given column and line is the product of 
the number at the left of the line and the number at 
the head of the column. 

In a table with two independent arguments, the 
nature of the function is usually given either in 
the title or at one side of the figures representing the 
function ; the nature of the first argument is given at 
the head or at one side of the column containing it ; 
while the nature of the second argument is given 
either at the beginning of the head-line of the table, 
or just above this head-line. 



756 ARRANGEMENT OF TABLES. 

There is a second method of arranging tables with 
two arguments, namely : to calculate a separate table 
of the ordinary sort for each value of one of the 
arguments. Thus Table 16, J., consists of two parts, 
one calculated for a value of the acceleration of grav- 
ity equal to 980, the other for the value 981 cm, per 
sec, per sec. A still greater number of such tables 
would be necessary to cover all variations in gravity 
(from 978 to 983) on the earth's surface. 

The only way in which it is practicable to repre- 
sent the value of a function depending upon three 
independent variables is by means of a series of 
tables containing two independent arguments, each 
table being calculated for a special value of the 
third variable. A complete 2-place table contain- 
ing three independent arguments, each varying from 
1 to 10, would ordinarily occupy abput the same 
space as a 4-place table with a single argument, 
varying from 1 to 1000, let us say 2 pages. A 
table with two independent arguments must occupy 
about 20 pages in order that 3 figures should be 
significant, and about 2000 pages to give significance 
to 4 figures. The addition of a third independent 
argument in the latter case would increase the table 
to about 2,000,000 pages. It is obvious that the use 
of tables containing more than 1 independent argu- 
ment is practically reduced to cases where a rough 
knowledge of a function is suflBcient (as in the cal- 
culation of corrections) or where one at least of the 
variables, like the acceleration of gravity on the 



PHYSICAL TABLES. 767 

earth's surface, or the ordinary condition of atmos- 
pheric temperature and pressure, is confined within 
narrow limits. 

PHYSICAL TABLES. 

We have seen that, in representing functions of two 
variables, one argument is usually printed at the left 
of the table, the other at the head of the table. A 
similar arrangement is adopted when it is desired to 
represent simultaneous variations in different physical 
quantities due to temperature, pressure, or any other 
single cause. The values of a given physical quan- 
tity are arranged either, as in Table 28, in a column 
opposite the values of the argument to which they 
correspond, or else, as in Table 31, in a line under- 
neath the corresponding values of the argument. 
The second argument in such tables is replaced hy 
names^ referring to a series of physical quantities. 
These are usually different properties of a given sub- 
stance, or a given property of different substances ; 
but the arrangement may be applied to any set of 
quantities which are affected by changes in a given 
variable. 

We have, furthermore, an arrangement peculiar to 
purely physical tables, in which one argument consists 
of a series of physical properties, while the other 
argument consists of a series of substances to which 
these properties belong. This arrangement is ad- 
opted in Tables 8, 9, 10, 11, 12, &c. The names 
of different substances are arranged in a column at 



768 ARRANGEMENT OF TABLES. 

the left of the table ; the names of different phys- 
ical properties are printed at the heads of a series 
of columns so as to form a line across the top of 
the table. The body of the table contains numer- 
ical values. The name of the property to which a 
given number relates is to be found at the head of 
the column containing that number ; the name of the 
substance to which it applies is to be found at the left 
of the table in line with the number in question. The 
names of the properties and of the substances should 
be such that, when combined together, they form com- 
plete definitions of the physical quantities to which 
the table relates. The numerical values are in each 
column reduced, when practicable, to the C. 6. S. 
system; when this is not practicable, a factor by 
which this reduction may be effected, is placed in the 
first line of the column. In any case the reduction 
consists simply in moving the decimal point. 



DIFFERENCES. 

The differences between adjacent numbers in a 
purely physical table (especially when, as in the cases 
which follow, an alphabetical order is observed) have 
in general no special significance. In mathematical 
tables, on the other hand, the use of such differences 
is exceedingly important. 

The difference between two adjacent numbers in a 
table should theoretically, if represented at all, be 
printed half-way between them as in VIIL 



DIFFERENCES. 759 

1 1 2 1 3 1 4 1 6 

6 5 6 6 5 

6 1 7 1 8 1 9 1 10 VIII 

6 5 6 C 6 

11 1 12 1 13 1 14 1 15 

It is, however, customary if a given line or column of 
differences is constant, or nearly constant, to omit this 
line or column, and instead to print the average value 
of the differences thus omitted near where the end of 
the line or column of differences would naturally have 
come. Table VIII. would thus assume one of the 
following forms: — 



IX. 























Dif. 




1 




2 


3 




4 




5 




1 




6 




7 


8 




9 




10 




1 




11 




12 


13 




14 




15 




1 


Dif. 


6 

1 




6 
2 


6 
8 




6 
4 




6 
5 




Dif 




6 




7 


8 




9 




10 




6 




11 




12 


13 




14 




15 




6 


Dif. 


1 

6 

11 


1 


1 

2 

7 
12 




1 

3 

8 

13 




1 

4 

9 

14 






5 
10 
15 


Dif. 


5 
Dif. 


1 


6 


1 


5 


1 


6 


1 




6 
















Dii 


t 


Dif. 


- 


1 


2 


3 


4 




5 


1 










6 


7 


8 


9 




10 


1 




6 






11 


12 


13 


14 




15 


1 




5 





X. 



XI. 



XIL 



Differences printed, as in IX., on a given line or in 
a given column relate accordingly to pairs of adjacent 



760 ARRANGEMENT OF TABLES. 

numbers in that line or column. On the other hand, 
differences printed, as in X., between two lines or be- 
tween two columns relate to pairs of adjacent numbers 
one in each line or one in each column. Either set of 
differences, if not needed, may of course be omitted. 
Table 3, D (page 806), corresponds, for instance, to 
form IX. without the lower line, or to form XII. 
without the right-hand column of differences. 

'Instead of printing a series of numbers in the col- 
umn of differences when they are exactly alike, it is 
customary to print only one of them, situated as 
nearly as possible in the middle of the space which 
the whole series would occupy. This method of rep- 
resenting differences is adopted in Tables 3 -A, 3 (7, 
3 G^, 4, 4 -A, 5, 5 -A, &c. The difference between any 
two consecutive values of the function is, in these 
tables, approximately equal to the nearest number in 
the column of differences. The use of this column 
of differences will be found to effect a considerable 
saving of time ^ in processes of interpolation. To ef- 
fect a still greater saving of time in these processes, a 
small table of '' proportional parts " has been printed 
jn.th^ table of logarithms (Table 6), beneath each 
difference. The use of proportional parts for in- 
terpolation will be explained below (see explanation 
of Table 1). 

1 It may be remarked that owing to necessary irregularities in the 
differences which most tables of functions contain, the most accurate 
results require that tliese differences should be calculated by actual 
subtraction in each case. 



USE AND EXPLANATION OP MATHEMATI- 
CAL AND PHYSICAL TABLES. 

Table 1 consists of products obtained by multiply- 
ing any of the whole numbers (from 1 to 100) in the 
left-hand column of either half of the table by the 
decimals .1, .2, .3, .4, .5, .6, .7, .8, .9 at the head of the 
table. The decimal part of the product is rejected in 
every case, the units being increased by 1 if the frac- 
tion is .5 or more. The table is useful in dividing 
differences into parts proportional to the numbers 1, 
2, 3, 4, 5, 6, 7, 8, 9, whence the name of the table. 
It may be used in connection with any of the tables 
which follow. Let us suppose that it is required to 
find the sine of 12°.34 in Table 4, page 814. We find 
the sine of 12°.3 (in the line with 12° and in the col- 
umn with .3) to be .2130, while the sine of 12°.4 is 
.2147. The first number (.2130) is too small ; the 
second (.2147) is too great. The difference between 
them is .0017, or 17 units in the last place, as indi- 
cated by the nearest number in the column of differ- 
ences. If 0°.l makes a difference of 17 units, 0^.04 
should make a difference of .04 -t- 0.1 X 17, that is, 
6.8 or (nearly) 7 units in the last place. The same 
remit may be found by seeking in Table 1 a number 

12 



762 ' EXPLANATION OF TABLES. [No. 1. 

opposite the difference (17) and under the figure (4) 
for which the interpolation takes place. The result 
(7 units in the last place) is to be added to the sine 
of 12^3, because the sines increase when the angles 
increase — in other words, because the differences 
are positive. The sine of 12°.34 is accordingly 
0.2180 + .0007 = 0.2137. 

Again, to find the reciprocal of 6.789, by Table 3 A, 
page 802, we observe that the reciprocal of 6.78 is 
.14749, while that of 6.79 is .14728. The difference 
between these reciprocals is — .00021, because the re- 
ciprocals decrease as the numbers increase. Opposite 
21 and under .9 in Table 1 we find 19 ; hence the 
answer is .14749 — 00019 = .14730. If we had used 
the nearest number (22) in the column of differences 
of Table 3 A., instead of the actual difference (21), 
we should have found similarly .14729 instead of 
.14730. The true reciprocal happens to lie between 
these two values. 

Table 1 can be used also in inverse processes. 
Let us suppose that it is required to find the cube 
root of 800, by Table 3 D, page 806. We notice 
that the cube of 9.28 is 799.2, just below 800, while 
the cube of 9.29 is 801.8, just above 800 ; the differ- 
ence being 26 units in the last place. . The difference 
between 799.2 and 800.0 is 8 units in the last place. 
In line with the number 26 in the left-hand column of 
Table 1, and over the number 8,^ we find .3. We see 

1 When the exact number cannot be found amongst the proper- 
iMtX parts we choose the one nearest to it. 



No. 2.] MATHBMA.TICAL TABLES. 763 

therefore that the cube of 9.283 would be 800.0 ; 
hence, conversely, 9,283 is the cube root of 800. 

The use of proportional parte is especially reoom* 
mended when accuracy in the last figure is important. 
The tables which follow have, however, been con- 
structed with such fulness that interpolation will 
generally be unnecessary, or readily carried on in the 
head. 

Tabi4£} 2 contains several functions often needed, 
and is intended for rough and rapid work. More 
exact values of the functions will be found in Tables 
3 A — 3 H, which follow. 

Column a contains the "reciprocals" of the num- 
bers in the first column from 1 to lOO. The recipro- 
cal of a number is defined as the quotient obtained 
when unity is divided by the number in question. 
Example : the reciprocal of 30 is .0333. 

Column h contains the square roots of numbers 
from 1 to 100. The square root of a number is de- 
fined as a number which multiplied by itself would 
give a product equal to the. original number. Ex- 
ample : the square root of 49 is 7.00. 

Column contains the squares of numbers from I 
to 100; that is, the products obtained when eacli 
number is multiplied by itself. Example i the square 
of 40 is 1000, 

Column d contains the cubes of numbers from 1 to 
100. The cube of a number is defined as the result 
of multiplying that number by the square of that num- 
ber 5 or as the result of multiplying that number three 
times into unity. Example : the cube of 6 is 125. 



764 EXPLANATION OF TABLES. [No. 3. 

Column e contains three-place logarithms (see under 
Table 6) from 0.1 to 10.0. Example : the logarithm 
of 2 is 0.301, correct to 3 places of decimals. 

Column f contains the circumferences of circles 
having diameters from .1 to 10.0. The circumference 
is given in the same units as the diameter. Example : 
given the diameter 2.0 c?w., the circumference is 
6.28 cm. 

Column g contains the areas of circles having diam- 
eters from .1 to 10.0. The area is given in units cor- 
responding to the unit of length employed in meas- 
uring the diameter. Example: given the diameter 
2.0 em,y the area of the circle is 3.14 9q. cm. 

Column h contains the volumes of spheres having 
diameters from .1 to 10.0. The volume is given in 
units corresponding to the unit of length employed 
in measuring the diameter. Example: given the 
diameter 2.0 cm., the volume of the sphere is 4.19 
cu. cm. 

Table 3 contains principally 8-place trigonometric 
functions, and is, like Table 2, intended for rough 
and rapid work. 

Column a contains angles from 0** to 90® ; covering 
in all a right-angle. 

Column h contains the tangents of angles. The 
tangent of an (acute) angle is defined, with reference 
to a right-angled triangle, as the ratio of the side 
opposite it to the (shorter) adjacent side. Example : 
the tangent of 15° is 0.268. 

Column c contains " arcs ; " that is, in a circle of 
radius unity, the length of the arcs intercepted by 



No. 3.] MATHEMATICAL TABLES. 765 

angles with their vertices at the centre of the circle. 
" Arcs " are also called the " circular measures " of 
angles. Example: 15° is equal to 0.262 in circular 
measure ; or the arc of 15° is 0.262. 

Column d contains the '* chords " of angles. The 
chord of an angle is defined, with reference to an 
isosceles triangle, as the ratio of the side opposite the 
vertical angle to either of the two equal sides. Ex- 
ample : the chord of 16° is 0.261. 

Column e contains natural sines. The sine of an 
angle is defined, with respect to a right-angled tri- 
angle, as the ratio of the side opposite that angle to 
the longest side, or hypotheuuse. Example : the sine 
of 16" is Q.259. 

Column/ contains natural cosines. The cosine of 
an angle is defined as the sine of the complement of 
that angle (see i). Example: the cosine of 15° is 
0.966. 

Column g contains rates of vibration corresponding 
to different arcs from 0° to 46^ through which for in- 
stance a pendulum is vibrating. The arcs are meas- 
ured from one side of the vertical to the other. The 
rate of vibration in a very small arc is taken as 1. 
Example I. : if a pendulum vibrates once a second in 
a very small arc, it will vibrate .99893 times a second 
in an arc of 15*^ (i. e. 1^ on each side of the verti- 
cal). Example II. : given the time of oscillation of 
a magnet equal to 10 seconds in an arc of 45° ; re- 
quired its time of oscillation in a very small arc. 
Answer, 10 X 99037 = 9.9037 sec. Column g con- 
tains also coversines from 46° to 90°. 



7C6 EXPLANATION OF TABLES. [No. 8. 

The coversiue of au angle it) defined as unity leas 
the sine of the angle* It is the same thing as the 
versine of the complement of the angle. Yersines 
and coversines measure various errors introduced into 
physical measurement when two lines which ought to 
be parallel or perpendicular are inclined at a given 
angle. The inclination of the two lines is to be 
found in column a or in column i as the case may be. 
Example I.: the shaft of a cathetometer (^ 262) 
makes an angle of 89° with the horizon ; required 
the error introduced in the measurement of vertical 
distances. Answer, .00015 parts \v\ 1, or ^JJ^^ of 1 %. 
Example II.: a magnet which should be horizontal 
dips 10°; required the error in estimating its mag* 
netism. Answer, .0162, or 1 ^^^ %. 

Column h contains secants, or the reciprocals of 
cosines. Example ; the secant of 16° is 1.085. 

Cohimn i contains the complements of the anglei 
contained in column a ', that is, the results of sub- 
tracting these angles from 90°. Example : the com- 
plement of 15° is 75°. 

It may be remarked that the cotangent of an angle 
lA the tangent of its complement ; the oochord of an 
angle is the chord of its complement ; the cosecant of 
an angle is the secant of its complement. These may 
all be found, accordingly, by Table 8. Examples : — 

The cotangent of 15^ = tangent of 76° = 8.732 
The cochord of 15** = chord of 75° = 1.218 
The cosecant of IS** = secant of 75° = 8.864 

To find any function of the complement of an angle, 



No. 3 F.] MATHEMATICAL TABLES. 767 

we have only to look up that angle in column t, in- 
stead of in column a. 

Table 8 A is essentially a 4-place table of recipro- 
cals from 1.00 to 11.09, carried out, however, to 6 
places between 6.00 and 9.99. Examples: the re- 
ciprocal of 2.73 is .8663 ; the reciprocal of 273 is 
.003668. 

Table 8 C is a 4-place table of squares from 1.00 
to 9.99, carried out to 5 places between 10.0 and 
11.09. Examples : the square of 8.14 is 9.860 ; the 
square of 81.4 is 986.0. The square rooc of 1.25 is 
is 1.12 nearly, or more exactly ^ 1.118 (see under 
Table 1). 

Table 8 D is a 4-place table of cubes from 1.00 to 
9.99, carried out to 6 places from 10.0 to 11.09. Ex- 
amples : the cube of 5.55 is 171.0 ; the cube of .555 is 
.1710. The cube root of 800 is 9.283 (see under 
Table 1). 

Table 3 F contains the circumferences of circles 
with diameters (diatn.) varying from 1000 to 10090 
by 10 units at a time. The results are carried out to 
units. The differences in this table are either 31 or 
32, from beginning to end. The mean difference is 
31.416. Proportional parts corresponding to this 
mean difference are printed at the bottom of the 
table. The circumference is given in units of the 
same magnitude as the diameter. Example I.: the 
circumference of a circle 3600 cm. in diameter is 
11810 cm. Example II. : given a circumference 
10,000 metres, the diameter is 3180 metres, nearly ; 
or more exactly, 8188 metres (see under Table 1). 



768 EXPLANATION OF TABLES. [No. 4 A. 

Table 3 G is a 4-place table containing the areas 
of circles corresponding to diameters (diam,) from 
10.0 to 100.9. The area is given in units correspond- 
ing to the unit of length in which the diameter is 
measured. Example I. : diameter = 15.0 cm.^ area 
:= 176.7 8q. cm. Example II. : diameter = 65.5 mm.^ 
area = 2419 sq, mm. = 24.19 sq. cm. Example III. : 
area = 4000 sq, cm.^ diameter = 71.4 cm., nearly ; 
more exactly, 71.36 cm, (see under Table 1). 

Table 3 H contailis the volumes of spheres cor- 
responding to diameters from 1.00 to 10.09. The 
volume is given in units corresponding to the unit of 
length in which the diameter is measured. Example 
I. : diameter = 11.1 mm. = 1.11 cm. : volume = 
.539 cu. cm, = 539 cu. mm. Example II. : volume = 
85.00 cu. cm.j diameter == 4.06 cm.^ nearly ; or more 
exactly, 4.058 cm. (see under Table 1). 

Table 4 is a 4-place table giving the natural sines 
of angles from 0°.0 to 89^9, when interpreted in the 
ordinary manner by means of the argument at the left 
and at the top of the page. Natural cosines may 
also be found by means of this table, by using the 
argument at the right and at the bottom of the page. 
Example I. : the sine of 30°.0 is 0.5000. Example 
11. : the cosine of 80°.0 is .8660. 

Table 4 A is a 4-place table giving the logarithmic 
sines (that is the logarithms of the sines) of angles 
from 0°.0 to 89°. 9, when read in the ordinary way. 
Logarithmic cosines may be found through the ar- 
gument at the right and bottom of the page. Ex- 
ample I. : the logarithm of the sine of 30° is 1.6990. 



No. 6.] LOGARITHMS. 769 

Example II. : the logarithm of the cosine of 30° is 
1.9375. 

Table 5 contains the natural tangents of angles 
from 0°.0 to 89°.9. Natural cotangents from 45^0 to 
89"*. 9 may also be found by using the argument at 
the right and bottom of the first half of the table. 
Below this limit, they are not given ; but they may 
be found by calculating the complement of the angle 
and looking up its tangent. Example I. : the tan- 
gent of 30° is 0.5774. Example II. : the cotangent 
of 22°.5 = tangent of 77°.5 = 4.511. 

Table 6 A is a 4-place table giving the loga- 
rithmic tangents (that is, the logarithms of the tan- 
gents) of angles when read in the ordinary way. 
Logarithmic cotangents may also be found by using 
the argument at the right and at the bottom of the 
page. Example I. : the logarithm of the tangent of 
30°.0 is 1.7614. Example II. : the logarithm of the 
cotangent of 30° is 0.2386. 

Table 6 is a 5-place table of the logarithms of 
numbers from 1,000 to 11,009. A decimal point 
is understood after the first figure of each number 
and before the first figure of each logarithm. Ex- 
ample : the logarithm of 2.000 is .30103. 

When the decimal point of a number does not 
follow the first figure, the corresponding logarithm 
consists of two parts. The first part is a whole 
number called the " characteristic " of the logarithm ; 
the second or decinlal part is called the '* mantissa." 

The " characteristic " of a logarithm is not to be 
found in Table 6, but is to be supplied by inspection. 



770 EXPLANATION OF TABLES. [No. 6. 

Its numerical value is equal to the number of spaces 
between the decimal point of the argument and the 
space following the first figure of the argument. 

Thus the logarithm of the number 1.11 has the 
characteristic 0; while the characteristics of 11.1 
and 111 are 1 and 2 respectivelj. The sign of the 
characteristic is positive if the decimal point is at the 
right of the first figure of the argument ; if it is at 
the left, the sign is negative. Thus the characteris- 
tic of the logarithm of .1111 is — 1., the character- 
istic of the logarithm of .01111 is — 2., &c. The 
negative sign is in practice written over the char- 
acteristic, as it affects this characteristic alone. 

It is a peculiarity of logarithms that the "man- 
tissa " is not affected by the location of the decimal 
point in the original number. The logarithm of 
1.111 (namely, 0.04571) is, for instance, the same as 
the logarithm of 1,111- (namely, 3.04571), as far as 
the mantissa is concerned. The mantissa or decimal 
part of the logarithm of any number may be found, 
accordingly, by Table 6, by considering only the fig- 
ures of which the number is composed. 

Initial and final ciphers may be thrown off ad libU 
turn in this process ; but ciphers in the middle of a 
number form an essential part of it. Thus in finding 
the decimal part of the logarithm of .000,100,100, 
we need to consider only the figures 1001, since these 
are preceded and followed only by ciphers ; but the 
ciphers between the first £Uid last figures cannot be 
neglected. The following logarithms from Table 6 
may also serve as examples : — 



No. 6] LOGABITHMS. 771 

The logarithm of 8.1416 ia 0.49716 

'« '' " 980 <* 2.99123 

^' 41,700,000 "7.62014 

^' *^ .00367 " 8.66487 

Conversely, in finding the number corresponding to a 
a given logarithm, we firtst obtain the figures of whioh 
the number is composed by considering simply the 
mantissa, or decimal part of the logarithm, and to 
these figures we add as many initial or final ciphers 
as may be needed ; then starting with the space at 
the right of the first figure (disregarding initial 
ciphers) we count off to the right if the characteristic 
of the logarithm is positive (or to the left if negative) 
a number of spaces equal to the characteristic in ques- 
tion, in order to locate the decimal point. In any 
case the num1>er of figures between the decimal 
point and the space following the first figure of the 
answer must be equal to the characteristic of the 
logarithm. 

Example I. : given the logarithm 0.14860, the fig- 
ures of the corresponding number are 1408; the char- 
acteristic of the logarithm being 0, the answer is 1.408. 
Example II. : given the logarithm 3,14860, the man- 
tissa being .14860 as before, we find the same figures, 
1408. Since the characteristic (3) is positive, the 
decimal point is at the right of the first figure, and 
since 3 figures must come between the decimal point 
and the space following the first figure, the answer is 
1,408. 



772 EXPLANATION OF TABLES. [No. 6. 

The following rules embody the most important 
applications of logarithms, — namely, to problems of 
multiplication and division. 

Hide 1. To multiply two or more numbers to- 
gether, find the logarithm of each and add the loga- 
rithms together. The number corresponding to their 
sum is the required product. Example : to multi- 
ply 2X4. 

The logarithm of 2 is 0.30108 

" " " 4 is 0.60206 

The sum of these logarithms is 0.90309, 

which is the logarithm of 8, the answer. Numbers 
involving more than 3 significant figures may be mul- 
tiplied together by the aid of logarithms with greater 
ease than by arithmetical processes. 

Mule 2. To divide onia number by another, find 
the logarithm of the first, subtract the logarithm of 
the second; the remainder is the logarithm of the 
answer. Example : to divide 4 by 8, 

The logarithm of 4 is 0.60206 

" '' '' 8 " 0.90309 

The difference is T.69997, 

which is the logarithm of 0.5, the answer. 

Mule 3. To find the value of a fraction with sev- 
eral factors, find the logarithm of each factor in the 
numerator, and add the logarithms together. Then 
find the logarithm of each term in the denominator, 
and add these logarithms together. Subtract the 



No. 7.] LOGARITHMS. 773 

latter sum from the former sum. The remainder is 
the logarithm of the answer. Example : to find the 
value of the fraction 

.2345 X 45.67 X 6,789 ,^,,_ 
1.234 X 34.56 X 667.8' 

(1) log. .2346 = 1.37014 (6) log. 1.234 = 0.09132 

(2) " 46.67 = 1.66963 (6) " 34.56 = 1.63867 

(3) " 6789 = 3.83181 (7) " 667.8 = 2.76420 

(4) sum = 4.86158 (8) sum = 4.38409 
(9) subtract 4.38409 

(10)remainder= 0.47749 = log. 3.002 +, ans. 

Bule 4. To raise a number to any power, find its 
logarithm, multiply by the power, and the product is 
the logarithm of the answer. Example : to find the 
4th power of 2. The logarithm of 2 is 0.30103 ; 
which multiplied by 4 gives 1.20412. This is the 
logarithm of 16, the answer. 

Bule 6. To extract any root of a number, find 
the logarithm of the number and divide by the root 
in question; the quotient is the logarithm of the 
answer. Example : to find the 12th root of 2. The 
logarithm of 2 is 0.30103 ; this divided by 12 gives 
.02509, which is the logarithm of 1.0595, the answer. 
(This is the value of the interval called 1 semitone 
on the tempered scale.) 

Table 7 contains the probability of an error's ex- 
ceeding limits bearing to the " probable error " (§ 50) 
the ratios represented in the left-hand column. The 
probability is expressed as so many chances in 1. 
Example I. : the probable error of a weighing is 1 



774 EXPLAIffATION OF TABLES. [Nog. g-ia. 

eentigram ; what are the chances of an error greater 
than 1 centigram? Answer, by definition, an even 
chance or 0.50000. Example II. : under the same 
circumstances, what are the chances of an error's ex- 
ceeding 2 centigrams? Answer, 0.17784, i. e; 17,734 
chances in 100,000, or about 1 chance in 6. Exam- 
ple III. : under the same circumstances, what are 
the chances of an errors exceeding 5 centigrams? 
Answer, 0.00075, or less than 1 in 1000.^ 

Tables 8, 9, 10, 11, and 12 contain (1) the names, 
(2) the chemical symbols, and (3) the atomic weights 
of various substances, and deal with the following 
physical properties : (4) the specific gravity (§ 69) of 
gases and vapors referred to hydrogen at the same 
temperature and pressure; (5) the density (§ 15) 
of substances at 0"* under the ordinary atmospheric 
pressure (6) the ** viscosity '' of liquids at about 
20"^, or the force in dynes required to maintain a 
relative velocity of 1 cm. per $ec. between two sur- 
faces 1 cm. square and 1 cm. apart ; (7) the ^^ surface 
tension " of liquids (^ 169) at about 20% or the force 
in dynes with which each iurface of a liquid film 
1 cm. broad tends to contract ; (8) the ** breaking 
strength" of solids, or the force in kilo-megadynes^ 
required to break a wire 1 9q. em, in cross section ; 
(9) the "crushing strength" of solids, or the force 
in kilo-megadynes required to crush a block 1 $q. cm. 

1 The chances reUte onlj to <' accidental erroiv" ({ 24). The 
chances of " mistakes " are not inclttded. 

^ 1 kilo-megadjne = 1.02 '' tonne weight," or 1 English ton weight, 
nearly. 



Nos. 8-12.1 GENERAL PfiOPfiHTIES. 775 

In cross section; (10) the "shearing Btrength" of 
solids, or the force in kilo-megadynes required to cut 
a wire 1 sq. cm. in cross section; (11) the "hard- 
ness " of solids according to Mohs' arbitrary scale 
(page 587) ; (12) the " simple rigidity " of solids, or 
the force in kilo-megadynes required to make two 
surfaces 1 em. square and 1 cm. apart move parallel 
to one another through a thousandth of a centimetre 
(.001 cm.) , (13) " Young's modulus/' or the force in 
kilo-megadynes required to pull two such surfaces 
apart through one thousandth of a centimetre (.001 
em.^ ; (14) the *• resilience of volume " or the pres- 
sure in kilo-megadynes required to compress a centi- 
metre cube by one cubic millimetre ; (15) the average 
cubical "coefficient of expansion" of substances^ 
(§ 83) between 0** and 100** under a constant pres- 
sure of 76 cm. of mercury ; (16) the •' melting-point " 
of solids, or the ** freezing-point'* of liquids on the 
Centigrade scale ; (17) the ** boiling-point " of liq- 
uids, or the "temperature of condensation" of 
vapors at the atmospheric pressure ; (18) the " criti- 
cal temperature" of liquids and vapors, — that is, the 
temperature at which the properties of the liquid and 
its vapor become indistinguishable ; (19) the " criti- 
cal pressure " of liquids and vapors, that is the pres- 
sure of the vapor of a liquid at the critical tempera- 
ture in megadynes per sq. cm.; (20) the "pressure 

1 Wli«n a chsnge of itat« takes plaee between (P snd 100^, the 
averages in question refer only to that part of the interval (0° to 
100°) in which the substance exists in the state named at the head 
of the table. 



776 EXPLANATION OF TABLES. [Noa. 8-12. 

of viapors " at 20°, in megadynes per sq. cm. ; (21) 
the average specific heat of substances^ (§ 86) be- 
tween 0° and 100°, under the *• constant pressure " 
of 76 cm. of mercury; (22) the average specific 
heat of substances between 0° and 100°, when pre- 
vented from expanding ; that is, confined to a " con- 
stant volume; " (23) the " latent heat of melting " of 
solids, or the "latent heat of liquefaction" of liquids; 
that is, the number of units of heat required to con- 
vert 1 gram of a solid, at its melting-point, into a liquid 
at the same temperature under a pressure of 1 atmos- 
phere ; (24) the " latent heat of vaporization " of liq- 
uids, or the " latent heat of condensation*' of vapors ; 
that is, the number of units of heat required to con- 
vert 1 gram of a liquid at the boiling-point into 
vapor at the same temperature under the atmospheric 
pressure; (25) the " heat conductivity" of substances, 
or the number of units of heat conducted in one 
second between two opposing faces of a centimetre 
cube differing 1° in temperature ; (26) the " electri- 
cal conductivity" of substances, or the current in 
amperes flowing between two opposing faces of a cen- 
timetre cube differing 1 microvolt (.000,001 volt) in 
electrical potential (§ 189).; (27) the "thermo-elec- 
tric heights" of conductors, or the electromotive force 
in microvolts developed by a thermo-electric junction 
of which one element is lead, corresponding to a dif- 
ference of temperature of 1° ; (28) ** electro-chemical 
equivalents,*' or the weight in milligrams of various 

^ See footnote, page 775. 



No8. 8-12.] GENERAL PROPERTIES. 77T 

elementary substances aflfected by a current of 10 am- 
pdres in 1 second ; (29) the specific inductive capacity 
of substances determined by currents alternating 
several hundred times per second (^ 256) ; (30) the 
minimum ^^extraordinary index of refraction " of opti- 
cal materials ; (81) the " ordinary index of refraction" 
of uniaxial crystals, or the " medium" index of refrac- 
tion of biaxial crystals; (32) the maximum '^ extraor- 
dinary index of refraction *' of diflferent substances — 
these three indices referring to the sodium (D) line ; 
(33) the ordinary (or medium) "index of disper- 
sion," or the difference between the ordinary (or 
medium) indices of refraction for the lines A and H 
of the solar spectrum ; and finally (34) the solubility 
of solids in water, expressed in per-cents by weight, 
and the solubility of gases, also in per-cents by 
weight, under a pressure of 1 atmosphere. 

The first line of each table contains factors by 
which the values given in the column below them 
may be red need to the c. G. 8. system. Thus the co- 
efficient of resilience of aluminum (Table 8) is 0.5 (?) 
X 1012 _ 500,000,000,000(?), and the thermoelectric 
height of copper is about 4 X 100 = 400 absolute 
units. 

Table 8 contains the properties of elementary 
substances. 

Table 9 contains the properties of solids remark- 
able especially for their strength or for other proper- 
ties rendering them suitable for building materials or 
for the construction of machines. 

13 



778 EXPLANATION OF TABLES. [No. 14 A. 

Table 9 A contains the properties of certain 
chemical salts and other substances in ordinary use. 

Table 10 contains the properties of solids remark- 
able for their optical or other allied properties. 

Table 11 contains properties of liquids. 

Table 12 contains properties of gases and vapors. 

Tables 13 A, B, and G, give the (maximum) pres- 
sure in megadynes per sq. cm. of the vapor arising 
from various liquids at different temperatures. 

Table 18 A contains substances which are for the 
most part gaseous at ordinary temperatures. 

Table 13 B contains more or less volatile liquids. 

Table 13 C gives the pressure of the vapor of 
mercury, sulphur, and water, including the vapor 
of water arising from sulphuric acid of diflferent 
strengths. 

Table 13 D contains the *' density of steam," or 
the maximum density of aqueous vapor at different 
temperatures. 

Table 14 gives the boiling-points of water corres- 
ponding to different barometric pressures from 68.0 
to 77.9 centimetres of mercury reduced to latitude 
45° (see Landolt and Bornstein, Table 20). Exam- 
ple : when the barometer stands at 7«5.0 cm., water 
boils at 99^63. 

Table 14 A gives dew-points (calculated from 
Regnault's data) corresponding to different degrees of 
temperature and " relative humidity." The *' dew- 
point" means that temperature at which moisture 
would (barely) be precipitated out of the air (as 
when dew is formed) ; the " relative humidity " is the 



No. 16 A.] METEOROLOGICAL DATA. 779 

proportion which the moisture contained in the air at 
a given temperature bears to the maximum possible 
amount which it can hold at that temperature. Ex- 
ample I. : the air of a room at 20"^ is half saturated 
with moisture (i. e. the relative humidity = 60 %); 
required the dew-point Answer, 9° Centigrade by 
Table 14 A. Example II. : sea air saturated at 9'' 
with moisture is warmed to 20*^ ; required the rela- 
tive humidity. Answer, 50 %. 

Table 15 shows at a given temperature (T) the 
maximum pressure (P) of aqueous vapor in centi- 
metres of mercury, the maximum density (D) of 
aqueous vapor, and the factor (Fj by which the diflfer- 
ence between the readings of a wet and a dry bulb 
thermometer must be multiplied in order to find the 
difference between the dew-point and the tempera- 
ture (T) of the air. The data have been taken from 
Kohlrausch, Table 13, Landolt and Bornstein, Tables 
18 a and 23, and from Everett's ^^ Units and Physical 
Constants," Art. 124. The first three columns are an 
amplification of results contained in Table 13. The 
last column is useful in hygrometry. Example: if 
the dry-bulb thermometer reads 20**, and the wet-bulb 
thermometer reads 15°, so that the difference be- 
tween them is 5®, we have (since F = 1.8), 5^ X 1.8 
= 9'', which subtracted from 20° gives 11"* for the 
dew-point. 

Table 16 A gives the specific heat of moist air at 
about 50°, corresponding to different dew-points un- 
der a constant pressure of 76 cm. of mercury. The 
specific heat of dry air at 0"* (.2383) is the mean be- 



780 EXPLANATION OF TABLES. [No. 16 B. 

tween the results obtained by Regnault and E. 
Wiedemann. The other specific heats have been cal- 
culated by interpolation between the specific heats of 
air and of steam (.4805). 

Table 15 B gives the velocity of sound in atmos- 
pheric air calculated for different degrees of temper- 
ature and relative humidity, allowing for the effect of 
moisture on the density of air and on the ratio of the 
two specific heats of air under constant pressure and 
Huder constant volume. The barometric pressure 
(which has hardly a perceptible influence on the re- 
sult) was assumed to be 76 cm. of mercury. 

Table 15 O contains coefficients of interdiffusion 
of gases. The values (due to Maxwell) are taken 
from Everett's '^Units and Physical Constants" (Art. 
131). If two reservoirs filled with different gases 
are connected by a tube 1 cm, long, the numbers in 
Table 16 C show the mean velocity in cm. per sec. 
with which a stream of gas flows through the tube 
from each reservoir into the other. 

Table 16 is intended for the reduction of baro- 
metric readings, when given in inches, to centimetres. 
The last line of the table contains ^* proportional 
parts " (see under Table 1). 

Tables 16 A and B are intended for the reduction 
of barometric readings in cm. of mercury at 0"* to meg- 
adynes per sq, cm. A is calculated for a valu« of the 
acceleration of gravity (ff) equal to 980 cm. per eec. per 
see. ; B for the value ff = 981. The two tables differ 
by about 10 units in the last place. For values of g 
between 980 and 981, or just outside of these limits^ 



No. 18 a.] BAROMETRIC TABLES. 781 

results may be easily obtained by interpolation. Ex- 
ample : g = 980.4 ; required the value of 1 atmos- 
phere (76 cm^ in megadynes jper sq* cm. Answer, 
1.0127 + 10 X .4 = 1.0131 megadynes per sq. cm. 

Table 17 gives the elevation in metres above the 
sea-level corresponding to different barometric pres- 
sures at lO'' Centigrade. It has been calculated for 
dry air in latitude 45° by the formula 

k = 190790 (log. 76— log. p) (1 + .000,000 1 K). 

It is used in estimating heights by the barometer. 
Example I.: the mean barometric pressure is 70.0 
em, at the top of a hill rising out of the sea, the sides 
of the hill having a mean temperature of about lO'' ; 
required the height of the hill. Answer, about 681 
metres. Example II. : the barometric pressures at 
a given instant are 75.1 cm. at the foot of a hill, and 
74.2 cm. at the top of the hill, — the mean temperature 
being about 10°; required the height of the hill. 
Answer, 199 — 99 = 100 metres. 

Tables 17 A and 17 B give corrections in per 
cent to be added to or subtracted from the results of 
Table 17, according to the mean temperature and 
dew-point between the observing stations. Thus for 
a mean temperature 23° and the dew-point -f- 8** add 
4.6 -}- 0.4 = 5.0 % to all results. This would make 
the height of the hill in Example II., 105 (instead of 
100) metres. 

Table 18 a gives the correction in centimetres to 
be subtracted (on acconnt of expansion) from the 
reading of s^ mercurial barometer provided with a 



782 EXPLANATION OF TABLES. No. 18 c] 

brass scale reaching from its zero in the surface of 
mercury in the reservoir to the free surface of mer- 
cury in the tube. In calculating this table, the coeflS- 
cient of expansion of mercury was assumed to be 
.000180 + .000,000,036 ^- the value .000019 was 
taken for the coefficient of expansion of brass. Ex- 
ample I. : the mercurial column is 76 cm, long, meas- 
ured by a brass scale, its temperature is 20% we 
subtract 0.245 cm., and find 75.755 cm, for the value 
at 0°. Example II. : same as I. except that a glass 
scale is used ; corrected value the same less .016 cm.y 
that is, 75.739 cm. 

Table 18 6 gives the mean correction to be added 
to the apparent height of the mercurial column on ac- 
count of " capillarity," that is, the tendency of capil- 
lary or in general %mall tubes to depress a mercurial 
column (see Everett, 46 A, and Pickering, Table 12). 
The correction depends, however, not only upon the 
internal diameter of the barometer tube at the point 
where the mercury stands, but also upon the height 
of the "meniscus," which is different according to 
the direction in which the mercurial column has 
been moving. Corrections corresponding to differ- 
ent heights of the meniscus are taken from Kohl- 
rausch, Table 15, 6th ed. The results in this 
table differ widely from those quoted in the 2d 
edition. 

Table 18 c contains corrections for the pressure of 
mercurial vapor. They have been obtained by aver- 
aging the results of Regnault, Hagen, and Hertz, 
quoted in Landolt and Bornstein, Table 27. The 



[No. 20. DENSITY OF GASES. 783 

results ill question differ in some cases even in regard 
to the position of the decimal point. 

On account of the great discrepancy between the 
results obtained by different observers, barometric 
readings, even when corrected by Tables 18 a, 18 6, 
and 18 e, are significant only as far as hundredths of 
a centimetre. 

Tables 18 d^ 18 e, 18 /, and 18 g^ contain factors 
for the reduction of either the density or the volume 
of a gas to 0° or to 76 cm. Example I. : the density 
of coal-gas being .0005 at 20® and 75.0 cw., required 
its density at 0° and 76 cm. Answer, .0005 X 1.0734 
X 1.0133 = .00054 +. Example II. : the volume of 
a gas at 20"^ and 75 cm. is 100 cu. cm. ; required its 
volume at 0*" and 76 cm. Answer, 100 X 0.9316 
X 0.9868 = 91.9 cu. cm. If the gas were collected 
over water at 20° we should subtract 1.74 cm. (see 
Table 15) from the apparent pressure (75 cm.) and 
find 73.26 cm. for the pressure of the gas. This 
would give a factor .9640 instead of .9868, and a 
result 89.8 cu. cm. in the example above. 

Table 19 contains the density (or weight of 1 
cu. cm.) of air corresponding to different tempera- 
tures and pressures, and has been taken from Kohl- 
rausch, 2d ed., Table 6. It was calculated from 
Regnault's observations for latitude 45°. 

Table 20 contains corrections for the results in 
Table 19 to be applied on account of moisture. Ex- 
ample : required the density of air at 20° and 76 cm. 
pressure when the dew-point is -f- 4° Centigrade, 
Answer, .001204 — .000004 = .001200. 



784 EXPLANATION OF TABLES. [No. 23. 

Table 20 A contains the weight of air displaced 
by 1 gram of brass of the density 8.4, and is useful 
in calculating effective weights (§ 64). Example : a 
body is balanced by 100 grams of brass in air of the 
density .001200 ; required the effective weight of the 
body. Answer, 100 grams minus 100 X 0.000143 
grams, or 99.9857 grams. 

Table 21 contains factors for reducing apparent 
weighings with brass weights to vacuo. The factors 
correspond to diiBferent densities of the substance 
weighed, as well as of the air in which the weighing 
takes place. Example : a piece of glass of the den- 
sity 2.6 is balanced by 100 grams of brass, in air of 
the density .00120 ; required its true weight in vacuo. 
Answer, 100 X 1.00034 = 100.034 grams. 

Table 22 contains " apparent specific volumes " of 
water ; that is, the space in cubic centimetres occupied 
by a quantity of water weighing apparently 1 gram 
when counterpoised in air with brass weights of the 
density 8.4. The apparent specific volumes corres- 
pond to different temperatures and different condi- 
tions of atmospheric density, and are useful especially 
in calculations of volume or capacity in hydrostatics. 
Example: a flask holds apparently 1000 grams (1 
litre, nearly) of water at 20°, when weighed in air of 
the density .00120 ; required the capacity of the flask. 
Answer, 1000 X 1.00279 = 1002.79 cu. cm. 

Table 23 contains true "specific volumes'* of 
water ; that is, the space in cubic centimeti*es occu- 
pied at various temperatures by a quantity of water 
weighing actually 1 gram in vacuo. These values 



No. 26.] DENSITY OF LIQUIDS. 785 

are reciprocals of those in Table 24, and are to be 
used for the calculation of volumes corresponding to 
true weights in vacuo. Example: a piece of steel 
displaces 100 grams of boiling water; required its 
volume. Answer, 100 X 1.04311 = 104.811 cu. cm. 

Table 28 A gives the true specific volume of mer- 
cury at different temperatures, and is used like Table 
23. In calculating this table Kegnault's value 
(13.596) for the density of mercury at 0° was used, and 
a coefficient of expansion .000180 + .000,000,036 L 

Table 23 B gives apparent specific volumes of 
mercury when balanced by brass weights of the den- 
sity 8.4 in air of the density .0012. It is used, like 
Table 22, to calculate volumes and capacities. Exam- 
ple : the apparent weight of mercury required to fill 
a tube at 20** is 100 grams ; required the capacity 
of the tube. Answer, 100 X 0.073812 = 7.3812 
cu. cm. 

Table 24 contains the density of mercury at dif- 
ferent temperatures. The values are reciprocals of 
those contained in Table 23 A. 

Table 25 contains the density of water at differ- 
ent temperatures. A mean value, 1.00001, was taken 
for the maximum density of water (Kupfer's value is 
1.000013). The relative densities lie between the 
estimates of Rossetti and Volkmann, founded upon 
observations by Despretz, Hagen, Jolly, Kopp, 
Matthiessen, Pierre, and Rossetti. 

Table 26 contains the density of commercial 
glycerine, calculated from observations made in the 
Jefferson Physical Laboratory. 



786 EXPLANATION OF TABLES. [No. 31 A. 

Table 27 contains the density of dilute alcohol 
corresponding to different temperatures and different 
strengths. The values are a mean between results 
obtained by numerous observers. 

Table 28 gives the density, at 15°, of acids and 
saline solutions corresponding to various strengths, 
and is useful in making tests with a densimeter. See 
Storer's " Dictionary of Solubilities." Example : the 
density of some sulphuric acid is 1.807 at (about) 
15°; required its strength. Answer (about) 88 %. 

Table 29 gives the boiling-points of solutions of 
various strengths estimated by interpolation from 
data contained in Storer's "Dictionary of Solubilities." 
It furnishes an independent (and in processes of con- 
centration by boiling a very convenient) method of 
estimating the strength of such solutions. Thus a 
solution of hydrate of sodium boiling at 120° is 
known to have a strength of about 40 %. 

Table 30 gives the specific heats of solutions of 
different strengths at about 20°. It is useful in cer- 
tain processes in calorimetry (see ^^ 99-100). The 
numbers were obtained by interpolation from results 
contained in Landolt and Bornstein, Tables 71 and 72. 
Those nearest the observed values are printed in 
heavier type. 

Table 31 A gives the electrical conductivity of 
solutions at about 18^ It shows the current in am- 
peres which an electromotive force of one volt would 
cause to flow through a metre-cube of the solutions in 
question, or through a voltameter with plates 1 deci- 
metre square and 1 cm. apart, filled with these solu- 



No. 31 F.] PROPERTIES OF SOLUTIONS. 787 

tions, neglecting the effects of polarization. The re- 
sults must be multiplied by l(Mi (.000,000,000,01) 
to reduce them to the c. 6. s. system. The relative 
values of different results are probably accurate 
within 5 or 10 per cent, but their absolute values are 
much less reliable. 

Table 31 B gives Refractive and Dispersive in- 
dices corresponding to the sodium (D) line for solu- 
tions of different strengths, and was obtained by 
interpolation from results quoted by Landolt and 
Bornstein. 

Table 31 C is intended to facilitate the preparation 
of solutions of any desired strength, and for the cal- 
culation of per cent contents from the ratio of two 
constituents. Example: how many parts of salt 
must be added to 100 of water to make a 20 % solu- 
tion ? Let A = salt ; B = water, — the answer is 
25 parts. Example II. : a solution contains 100 parts 
of sulphuric acid to 150 of water; required its 
strength. Let A = water, B = sulphuric acid ; the 
answer is : 60 % water, 40 % sulphuric acid. 

Table 31 D gives coeflScients of diffusion of saline 
solutions in water at about 20°. The values were 
calculated from Graham's data quoted in Cooke's 
" Chemical Physics." Example : how much common 
salt would escape l3y diffusion into pure water from a 
20 % solution in 600,000 seconds through a layer 1.2 
cm. thick and 8 sq. cm. in cross section ? Answer, 
20 % of 600,000 X 8 X .000,0046 -5- 1.2 = 3.68 grams. 

Tables 31 E and F give the rotation in degrees of 
the plane of polarization of different kinds of light 



788 EXPLANATION OF TABLES. [No. 31 L. 

corresponding to the Fraunhofer lines A to H. E re- 
fers to dilute solutions having such a depth that a 
beam of light passing through an orifice 1 cm, square 
meets just one gram of the dissolved substance.^ F 
refers to the effect of plates 1 cm, thick. 

Table 31 G relates to the effect of a magnetic field 
in rotating the plane of polarization of light parallel 
to the lines of force. 

Table 81 H relates to (1) Magnetic Susceptibil- 
ity, (2) Saturation, and (3) Permanent Magnetism, — 
that is, the magnetic moment of a unit cube of differ- 
ent materials (1) in a unit magnetic field, (2) in an 
infinite magnetic field, and (3) in space after the 
magnetizing influence has been removed. The re- 
sults are taken from Everett and Ganot. 

Table 31 I contains some of Weisbach's results for 
the coefficient of friction of water moving with dif- 
ferent velocities through tubes not far from 1 cm, in 
diameter. The results have been reduced to the 
the c. G. s. system. 

Table 31 J gives coefficients of friction of solids 
on solids, taken from De Laharpe's ** Notes et Form- 
ules de Tlng^nieur." 

Table 31 K contains coefficients of reflection, ab- 
sorption, and transmission of radiant heat, from 
Ganot's Physics. 

Table 31 L contains estimates (by the author) of 
the heat radiated at different temperatures by 1 9q. 

1 The rotation is proportional, within more or less narrow limits, 
to the strength of the solution; but may vary widely outside of these 
limits. Cases of reyersal even occur. See Landolt and Bomsteia 



No. S3.] HEAT OF COMBUSTION. 789 

cm. of blackened or perfectly radiating surface sur- 
rounded by perfectly absorbing walls, or space at 0**. 
The table was calculated by the formula — 

q = log.-i (.0013 X (e° + 273^) — 1.8249) — .034, 

which was found to reconcile various well-known 
facts. Example : how much heat is required to main- 
tain 1 8q. cm. of platinum at its melting-point (1900°) 
for 1 sec? Answer, 10 (?) units.^ 

Tables 82 A and 32 B give heats of combustion' 
in oxygen and in chlorine respectively, from data 
quoted by Everett, by Landolt and Bornstein, and by 
other authorities. The chemical reactions are not in 
all cases such as actually take place ; but the table 
gives the heat which it is supposed would be developed 
if the reactions did take place. The last column 
gives the electromotive forces developed by or neces- 
sary to undo some of the reactions. Example: 2 
grams of hydrogen uniting with 16.0 grams of oxy- 
gen give out 69,000 units of heat, or 34.500 units 
per gram of hydrogen. This is equivalent to 1440 
megergs per mgr. of hydrogen consumed. To de- 
compose water, an electromotive force of 1.49 volts 
is required. 

Table 33 gives " heats of combination " involving 
more complicated chemical reactions than those which 
take place in simple combustion. 

1 This corresponds to 8 + Tolt-amp^res per candle>power. 
3 The heat of combustion of many substances can be inferred only 
from indirect processes. See experiment 38. 



790 EXPLANATION OF TABLES. [No. 38. 

Table 34 gives contact differences of electrical po- 
tential in volts. The data are taken from Everett's 
"Units and Physical Constants," Art. 206. Example : 
a piece of zinc is brought into contact with a piece of 
copper ; required the difference of electrical potential. 
Answer, the zinc is positively electrified with respect 
to the copper; the difference of potential is 0.750 
volts. 

Table 35 grves the electromotive force in volts of 
voltaic cells of various sorts. 

Table 36 gives the relation between electromotive 
forces and the length in mm, of the spark which 
they produce in ordinary atmospheric air, calculated 
from Everett, Art. 192. Example: an induction 
machine produces sparks 2.5 mm, long ; required the 
difference of potential between its poles at the in- 
stant. Answer, 9000 volts. Only the first two fig- 
ures are significant in this answer. 

Tables 37 a and h give specific resistances of con- 
ductors and insulators at 0°. The last column gives 
the per cent of increase of all these resistances due 
to a rise of temperature of 1® Centigrade. 

Table 38 gives the specific resistances of electro- 
lytes corresponding to various strengths. The resist- 
ances are in ohms, and apply to a centimetre cube of 
the liquid. The probable error of the results is about 
10 %. Relative values are probably not so inaccurate. 
Example : required the resistance of a cubical Dan- 
iell cell, with a plate of copper 10 X 10 cm., separated 
by a layer of 20 % (crystallized) sulphate of copper 
5 cm. deep, and by a layer of 20 % (crystallized) sul- 



No. 43.] ARBITRARY SCALES. 791 

phate of ziuc, also 5 cm. deep, from a plate of zinc 10 
X 10 cm. Answer : the resistance of the copper so- 
lution is 20 X 5 -5- (10 X 10) = 1 ohm ; that of the 
zinc solution is the same ; hence the resistance of the 
battery is 2 ohms. 

Table 39 gives a comparison between the Fahren- 
heit and Centigrade thermometers. Example : 98^.6, 
F = 87°.0, C. 

Table 40 (Pickering, Table 14) gives a comparison 
of hydrometer scales. Example: 40 Beaum^ for 
liquids lighter than water corresponds to the density 
0.830. 

Table 41 gives lengths of waves of light in air, 
intermediate between the numerous results quoted 
by Landolt and Bornstein. The probable error is 
about 1 unit in the last figure. Example : the Fraun- 
hofer lines Di and Dj, together designated Na (or D), 
are due to sodium (symbol, Na) and occur in the yellow 
of the spectrum. They correspond to number 50 on 
Bunsen's scale, to numbers 1003 and 1007 on (Bun- 
sen and) Kirchoffs scale, and have the wave-lengths 
0.00005896 cm. and 0.00005890 cm. respectively. 

Table 42 A refers to the imperial wire gauge adopt- 
ed by the Board of Trade (Stewart & Gee, I. B.). 

Table 42 B gives the Birmingham wire gauge 
(b. w. g.). The results are intermediate between 
those quoted in English, French, and German books. 
The probable error is about 1 unit in the last figure. 

Table 43 gives the number of vibrations corres- 
ponding to a series of musical notes on the tempered 
or isotonic scale, one half of a semitone apart. The 



792 EXPLANATION OF TABLES. [No. 44 C 

designation of some of these notes is given iu the 
left-hand or in the right-hand column. The former 
is to be used for "physical pitch," in which all 
powers of the number 2 represent the note C ; the 
right-hand column may be used for notes given by 
American instruments tuned to "concert pitch." 
The numbers between those corresponding to a given 
note in the first and last columns may be taken to 
represent the same note according to the .old Stutt- 
gart standard of pitch (A = 440, C = 264). Ex- 
ample : the " middle C " of an American piano (in 
the little octave), makes about 135.6 vibrations per 
second, and corresponds to C$ physical pitch. 

Table 44 A gives reductions of minutes and 
seconds to thousandths of a degree. The num- 
ber of minutes is first sought; the tenths of a 
degree will be found next to it. Then in the 
same section of the table (there are 6 sections) the 
nearest number of seconds is found, and next to 
it the hundredths and thousandths of a degree. Ex- 
ample : 23^2713" = 23^ + 0°.4 + 0°.064, nearly, or 
23^454. 

Table 44 B gives the correction to be added to 
dates in diflferent years to compare them with the 
year 1891. Thus, Jan. 1, lOh. Om. Os., A. M., 1899 ; 
corresponds to Jan. 1, lOh. Om. Os. -|- Ih. 29m. 28s. 
A.M. = Jan. 1, llh. 29m. 28s. A.M. 1891. The dec- 
lination of the sun and the equation of time will, for 
instance, be the same on these two dates. 

Table 44 G gives the gain of sidereal over mean 
solar time. 



No. 50.] ASTRONOMICAL TABLES, ETC. 793 

Table 44 D gives the sidereal time at Greenwich 
meaD noon for the 10th, 20th, and last day of every 
month of the year 1891. 

Table 44 E gives the semidiameter of the sun at 
different times in the year. 

Table 44 F gives the declination of the sun at 
Greenwich mean noon for the year 1891. The sign 
of the declination is to be found at the head of the 
several columns (+ north, — south). 

Table 44 G gives the ** equation of time" at 
Greenwich mean noon for the year 1891. The signs 
-f and — at the head of the columns and elsewhere 
show whether the sun is '' fast " or " slow " respect- 
ively ; + indicates that the sun passes the meridian 
before noon ; — after noon. 

Table 44 H gives certain astronomical data relat- 
ing to the solar system. 

Table 45 gives the Right Ascension and Declina- 
tion of some of the most important stars. 

Table 46 gives latitudes, longitudes, and eleva- 
tions of certain important places. 

Tables 47 and 48 give respectively the acceleration 
of gravity and the length of the seconds pendulum 
corresponding to different latitudes. Example : since 
the latitude of the Jefferson Physical Laboratory of 
Harvard College is 42^22^' or 42^.38, the acceleration 
of gravity is 980.37 cm. per sec, per see. 

Table 49 A and 49 B relate to the reduction of 
measures to and from the c. G. s. system. 

Table 50 contains mathematical and physical con- 
stants in frequent use. 

14 



SOURCES OF AUTHORITY. 

Tables 1-8 H were prepared, in so far as possible, 
from existing tables, by rejecting decimal places when 
necessary. More than 3,000 values (including all 
doubtful cases) were confirmed or determined by an 
independent calculation. The results were printed 
with the ordinary prex5autions to avoid typographical 
errors. Tables 4-5 A were obtained by transposing 
Pickering's tables 6-9. 

The logarithms of numbers from 1,000 to 10,000, 
in Table 6, were printed directly from a copy of the 
tables arranged by Mr. Oliver Whipple Huntington, 
of Harvard College. The proofs were compared 
with Bowditch's 5-place tables (Government Printing 
OflBce, Washington, 1882). The logarithms of num- 
bers from 10,000 to 11,000 were obtained by rejecting 
figures in Chamber's 7-8-place tables. A special in- 
vestigation was made in cases where the rejected 
figures were 50 or 500. Stereotype-proofs of all the 
logarithms were compared with the tables of Gauss. 
The table of probabilities as far as 5.0 is due to 
Chauvenet. The remainder of the table was the 
result of special calculation. 



SOURCES OF AUTHOUITY. 795 

The physical tables (Nos. 8 to 60) were compiled 
for the most part by the aid of results contained in 
Landoltand Bornstein's "Physico-Chemical Tables/'^ 
to which the reader is referred for a full exposition of 
the evidence upon which the selection of values has 
been made. The author wishes to thank Professors 
Laudolt and Bomstein for looking over his manu- 
script, for several useful suggestions, and for their 
kind permission to utilize their results. 

The author has quoted numerous data from Ever- 
ett's '* Units and Physical Constants '* (Macmillan, 
1886). He has also made use of information given by 
Professor Everett in choosing the unusually low value 
(4.17 X 10^ for the mechanical equivalent of heat. 

Among other books from which results have been 
taken are the following : Cooke's Chemical Philos- 
ophy, Deschanel's Natural Philosophy, Ganot's Phy- 
sics, Hoffmann's Tabellen fUr Chemiker, Kohlrausch's 
Leitfaden der Praktischen Physik, das Nautisches 
Jahrbuch, 1891, Pickering's Physical Manipulation, 
Stewart and Gee's Practical Physics, Storer's Dic- 
tionary of Solubilities, Trowbridge's New Physics, 
and Weisbach's Mechanics. 

These and other sources of authority have been 
acknowledged in connection with the explanation of 
the tables above ; but it was found impossible, in the 
limited space which could be devoted to the tables, 
to give authority for the separate data. It was, 

1 Physikaliflch-ChemfBche Tabellen Ton Dr. H. Landolt und Dr. 
Richard B5mstein, Professoren. Verlag von Julias Springer, Montbi- 
jou Platz 3» Berlin. 



796 SOURCES OF AUTHORITY. 

moreover, considered inexpedient to present to stu- 
dents, who would naturally be unaccustomed to 
weighing evidence, the conflicting statements from 
which the probable values of many of the physical 
constants have to be estimated by scientific men. 

Care has been taken, in all such cases, to give re- 
sults intermediate between those obtained by diflferent 
observers. To do this, a considerable number of 
figures was sometimes required ; but the use of fig- 
ures, not really significant^ has been in so far as pos- 
sible avoided. The last figure quoted in the results 
is in general the only one in regard to which a differ- 
ence of opinion was found to exist. 

It is regretted that, owing to the necessity of 
entrusting the composition to foreign printers, obvi- 
ous imperfections of type will be found, especially in 
the mathematical tables. In the expectation of re- 
printing these tables at no distant date, corrections 
and suggestion^ will be most gladly received. 



Table 1. Proportional Parts. 797 

M a *2 43 A «5 «6 47 «8 «0 M A ^2 ^S «4 «5 «6 «7 48 .0 

5 10 15 20 25 30 35 40 45 

5 10 15 20 26 31 36 41 46 

5 10 16 21 26 31 36 42 47 

5 11 16 21 27 32 37 42 48 

5 11 16 22 27 32 38 43 49 

6 11 17 22 28 33 39 44 50 
6 11 17 22 28 34 39 45 50 
6 11 17 23 29 34 40 46 51 
6 12 17 23 29 35 41 46 52 
6 12 18 24 30 35 41 47 53 

6 12 18 24 30 36 42 48 54 
6 12 18 24 31 37 43 49 55 
6 12 19 25 31 37 43 50 56 
6 13 19 25 Sf2 38 44 50 57 

6 13 19 26 32 38 45 51 58 

7 13 20 26 33 39 46 52 59 
7 13 20 26 33 40 46 53 59 
7 13 20 27 34 40 47 54 60 
7 14 20 27 34 41 48 54 61 
7 14 21 28 35 41 48 55 62 

7 14 21 28 35 42 49 56 63 
7 14 21 28 36 43 50 57 64 
7 14 22 29 36 43 50 58 65 
7 15 22 29 37 44 51 58 66 

7 15 22 30 37 44 52 59 67 

8 15 23 30 38 45 53 60 68 
8 15 23 30 38 46 53 61 68 
8 15 23 31 39 46 54 62 69 
8 16 23 31 39 47 55 62 70 
8 16 24 32 40 47 55 63 71 

8 16 24 32 40 48 56 64 72 
8 16 24 32 41 49 57 65 73 
8 16 25 33 41 49 57 66 74 
8 17 25 33 42 50 58 66 75 

8 17 25 34 42 50 59 67 76 

9 17 26 34 43 51 60 68 77 
9 17 26 34 43 52 60 69 77 
9 17 26 35 44 52 61 70 78 
9 18 26 35 44 53 62 70 79 
9 18 27 36 45 53 62 71 80 

9 18 27 36 45 54 63 72 81 
9 18 27 36 46 55 64 73 82 
9 18 28 37 46 55 64 74 83 
9 19 28 37 47 56 65 74 84 
9 19 28 38 47 56 66 75 85 

10 19 29 38 48 57 67 76 86 
10 19 29 38 48 58 67 77 86 \ 
10 19 29 39 49 58 68 78 87 
10 20 29 39 49 59 69 78 88 
10 20 30 40 50 59 69 79 89 
10 20 30 40 50 60 70 80 90 





1 

2 
8 

4 

















1 
1 
1 



11111 
11112 2 
12 2 2 2 3 
2 2 2 3 3 4 


50 
51 
52 
63 
54 


5 
6 

7 
8 
9 




2 

2 


2 
2 
2 
2 
3 


2 3 3 4 4 6 

2 3 4 4 5 5 

3 4 4 5 6 6 

3 4 6 6 6 7 

4 5 6 6 7 8 


55 
56 
57 
58 
59 


10 
11 
12 
13 
U 




2 
2 
2 
3 
3 


3 
3 
4 
4 
4 


4 6 6 7 8 9 

4 6 7 8 9 10 

5 6 7 8 10 11 

5 7 8 9 10 12 

6 7 8 10 11 13 


60 
61 
62 
63 
64 


15 
16 
17 
18 
10 


2 

2 
2 
2 
2 


3 
3 
3 
4 
4 


5 
5 
5 
6 
6 


6 8 9 11 12 14 

6 8 10 11 13 14 

7 9 10 12 14 15 

7 9 11 13 14 16 

8 10 11 13 15 17 


65 
66 
67 
68 
69 


20 
21 
22 
23 
24 


2 
2 
2 
2 
2 


4 
4 
4 
5 

6 


6 8 10 12 14 16 18 

6 8 11 13 15 17 19 

7 9 11 13 15 18 20 
7 9 12 14 16 18 21 
7 10 12 14 17 19 22 


70 
71 
72 
73 
74 


25 
26 
27 
28 
29 


3 
3 
3 
3 
3 


5 
6 
5 
6 
6 


8 10 13 15 18 20 23 
8 10 13 16 18 21 23 
8 11 14 16 19 22 24 

8 11 14 17 20 22 25 

9 12 15 17 20 23 26 


75 
76 
77 
78 
79 


30 
31 
32 
33 
34 


3 
3 
3 

a 
3 


6 9 12 15 18 21 24 27 
6 9 12 16 19 22 25 28 

6 10 13 16 19 22 26 29 

7 10 13 17 20 23 26 30 
7 10 14 17 20 24 27 31 


80 
81 
82 
83 
84 


35 
86 
87 
38 
39 




7 11 14 18 21 25 28 32 
7 11 14 18 22 25 29 32 

7 11 15 19 22 26 30 33 

8 11 15 19 23 27 30 34 
8 12 16 20 23 27 31 35 


85 
86 
87 
88 
89 


40 
41 
42 
43 
44 




8 12 16 20 24 28 32 36 
8 12 16 21 25 29 33 37 

8 13 17 21 25 29 34 38 

9 13 17 22 26 30 34 39 
9 13 18 22 26 31 35 40 


90 
91 
92 
93 
94 


45 
46 
47 
48 

49 
SO 


6 9 14 18 23 27 32 36 41 
5 9 14 18 23 28 32 37 41 
5 9 14 19 24 28 33 38 42 
S 10 14 19 24 29 34 38 43 

5 10 15 20 25 29 34 39 44 

6 10 15 20 26 30 35 40 45 


95 
96 
97 
98 
99 
iOO 



798 Powen. Table 2. 






eo 


0.00 


• 





60 


.0200 


7.07 


2600 


125000 


1 


1.000 


1.00 


1 


1 


61 


196 


7.14 


2601 


132651 


2 


0.500 


1.41 


4 


s 


62 


192 


7.21 


2704 


140608 


8 


333 


1.73 


9 


27 


63 


189 


7.28 


2809 


148877 


4 


250 


2.00 


16 


64 


64 


185 


7.35 


2916 


157464 


6 


0.200 


2.24 


25 


125 


65 


.0182 


7.42 


S025 


166375 


6 


167 


2.45 


36 


216 


66 


179 


7.48 


3136 


175616 


7 


143 


2.65 


49 


343 


67 


176 


7.65 


3249 


185193 


a 


125 


2.83 


64 


612 


68 


172 


7.62 


3364 


195112 


9 


111 


300 


81 


729 


69 


169 


7.68 


3481 


205379 


10 


0.100 


3.16 


100 


1000 


60 


.016? 


7.75 


3600 


216000 


11 


.0909 


3.32 


121 


1331 


61 


164 


7.81 


3721 


226981 


12 


833 


346 


144 


1728 


62 


161 


7.87 


3844 


238328 


13 


769 


3.61 


169 


2197 


63 


159 


7.94 


3969 


250047 


U 


714 


3.74 


196 


2744 


64 


156 


800 


4096 


262144 


15 


.0667 


3.87 


225 


3375 


65 


.0154 


8.06 


4225 


274626 


16 


625 


4.00 


256 


4096 


66 


152 


8.12 


4356 


287496 


17 


688 


4.12 


289 


4913 


67 


149 


8.19 


4489 


300763 


18 


556 


4.24 


324 


6832 


68 


147 


8.25 


4624 


314432 


19 


526 


4.36 


361 


6859 


69 


145 


8.31 


4761 


328509 


20 


.0500 


4.47 


400 


8000 


70 


.0143 


8.37 


4900 


343000 


21 


476 


4.58 


441 


9261 


71 


141 


8.43 


5041 


357911 


22 


455 


4.69 


484 


10648 


72 


139 


8.49 


6184 


373218 


23 


435 


4.80 


529 


12167 


73 


137 


854 


5329 


389017 


24 


- 417 


490 


676 


13824 


74 


135 


8 60 


5476 


405224 


25 


.0400 


5.00 


625 


15625 


75 


0133 


8.66 


6625 


421875 


26 


385 


5.10 


676 


17576 


76 


132 


8.72 


5776 


438976 


27 


370 


6.20 


729 


19683 


77 


130 


877 


5929 


456533 


28 


357 


5.29 


784 


21952 


78 


128 


8 83 


6084 


474552 


29 


345 


5.39 


841 


24389 


79 


127 


8.89 


6241 


493039 


80 


.0333 


5.48 


900 


27000 


80 


.0125 


8.94 


6400 


612000 


31 


323 


5.57 


961 


29791 


81 


123 


0.00 


6561 


631441 


82 


313 


6.66 


1024 


32768 


82 


122 


9.06 


6724 


651368 


83 


303 


6.74 


1089 


35937 


83 


120 


9.11 


6889 


571787 


81 


294 


6.83 


1156 


39304 


84 


119 


9.17 


7056 


692704 


85 


.0286 


6.92 


1325 


42875 


85 


.0118 


9.22 


7225 


614125 


86 


278 


6.00 


1296 


46666 


86 


116 


9.27 


7396 


636056 


87 


270 


6.08 


1369 


60653 


87 


115 


9.33 


7569 


658503 


88 


263 


6.16 


1441 


64872 


88 


114 


9.38 


7744 


681472 


89 


256 


6.24 


1521 


69319 


89 


112 


9.43 


7921 


704969 


40 


.0250 


6.32 


1600 


64000 


90 


.0111 


9.49 


8100 


729000 


41 


244 


640 


1681 


68921 


91 


110 


9.64 


8281 


753671 


42 


238 


6.48 


1764 


74088 


92 


109 


9.59 


8464 


778688 


48 


233 


6.56 


1849 


79507 


93 


108 


9.64 


8649 


804357 


44 


227 


6.63 


1936 


85184 


94 


106 


9.70 


8836 


830584 


4S 


.0222 


6.71 


2025 


ftn25 


96 


.0105 


0.75 


9025 


857376 


46 


217 


6.78 


2116 


97336 


96 


101 


9.80 


9216 


884736 


47 


213 


6.86 


2209 


103823 


97 


103 


9.85 


9409 


912678 


48 


208 


6.93 


2304 


110692 


98 


102 


9.90 


9604 


941192 


49 


204 


7.00 


2401 


117649 


09 


101 


9.95 


9301 


970299 



50 .0200 7.07 2500 125000 100 .0100 10.00 10000 1000000 



Table 2. 






Circles, eto. 








799 


Diaa- 


$. Log- 


/. Cireui- 


;. trit 


k. Vilaai 


iru- 


t. li|> I 


f. CIrna. 


f. tnt k. VilMM 


itir 


arltho 


firiaci 


II Circli 


If Sfhiri 


•tir 


tritka 


Iirittl 


olCircto ifSilwri 


.0 


— oo 


0.00 


0.00 


.000 


s.o 


0.609 


15.71 


19.6 


65 


.1 


rooo 


31 


01 


.001 


6.1 


708 


16.02 


20.4 


- 69 


.2 


301 


63 


03 


.004 


6.2 


716 


16.34 


21.2 


74 


.3 


477 


94 


07 


.014 


6.3 


724 


16.65 


22.1 


78 


.4 


602 


1.26 


13 


.034 


6.4 


732 


16.96 


22.9 


82 


.5 


T699 


1-57 


0.20 


.065 


6.6 


0.740 


17.28 


23.8 


87 


.0 


778 


1.88 


28 


.113 


6.6 


748 


17.59 


24.6 


92 


.7 


845 


2.20 


38 


.180 


6.7 


756 


17.91 


25.5 


97 


.8 


903 


2.51 


50 


.268 


6.8 


763 


18.22 


26.4 


102 


.9 


954 


2.83 


64 


.382 


6.9 


771 


18.54 


27.3 


108 


1.0 


0.000 


3.14 


0.79 


0.S2 


6.0 


0.778 


18.85 


28.3 


113 


1.1 


041 


3.46 


0.95 


70 


61 


785 


19.16 


29.2 


119 


1.2 


079 


3.77 


1.13 


90 


«.a 


792 


19.48 


30.2 


125 


1.3 


114 


4.08 


1.38 


1.15 


6.3 


799 


19.79 


31.2 


131 


1.4 


146 


4.40 


1.54 


1.44 


6.4 


806 


20.11 


32.2 


137 


1.5 


0.176 


4.71 


1.77 


1.77 


66 


0.818 


20.42 


33.2 


144 


1.6 


204 


5.03 


201 


2.14 


6.6 


820 


20.73 


34.2 


161 


1.7 


230 


5.34 


2.27 


2.57 


6.7 


826 


21.05 


35.3 


157 


1.8 


255 


5.65 


S.54 


3.05 


6.0 


833 


21.36 


36.3 


165 


1.9 


279 


5.97 


2.81 


3.59 


6.9 


839 


21.68 


37.4 


172 


2.0 


0.301 


6.28 


3.14 


4.19 


7.0 


0846 


21.99 


38.6 


180 


21 


322 


6.60 


3.46 


4.85 


7.1 


851 


2231 


39.6 


187 


2.2 


342 


6.91 


3.80 


6.58 


7.2 


857 


22.62 


40.7 


196 


2.3 


362 


7.23 


4.15 


637 


78 


863 


22.93 


41.9 


204 


2.4 


380 


7.54 


4.52 


721 


7.4 


869 


23.25 


43.0 


212 


2.5 


0.398 


7.85 


4.91 


8.2 


7.5 


0.875 


23.66 


44.2 


221 


2.6 


415 


8.17 


5.31 


9.2 


7.6 


881 


23.88 


45.4 


230 


2.7 


431 


8.48 


6.73 


10.3 


7.7 


886 


2419 


46.6 


239 


2.8 


447 


8.80 


6.16 


11.5 


7.8 


892 


24.50 


47.8 


248 


2.9 


462 


9.11 


6 61 


12.8 


7.9 


898 


2482 


49.0 


268 


3.0 


0.477 


9.42 


7.C7 


14.1 


8.0 


0.903 


25.13 


50.3 


268 


3.1 


491 


9.74 


7.55 


16.6 


8.1 


91)8' 


25.45 


615 


278 


3.*.^ 


505 


1005 


8.04 


17.2 


C.2 


914 


25.76 


52 8 


289 


3.3 


519 


10.37 


8.56 


18.8 


8.3 


919 


2.6.08 


•641 


299 


3.4 


531 


10.68 


9.08 


20.6 


8.4 


924 


26 39 


65.4 


310 


3.5 


0.544 


11.00 


9.6 


22.4 


8.5 


0.929 


26.70 


56.7 


822 


36 


556 


11.31 


10 2 


24.4 


8.6 


934 


27.02 


58.1 


333 


3.7 


568 


11.62 


10.8 


26 6 


8.7 


940 


27.33 


59.4 


345 


3.8 


580 


11.94 


11.3 


28.7 


8.8 


941 


27.65 


60 8 


357 


3.9 


591 


12.25 


11.9 


31.1 


8.9 


949 


27.96 


62.2 


369 


4.0 


0.602 


12.67 


12.6 


33.6 


9.0 


0.954 


28.27 


63.6 


882 


4.1 


613 


12.88 


13.2 


36.1 


9.1 


959 


28.69 


65.0 


396 


4.2 


623 


13.19 


13 9 


38.8 


0.2 


964 


28.90 


66.6 


408 


4.3 


633 


13.51 


14.6 


416 


9.3 


968 


29.22 


67.9 


421 


4.4 


643 


13.82 


15.2 


44.6 


9.4 


973 


29.53 


69.4 


435 


4.5 


0.653 


14.14 


15.9 


47.7 


05 


0.978 


29.86 


70.9 


449 


4.6 


663 


14.45 


16.6 


610 


96 


982 


30.16 


72.4 


463 


4.7 


672 


14.77 


17 3 


64.4 


9.7 


987 


3047 


73.9 


478 


4.8 


681 


15 08 


18.1 


57.9 


98 


991 


30.79 


75.4 


493 


4.9 


690 


15.39 


18.9 


61.6 


99 


996 


31.10 


77.0 


608 


6.0 


0.699 


15.71 


19.6 


65.4 


iO.O 


1.000 


31.42 


78 5 


524 



800 Trigonometrio Funotions. Tabled. 

a.Aigli. 5.Taigiit §, Are. 4. ChBrl •. Siai. f. C&sioi. ff, Rato of ft.Sicaot. <.CoinpIi 

Vibratioo. miBt. 



0» 


0.000 


0.000 


0.000 


0.000 


1.000 


1.00000 


1.000 


90» 


1 


017 


017 


017 


■017 


1.000 


1.00000 


1.000 


80 


2 


035 


035 


035 


035 


0.999 


0.99998 


1.001 


88 


8 


052 


052 


032 


052 


999 


99996 


1.001 


87 


4 


070 


070 


070 


070 


998 


99992 


1.002 


86 


S 


0.087 


0.087 


0.087 


0.087 


0.996 


0.99988 


1.004 


85 


6 


105 


105 


105 


106 


995 


99983 


1.006 


84 


7 


123 


122 


122 


122 


993 


99977 


1.008 


83 


8 


141 


140 


140 


139 


990 


99970 


1.010 


8S 


9 


158 


157 


157 


156 


988 


99961 


1.012 


81 


10 


0.176 


0.175 


0.174 


0.174 


0.986 


0.99952 


1.015 


80 


11 


194 


192 


192 


191 


982 


99942 


1.019 


79 


12 


213 


209 


209 


208 


978 


99931 


1.022 


78 


13 


231 


227 


226 


225 


974 


99920 


1.026 


77 


14 


249 


244 


244 


242 


970 


99907 


1.031 


76 


15 


0.268 


0.262 


0.261 


0.259 


0.966 


0.99893 


1.035 


75 


16 


287 


279 


278 


276 


961 


99878 


1.040 


74 


11 


306 


297 


296 


292 


956 


99862 


1.046 


78 


18 


325 


314 


313 


309 


951 


99846 


1.051 


72 


19 


344 


332 


330 


326 


946 


99828 


1.058 


71 


20 


0.364 


0.349 


0.347 


0342 


0.940 


0.99810 


1.064 


70 


21 


384 


367 


364 


36jS 


934 


99790 


1.071 


69 


22 


404 


384 


382 


876 


927 


99770 


1079 


68 


23 


424 


401 


399 


391 


921 


99749 


1.086 


67 


24 


445 


419 


416 


407 


914 


99726 


1.095 


66 


25 


0466 


0.436 


0.433 


0.423 


0.906 


0.99703 


1.103 


65 


26 


488 


454 


450 


438 


899 


99678 


1.113 


64 


27 


610 


471 


467 


454 


891 


99653 


1.122 


63 


28 


532 


489 


484 


469 


883 


99627 


1.133 


62 


29 


551 


506 


601 


485 


876 


99600 


1.143 


61 


80 


0.577 


0.524 


0.618 


0.500 


0.866 


0.99572 


1.156 


60 


81 


601 


541 


534 


015 


857 


99543 


1.167 


59 


a-i 


625 


559 


651 


630 


848 


99513 


1.179 


68 


88 


649 


576 


568 


545 


839 


99482 


1.192 


57 


84 


675 


693 


685 


659 


829 


99450 


1.206 


56 


85 


0.700 


0.611 


0.601 


0.674 


0.819 


0.99417 


1.221 


55 


86 


727 


628 


618 


588 


809 


99384 


1236 


64 


87 


754 


646 


635 


602 


799 


99349 


1.252 


53 


88 


781 


663 


651 


616 


788 


99314 


1.269 


62 


89 


810 


681 


668 


629 


777 


99277 


1.287 


51 


40 


0.839 


0.698 


0.684 


0.643 


0.766 


0.99239 


1.305 


60 


41 


869 


716 


700 


656 


756 


99200 


1.325 


49 


42 


900 


733 


717 


669 


743 


99161 


1.346 


48 


43 


933 


750 


733 


682 


731 


99121 


1.267 


47 


44 


966 


768 


749 


695 


719 


99079 


1.390 


46 


45» 


l.OCD 


0.786 


0.766 


0.707 


0.707 


0.99037 


1.414 


45* 



Table 3. Trigonometric Functions. 801 

a.Aiglf. ^Tugut. e, kt. 4. Cliorl •. Siai. A Cain. y. Cmrsioi *.Sicait iCiaplt. 



45« 


1.000 


0.785 


0.765 


0.707 


0.707 


0.293 


1.414 


45« 


46 


1.036 


0.803 


781 


719 


695 


281 


1.440 


44 


47 


1.072 


0.820 


797 


731 


682 


269 


1.466 


43 


48 


i.ni 


0.838 


813 


743 


669 


257 


1.494 


42 


49 


1.150 


0.855 


829 


755 


656 


246 


1524 


41 


SO 


1.192 


0.873 


0.845 


0.766 


0643 


0.234 


1.556 


40 


51 


1.235 


0.890 


861 


777 


629 


223 


1.589 


89 


62 


1.280 


0.908 


877 


788 


616 


212 


1.624 


38 


63 


1.327 


0.925 


892 


799 


602 


201 


1.662 


37 


64 


1.376 


0.942 


908 


809 


588 


191 


1701 


86 


66 


1.428 


0.960 


0.923 


0.819 


0.674 


0.181 


1.743 


86 


66 


1.483 


0977 


939 


829 


659 


171 


1.788 


84 


6'? 


1.540 


0.995 


954 


839 


645 


161 


1.836 


33 


68 


1.600 


1.012 


970 


848 


630 


162 


1.887 


82 


69 


1.664 


1.030 


985 


857 


616 


143 


1942 


81 


60 


1.732 


1.047 


1.000 


0.866 


0.500 


0.134 


2.000 


30 


61 


1.804 


1.065 


1.015 


875 


485 


125 


2.063 


29 


62 


1.881 


1.082 


1.030 


883 


469 


117 


2.130 


28 


63 


1.963 


1.100 


1.045 


891 


454 


109 


2.203 


27 


«4 


2.050 


1.117 


1.060 


899 


438 


101 


2.281 


26 


66 


2.145 


1.134 


1.075 


0906 


0.423 


0.094 


2 366 


25 


66 


2.246 


1.152 


1.089 


914 


407 


086 


2.459 


24 


67 


2.356 


1.169 


1.104 


921 


391 


079 


2559 


23 


68 


2.475 


1.187 


1.118 


927 


376 


073 


2.669 


22 


69 


2.605 


1.204 


1.133 


934 


358 


066 


2.790 


21 


70 


2.747 


1.222 


1.147 


0.940 


0.342 


0.060 


2.924 


20 


71 


2.904 


1.239 


1.161 


946 


326 


054 


3.072 


19 


72 


3.078 


1.257 


1.176 


951 


309 


049 


3.236 


18 


73 


3.271 


1.274 


1.190 


956 


292 


044 


3.420 


17 


74 


3.487 


1.292 


1.204 


961 


276 


039 


3.628 


16 


75 


3.732 


1.309 


1218 


0.966 


0.259 


0.034 


3864 


15 


76 


4.011 


1.326 


1.231 


970 


242 


030 


4.134 


14 < 


77 


4.331 


1344 


1.245 


974 


225 


026 


4.445 


13 


73 


4.705 


1.361 


1.259 


978 


208 


022 


4.810 


12 


79 


6.145 


1.379 


1.272 


982 


191 


018 


6.241 


11 


80 


5.671 


1.396 


1.286 


985 


0.174 


0.0152 


6.759 


10 


81 


6.314 


1414 


1299 


988 


156 


0123 


6392 


9 


83 


7.115 


1.431 


1.312 


990 


139 


0097 


7.185 


8 


83 


8.144 


1.449 


1.325 


993 


122 


0075 


8.206 


7 


8i 


9.514 


1.466 


1338 


995 


105 


0055 


9.567 


6 


.85 


1143 


1.484 


1.351 


0.996 


0.087 


0.00381 


11.47 


S 


86 


1430 


1.501 


1364 


998 


070 


00244 


14.34 


4 


87 


19.08 


1.518 


1.377 


999 


052 


00137 


19.11 


8 


88 


28.64 


1.536 


1.389 


999 


035 


00061 


28.65 


2 


89 


67.29 


1.553 


1402 


1.000 


017 


00015 


67.30 


1 


90* 


oo 


1.571 


1.414 


1.000 


0.090 


0.00000 


oe 


«• 



• 



802 BeciprooaLi. Table 3, a. 

JI6 128456780 IL 

1.0 1.0000 9901 9804 9709 9615 9524 9434 9346 9259 9174 ^ 

1.1 0.9091 9009 8929 8850 8772 8696 8621 8547 8475 8403 « 

1.2 8333 8264 8197 8130 8065 8000 7937 7874 7813 7752 ^ 

1.3 7692 7634 7576 7519 7463 7407 7353 7299 7246 7194 » 

1.4 7143 7092 7042 6993 6944 6897 6849 6803 6757 6711 ^ 

1.5 0.6667 6623 6579 6536 6494 6452 6410 6369 6329 6289 ^ 

1.6 6250 6211 6173 6135 6098 6061 6024 59S8 5952 5917 " 

1.7 5882 5848 5814 5780 5747 5714 5682 5650 5618 5587 ^ 

1.8 5556 5525 5495 5464 5435 5405 5376 5348 5319 5291 » 

1.9 5263 5236 5208 5181 5155 5128 5102 5076 5051 5025 » 

2.0 0.5000 4975 4950 4926 4902 4878 4854 4831 4808 4785 ^ 

2.1 4762 4739 4717 4695 4673 4651 4630 4608 4587 4566 » 

2.2 4545 4525 4505 4484 4464 4444 4425 4405 4386 4367 ^ 

2.3 4348 4329 4310 4292 4274 4255 4237 4219 4202 4184 '* 

2.4 4167 4149 4132 4115 4098 4082 4065 4049 4032 4016 '^ 

2.5 0.4000 3984 3968 3953 3937 3922 3906 3891 3876 3861 '» 

2.6 3846 3831 3817 3802 3788 3774 3759 3745 3731 3717 " 

2.7 3704 3690 3676 3663 3650 3636 3623 3610 3597 3584 " 

2.8 3571 3559 3546 3534 3521 3509 3496 3484 3472 3460 M 

2.9 3448 3436 3425 3413 8401 3390 3378 3367 3356 3344 M 

3.0 0.3333 3322 3311 3300 3289 3279 3268 3257 3247 3236 ^^ 

3.1 3226 3215 3205 3195 3185 3175 3165 3155 3145 3135 

3.2 3125 3115 3106 3096 3086 3077 3067 3058 3049 3040 

3.3 3030 3021 3012 3003 2994 2985 2976 2967 2959.2950 * 

8.4 2941 2933 2924 2915 2907 2899 2890 2882 2874 2865 ' 

3.5 0.2857 2849 2841 2833 2825 2817 2809 2801 2793 2786 :« 

8.6 2778 2770 2762 2755 2747 2740 2732 2725 2717 2710 

3.7 2703 2695 2688 2681 2674 2667 2660 2653 2646 2639 i^ 

3.8 2632 2625 2618 2611 2604 2597 2591 2584 2577 2571 

3.9 2564 2558 2551 2545 2538 2532 2525 2519 2513:2506 ' 

4.0 0.2500 2494 2488 2481 2475 2469 2463 2457 2451 2445 « 

4.1 2439 2433 2427 2421 2415 2410 2404 2398 2392 2387 * 

4.2 2381 2375 2370 2364 2358 2353 2347 2342 2336 2331 , 

4.3 2326 2320 2315 2309 2304 2299 22D4 2288 2283 2278 ' 

4.4 2273 2268 2262 2257 2252 2247 2242 2237 2232 2227 » 

4.5 0.2222 2217 2212 2208 2203 2198 2193 2188 2183 2179 • 

4.6 2174 2169 2165 2160 2155 2151 2146 2141 2137 2132 . 

4.7 2128 2123 2119 2114 2110 2105 2101 2096 2092 2(;c)8 

4.8 2083 2079 2075 2070 2066 2062 2058 2053 2049 2045 

4.9 2041 2037 2033 2028 2024 2020 2016 2012 2008 2004 

5.0 0.2000 1996 1992 1988 1984 1980 1976 1972 1969 1965 « 

5.1 1961 1957 1953 1949 1946 1942 1G38 1934 1931 1927 

5.2 1923 1919 1916 1912 1908 1905 1901 1898 1894 1890 ' 

5.3 1887 1883 1880 1876 1873 1869 1866 1862 1859 1855 

5.4 1852 1848 1845 1842 1838 1835 1832 1828 1825 1821 

5.5 0.1818 1815 1812 1808 1805 1802 1799 1795 1792 1789 

5.6 1786 1783 1779 1776 1773 1770 1767 1764 1761 1757 

6.7 1754 1751 1748 1745 1742 1739 1736 1733 1730 1727 » 

5.8 1724 1721 1718 1715 1712 1709 1706 1704 1701 1698 

5.9 1695 1692 1689 1686 1684 1681 1678 1C75 1672 1669 
6.0 0.1607 1664 1661 1658 1656 1653 1650 1647 1645 1642 



Table 3, A. 






Beoiprooals. 






803 


M 


1 


2 


3 4 5 6 


7 


8 


9 Oil. 



6.0 0.16667 16639 16611 16584 16556 16529 16502 16474 16447 16420 ^ 

6.1 16393 16367 16340 16313 16287 16260 16234 16207 16181 16155 '' 

6.2 16129 16103 16077 16051 16026 16000 15974 15949 15924 15898 ^"^ 

6.3 15873 15848 15823 15798 15773 15748 15723 15699 15674 15649 '' 

6.4 15625 15601 15576 15552 15528 15504 15480 15456 15432 15408 '' 

6.5 0.15385 15361 15337 15314 15291 15267 15244 15221 15198 15175 ^ 

6.6 15152 15129 15106 15083 15060 15038 15015 14992 14970 14948 ""^ 

6.7 14925 14903 14881 14859 14837 14815 14793 14771 14749 14728 ^ 

6.8 14706 14684 14663 14641 14620 14599 14577 14556 14535 14514 " 

6.9 14493 14472 14451 14430 14409 14388 14368 14347 14327 14306 >' 

7.0 0.14286 14265 14245 14225 14205 14184 14164 14144 14124 14104 »> 

7.1 14085 14065 14045 14025 14006 13986 13966 13947 13928 13908 

7.2 13889 13870 13850 13831 13812 13793 13774 13755 13736 13717 » 

7.3 13699 13680 13661 13643 13624 13605 13587 13569 13550 13532 

7.4 13514 13495 13477 13459 13441 13423 13405 13387 13369 13351 " 

7.5 013333 13316 13298 13280 13263 13245 13228 13210 13193 13175 

7.6 13158 13141 13123 13106 13089 13072 13055 13038 13021 13004 " 

7.7 12987 12970 12953 12937 12920 12903 12887 12870 12853 12837 

7.8 12821 12804 12788 12771 12755 12739 12723 12706 12690 12674 >* 

7.9 12658 12642 12626 12610 12594 12579 12563 12547 12531 12516 

8.0 0.12500 12484 12469 12453 12438 12422 12407 12392 12376 12361 

8.1 12346 12330 12315.12300 12285 12270 12255 12240 12225 12210 '^ 

8.2 12195 12180 12165 12151 12136 12121 12107 12092 12077 12063 

8.3 12048 12034 12019 12005 11990 11976 11962 11947 11933 11919 

8.4 11905 11891 11876 11862 11848 11834 11820 11806 11792 11779 '« 

8.5 0.11765 11751 11737 11723 11710 11696 11682 11669 11655 11641 

8.6 11628 11614 11601 11587 11574 11561 11547 11534 11521 11507 

8.7 11494 11481 11468 11455 11442 11429 11416 11403 11390 11377 '' 

8.8 11364 11351 11338 11325 11312 11299 11287 11274 11261 11249 

8.9 11236 11223 11211 11198 11186 11173 11161 11148 11136 11123 

9 0.11111 11099 11086 11074 11062 11050 11038 11025 11013 11001 
9 1 10989 10977 10965 10953 10941 10929 10917 10905 10893 10881 " 

9.2 10870 10858 10846 10834 10823 10811 10799 10787 10776 10764 

9.3 10753 10741 10730 10718 10707 10695 10684 10672 10661 10650 
9 4 10638 10627 10616 10604 10593 10582 10571 10560 10549 10537 

9.5 0.10526 10515 10504 10493 10482 1047^1 10460 10449 10438 10428 '' 

9.6 10417 10406 10395 10384 10373 10363 10352 10341 10331 10320 
9 7 10309 10299 10288 10277 10267 10256 10246 10235 10225 10215 
9.8 10204 10194 10183 10173 10163 10152 10142 10132 10121 10111 
9 9 10101 10091 10081 10070 10060 10050 10040 10030 10020 10010 

10.0 0.10000.09990 9980 9970 9960 9950 9940 9930 9921 9911 '<> 

10.1 .09901 9891 9S81 9872 9862 9852 9843 9833 9823 9814 

10.2 9804 9794 9785 9775 9766 9756 9747 9737 9728 9718 

10.3 9700 9699 9690 9681 9671 9662 9653 9643 9634 9625 

10.4 9615 9606 9597 9588 9579 9569 9560 9551 9542 9533 

10.5 0.09524 9515 9506 9497 9488 9479 9470 9461 9452 9443 > 

10.6 9434 9425 9416 9407 9398 9390 9381 9372 9363 9355 

10.7 9346 9337 9328 9320 9311 9302 9294 9285 9276 9268 

10.8 9259 9251 9242 9234 9225 9217 9208 9200 9191 9183 

10.9 9174 9166 9158 9149 9141 9132 9124 9116 9107 9099 
11.0 0.09091 9083 9074 9066 9058 9050 9042 9033 9025 9017 • 



804 Squares. Table a. c. 

1.0 1.000 1.020 1.040 1.061 1.082 1.103 1.124 1.145 1.166 1.188''^ 

1.1 1.210 1.282 1.254 1.277 1.300 1.323 1.346 1.369 1.392 1.416 » 
1.3 1.440 1.464 1.488 1.513 1.538 1.563 1.588 1.613 1.638 I.C64 » 

1.3 1.690 1.716 1.742 1.769 1.796 1.823 1.850 1.877 1.S04 1.932** 

1.4 1.960 1.988 2.016 2.045 2.074 2.103 2.132 2.161 2.190 2.^20^ 

1.5 2.250 2.280 2.310 2.341 2.372 2.403 2434 2.465 2.496 2.528 '> 

1.6 2.560 2 592 2.624 2.657 2.690 2.723 2 756 2.789 2.822 2.856 " 
1> 2.890 2.924 2.958 2.993 3.028 3.063 3 098 3.133 3.168 3.204 » 

18 3.240 3.276 3.312 ^.349 3.386 3.423 3.460 3.497 3.534 3.572 "^ 

19 3.610 3.648 ^.686 3.725 3.764 3.803 3.842 3.881 3.920 3.960 ** 

2.0 4.000 4.04) 1.080 4.121 4.162 4.203 4.244 4.285 4.326 4.368 '^ 

2.1 4.410 4.452 1494 4.537 4.580 4.623 4.666 4.709 4.752 4.796*' 

2.2 4.840 4 884 4.928 4.973 5.018 5.063 5.108 5.153 5.198 5.244 "> 
2 8 5.290 5.338 5.382 5.429 5.476 5.523 5.570 5.617 5.664 5.712*' 
2.4 5.760 5.808 5.856 5.905 5.954 6.003 6.052 6.101 6.150 6.200^ 



2.5 


6.250 


6.300 


6.350 


6.401 


6.452 


6.603 


6.564 


6.605 


6.656 


6 708" 


2.6 


6.760 


6.812 


6.864 


6.917 


6.970 


7.023 


7.076 


7.129 


7.182 


7 286" 


2.7 


7.290 


7.344 


7.398 


7.453 


7.508 


7.563 


7.618 


7.673 


7.728 


7.784 » 


2.8 


7.840 


7.896 


7.952 


8.009 


8.066 


8.123 


8.180 


8.237 


8.294 


8 352« 


2.9 


8.410 


8.468 


8.526 


8.685 


8.644 


8.703 


8.762 


8.821 


8.880 


8.940 " 


3.0 


9.000 


9.060 


9.120 


9.181 


9.242 


9.303 


9.364 


9.425 


9.486 


9.648 « 


3.1 


9.610 


9.672 


9.734 


9.797 


9.860 


9.923 


9.986 


10.05 


10.11 


10.18- 


3.2 


10.24 


10.30 


10.37 


10.43 


10.50 


10.56 


10.63 


10.69 


10.76 


10.82 


3.3 


10.89 


10.96 


11.02 


11.09 


11.16 


11.22 


11.29 


11.36 


11.42 


11.49 


3.4 


11.56 


11.63 


11.70 


11.76 


11.83 


11.90 


11.97 


12.04 


12.11 


12.18 


3.5 


12.25 


12.32 


12.39 


12.46 


12.53 


12.60 


12.67 


12.74 


12.82 


12.89 ' 


3.6 


12.96 


13.03 


13.10 


13.18 


13.25 


13.32 


13.40 


13.47 


1364 


13.62 


3.7 


13.69 


13.76 


13.84 


13.91 


13.99 


14.06 


14.14 


14.21 


14.29 


14.36 


8.8 


14.44 


14.52 


14.69 


14.67 


14 75 


14.82 


14.90 


14.98 


16.05 


16.1S 


3.9 


15.21 


15.29 


15.37 


16.44 


15.62 


15.60 


15.68 


15.76 


16.84 


15.92 


4.0 


16.00 


16.08 


16.16 


16.24 


16.32 


16.40 


16.48 


16.66 


16.65 


16.78 ' 


4.1 


16.81 


16.89 


16.97 


17.06 


17.14 


17.22 


17.31 


17.39 


17.47 


17.66 


4.2 


17.64 


17.72 


17.81 


17.89 


17.98 


18.06 


18.15 


18.23 


18.32 


18.4* 


4.3 


18.49 


18.58 


18.66 


18.75 


18.84 


18.92 


19.01 


19.10 


19.18 


19.27 


4.4 


19.36 


19.45 


19.54 


19.62 


19.71 


19.80 


19.89 


19.98 


20.07 


20.16 


4.5 


20.25 


20.34 


20.43 


20.52 


20.61 


20.70 


20.79 


20.88 


20.98 


21.07 • 


4.6 


21.16 


21.25 


21.34 


21.44 


21.63 


21.62 


21.72 


21.81 


21.90 


22.00 


4.7 


22.09 


22.18 


22.28 


22 37 


22.47 


22.66 


22.66 


22.75 


22.85 


22.94 


4.8 


23.04 


23.14 


23.23 


23.33 


23.43 


23.52 


23.62 


23.72 


23.81 


23.91 


4.9 


2401 


2411 


24.21 


24.30 


24 40 


24.50 


24.60 


24.70 


24.80 


24.90 


5.0 


25.00 


25.10 


25.20 


25.30 


25.40 


25.60 


25.60 


25.70 


25.81 


25.91 w 


5.1 


26.01 


26.11 


26.21 


26.32 


26.42 


26.52 


26.63 


26.73 


26.83 


26.94 


5.2 


27.04 


27.14 


27.25 


27.35 


27.46 


27.56 


27.67 


27.77 


27.88 


27.98 


5.3 


28.09 


28.20 


28.30 


28.41 


28.52 


28.62 


28.73 


28.84 


28.94 


29.05 


5.4 


29.16 


29.27 


29.38 


29.48 


29.59 


29.70 


29.81 


29.92 


30.03 


SV.14 


6.6 


30.25 


30.36 


80.47 


30.68 


80.69 


80.80 


30.91 


31.02 


31.14 


81.26 " 


5.6 


31.36 


31.47 


31.68 


31.70 


31.81 


31.92 


32.04 


32.16 


32.26 


82.38 


5.7 


82.49 


32.60 


32.72 


32.83 


82.96 


33.06 


33.18 


33.29 


83.41 


83.52 


68 


83.64 


33.76 


33.87 


33.99 


34.11 


34.22 


34.34 


84.46 


84.67 


34.69 


59 


34.81 


34.93 


95.05 


36.16 


35.28 


3).40 


35 52 


85.64 


35.76 


35 88 


60 


36.00 


361? 


tt6.24 


36.36 


36.48 


36.60 


36 72 


86.84 


S6.97 


87 09 



Table 3, c. Squares. 805 

1^2345 6 7 8 9 Off. 

6.0 36.00 3612 36.24 36 36 36.48 36.60 36.72 36.84 36.97 37.09 * 

6.1 37.21 37.33 37.45 37.58 37.70 87.82 37.95 38.07 38.19 38.32 

6.2 38.44 38.56 38.69 38.81 38.94 39.06 39.19 39.31 39.44 39.56 

6.3 39.69 39.82 39.94 40.07 40.20 40.32 40.45 40.58 40.70 40.83 

6.4 40.96 41.09 41.22 41.34 41.47 41.60 41.73 41.86 41.99 42.12 



6.5 42.25 


42.38 


42.51 


42.64 


42.77 


42.90 


43.03 


43.16 


43.30 


43.43" 


6.6 43 56 


43.69 


43.82 


43.96 


44.09 


44.22 


4436 


44.49 


44 62 


44.76 


6.7 44.89 


45 02 


45.16 


45.29 


45.43 


45.56 


45.70 


45.83 


45.97 


46.10 


6.8 46.24 


46.38 


46.51 


46.65 


46.79 


46.92 


47.06 


47.20 


47.33 


47.47 


6.9 47.61 


47.75 


47.89 


48.02 


48.16 


48.30 


48.44 


48.58 


48.72 


48.86 


7.0 49.00 


49.14 


49.28 


49.42 


49.56 


49.70 


49.84 


49.98 


50.13 


50.27" 


7.1 50.41 


50.55 


50.69 


50.84 


50.98 


51.12 


51.27 


51.41 


51.55 


51.70 


7.2 51,84 


51.08 


52.13 


52.27 


52.42 


52.56 


52.71 


52.85 


53.00 


53.14 


7.3 53.29 


53.44 


53.58 


53.73 


53.88 


54 02 


54.17 


54.32 


54.46 


54.61 


7.4 54 76 


54.91 


55.06 


55.20 


55.35 


55.50 


55.65 


55.80 


55.95 


56.10 


7.5 56.25 


56 40 


56.55 


56.70 


56.85 


57.00 


57.15 


57.30 


57.46 


67.61" 


7.6 57.76 


57.91 


58.06 


58.22 


58.37 


58.52 


58.68 


58.83 


58.98 


59.14 


7.7 59.29 


59.44 


59.60 


59 75 


59.91 


60.06 


60.22 


60.37 


60.53 


60.68 


7.8 60.84 


61.00 


61.15 


61.31 


61.47 


61.62 


61.78 


61.94 


62.09 


62 25 


7.9 62.41 


62.57 


62.73 


62.88 


63.04 


63.20 


63.36 


63.52 


63.68 


63.84 



8.0 64.00 64.16 64.32 64.48 64.64 64.80 64.96 65.12 65.29 65 45" 

8.1 65.61 65.77 65.93 66.10 66.26 66 42 66.59 66.75 66.91 67 08 

8.2 67.24 67.40 67.57 67.73 67.90 68.06 68.23 68.39 68.56 68.72 

8.3 68.89 69 06 69.22 69.39 69.56 69.72 69.89 70.06 70.22 70.39 

8.4 70.56 70 73 70.90 71.06 71.23 71.40 71.57 71.74 71.91 72 08 

8.5 72.25 72.42 72.59 72.76 72.93 73.10 73.27 73.44 73.62 73.79" 

8.6 73.96 74.13 74.30 74.48 74.65 74.82 75.00 75.17 75.34 75.52 

8.7 75.69 75.86 76.04 76.21 76.39 76.56 76.74 76.91 77.09 77 20 

8.8 77.44 77.62 77.79 77.97 78.15 78.32 78.50 78.68 78.85 79 03 

8.9 79.21 79.39 79 57 79.74 79.92 80.10 80.28 80.46 80.64 80.82 

9.0 81.00 81.18 81.36 81.54 8172 81.90 82.08 82.26 82.45 82.63 >• 

9.1 82.81 82.99 83.17 83.36 83.54 83.72 83.91 84.09 84.27 84.46 

9.2 84.64 84.82 85.01 85.19 85.38 85.56 85.75 85.93 86.12 86.30 

9.3 86.49 86.68 86.86 87.05 87.24 87.42 87.61 87.80 87.98 88.17 

9.4 88.36 88.55 88.74 88.92 89.11 89.30 89.49 89.68 89.87 9006 

9.5 90.25 90.44 90.63 90.82 91.01 91.20 91.39 91.58 91.78 91.97" 

9.6 92.16 92.35 92.54 92.74 92.93 93.12 93.32 93.51 93.70 93.90 

9.7 94.09 94.28 94.48 94.67 94.87 95 06 95 26 95.45 95.65 95.84 

9.8 96.04 96.24 9643 96.63 96.83 97.02 97.22 97.42 97.61 97.81 

9.9 98.01 98.21 98.41 98.60 98.80 99.00 99.20 99.40 99.60 99.80 

10.0 100.00 100.20 10040 100.60 100.80 101.00 101.20 101.40 101.61 101.81 w 
10.1 102.01 102 21 102.41 10262 102.82 103.02 103.23 103.43.103.63 103.84 
10.2 104.04 104.24 104.45 104 65 104.86 105.06 105.27 105.47 105.68 105.88 
10.3 106.09 106.30 106.50 106-71 106.92 107.12 107.33 107.54 107.74 107.95 
10.4108.16 108.37 108.58 108-78 108.99 109.20 109.41 109.62 109.83 110.04 

10.5 110.25 110.46 110.67 110.88 111.09 111.30 111.51 111.72 111.94 112.15tt 
10.6 112.36 112.57 112.78 113.00 113.21 113.42 113.64 113.85 114.06 114.28 
10.7 114.49 114.70 114.92 115.13 115 35 115.56 115.78 115.99 116 21 116.42 
10.8116.64 116.86 117.07 117.29 117.51 117.72 117.94 118.16 118.37 118.59 
10.9 118.81 119.03 119.25 119.46 119.68 119.90 120.12 120.34 120.56 120.78 
11.0 121.00 121.22 121.44 121.66 121.88 122.10 122.32 122.54 122.77 122.99 *« 



806 Cubes. Table 3, D. 

MO 1 8 84 567 89iir. 

J.O 1.000 1.030 1.061 1.093 1.125 1.158 1.191 1.225 1.260 1.295 •' 

1.1 1.331 1368 1.405 1.443 1.482 1.521 1.561 1.602 1.643 1.685 *> 

1.2 1.728 1.772 1.816 L861 1.907 1.953 2.000 2.048 2,097 2.147 *^ 

1.3 2.197 2.248 2.300 2.353 2.406 2.460 2.515 2.571 2.628 2 686 ^ 

1.4 2.744 2.803 2.863 2.924 2.986 3.049 3.112 3.177 3.242 3.308 •' 

1.5 3.375 3.443 3.512 3.582 3.652 3.724 3.796 3.870 3.944 4.020 ^' 

1.6 4.096 4.173 4.252 4.331 4.411 4492 4.574 4.657 4.742 4.827 " 

1.7 4.913 5.000 5.088 5.178 5.268 5.359 5.452 5.545 5.640 5.735 >' 

1.8 5.832 5.930 6.029 6.128 6.230 6.332 6.435 6.539 6.645 6.75P'' 

1.9 6.859 6.968 7.078 7.189 7.301 7.415 7.530 7.645 7.762 7.881"* 

2.0 8.000 8.121 8.242 8.365 8.490 8.615 8.742 8.870 8.999 9.129''* 

2.1 9.261 9.394 9.528 9.664 9.800 9 938 10.08 10.22 10.36 10.50 - 

2.2 10.65 10.79 10.94 11.09 11.24 11.39 1154 11.70 11.85 12.01 '' 

2.3 12.17 12.33 12.49 12.65 12.81 12.98 13.14 13.31 13 48 13.65 '< 

2.4 13.82 14.00 14.17 14.35 14.53 14.71 14.89 15.07 15.26 15.44 " 



8.5 


16.63 


15.81 


16.00 


16.19 


16.39 


16.58 


16.78 


J6.97 


17.17 


17.37 »• 


2.6 


17.58 


17.78 


17.98 


18.19 


18.40 


18.61 


18.82 


19.03 


19.25 


19.47 »' 


2.7 


19.68 


19.90 


20.12 


20.35 


20.57 


20.80 


21.02 


21.25 


21.48 


21.72 •» 


28 


81.95 


22.19 


22.43 


22.67 


22.91 


23.15 


23.39 


23.64 


23.89 


24.14 " 


2.9 


24.39 


24.64 


24.90 


25.15 


25.41 


25.67 


25 93 


26.20 


26.46 


26.73 ** 



3.0 27.00 27.27 27.54 27.82 28.09 28.37 28.65 28.93 29.22 29.50 «• 

3.1 29.79 30.08 30.37 30.66 30.96 31.26 31.55 31.86 32.16 32.46 ^ 

8.2 32.77 33.08 33.39 33.70 34.01 34.33 34.65 34.97 35.29 35.61 " 

3.3 35.94 36.26 36.59 36.93 37.26 37.60 37.93 38.27 38 61 38.96 '« 

8.4 39.30 39.65 40.00 40.35 40.71 41.06 41.42 41.78 42.14 42.51 '< 

8.5 42 88 43.24 43.61 43.99 44.36 44.74 45.12 45 50 45.88 46 27 «" 

8.6 46.66 47.05 47.44 47.83 48.23 48.63 49.03 49.43 49.84 50.24 «<» 

3.7 50.65 51.06 51.48 51.90 52.31 52.73 63.16 53.58 54.01 54.44 *• 

8.8 54.87 65.31 65.74 66.18 66.62 57.07 57.51 67.96 68.41 68.86 «' 

8.9 69.32 59.78 60.24 60.70 61.16 61.63 62.10 62.67 63.04 63.52 '^ 

4.0 64.01) 64.48 64 96 65.45 66.94 66.43 66.92 67.42 67.92 68.42 '• 

4.1 6892 69.43 69.93 70.44 70.96 71.47 71.99 72.51 73.03 73.56 " 

4.2 74.09 74.62 75.15 75.69 76.23 76.77 77.31 77.85 78.40 78.95 " 

4.3 79.51 80.06 80.62 81.18 81.76 82.31 82.88 83.45 84.03 84.60 " 

4.4 85.18 85.77 86.35 86 94 87.53 88.12 88.72 89.31 89 92 90.52 <^ 

4.5 91.13 91.73 92.35 92.96 93.58 94 20 94.82 95.44 96.07 96.70 «> 

4.6 97.34 97.97 98.61 99 25 99.90 100.6 101.2 101.8 102.6 103.2 - 

4.7 103.8 104.6 105.2 105.8 106.6 107.2 107.9 108.5 109.2 109.9 ' 

4.8 110.6 111.3 112.0 112.7 113.4 114.1 114.8 115.6 1162 116.9 ' 

4.9 117.6 118.4 119.1 119.8 120.6 121.3 122.0 122.8 123.6 124.3 ' 

5.0 125.0 125.8 126.6 127.3 128.0 128.8 129.6 130.3 131.1 131.9 " 

5.1 132.7 133.4 134.2 135.0 135.8 136 6 137.4 138.2 139.0 139.8 ' 

5.2 140.6 141.4 142.2 143.1 143.9 144.7 145.6 146.4 147.2 148.0 ' 

5.3 148.9 149.7 160.6 151.4 152.3 153.1 154.0 154.9 166.7 166.6 ' 

5.4 157.5 158.3 159.2 160.1 161.0 161.9 162.8 163.7 164.6 165.6 * 

5.5 166.4 1673 168.2 169.1 170.0 171.0 171.9 172.8 173.7 174.7 ' 

5.6 175.6 176.6 177.6 178.5 179.4 180.4 181.3 182.3 183.3 184.2 '"^ 

5.7 185.2 186.2 187.1 188.1 189.1 190.1 191.1 192.1 193.1 194.1 '"^ 

5.8 195 1 196.1 197.1 198.2 199.2 2002 201.2 202.3 203.3 204.3 '"^ 

5.9 205 4 206.4 207.6 208.6 209.6 210.6 211.7 212.8 213.8 214.9 " 
6.0 216.0 217.1 218.2 219.3 220.3 221.4 222.6 223.6 224.8 225.9 '' 



Table 3, D. Cubet. 807 

f 28 45 789 on. 

6.0 216.0 217.1 218.2 219.3 220.3 221.4 222.5 223.6 224.8 225.9 '« 

6.1 227 228.1 229.2 230.3 231.5 232.6 233.7 234.9 236.0 237.2" 

6.2 238.3 239.5 240.6 241.8 243.0 244.1 245.3 246.5 247.7 248.9 '> 
C.3 250.0 251.2 252.4 253.6 254.8 256.0 257.3 258.5 259.7 260.9" 

6.4 262.1 263.4 264.6 265.8 267.1 268.3 269.6 270.8 272.1 273.4 '* 

6.5 274.6 275.9 277.2 278.4 279.7 281.0 282.3 283.6 284.9 286.2 ^ 

6.6 287.5 288.8 290 1 291.4 292.8 294.1 295.4 296.7 298.1 299.4 >• 

6.7 300.8 302.1 303.5 304.8 306.2 307.5 308.9 310.3 311.7 313.0 ^^ 

6.8 314.4 315.8 317.2 318.6 320.0 321.4 322.8 324.2 325.7 327.1 '« 

6.9 328.5 329.9 331.4 332.8 334.3 335.7 3ii7.2 338.6 340.1 341.5 ^» 

7.0 343.0 344.5 345.9 847.4 348.9 350.4 351.9 353.4 354.9 356.4 » 

7.1 357.9 359.4 360.9 362.5 364.0 365.5 367.1 368.6 370.1 371.7 » 

7.2 373.2 374.8 376.4 377.9 379.5 381.1 382.7 384.2 385.8 387.4 »• 

7.3 389.0 390.6 392.2 393.8 395.4 397.1 398.7 400.3 401.9 403.6 " 

7.4 405.2 406.9 408.5 410.2 411.8 413.5 415.2 416.8 418.5 420.2 '• 

7.5 421.9 423.6 425.3 427.0 428.7 430.4 432.1 433.8 435.5 437.2 " 

7.6 439.0 440.7 442.5 444.2 445.9 447.7 449.5 451.2 453.0 454.8 ^^ 

7.7 456 5 458.3 460.1 461.9 463.7 465.5 467.3 469.1 470.9 472.7" 

7.8 474 6 476.4 478.2 480.0 481.9 483.7 485.6 487.4 489.3 491.2 " 

7.9 493.0 494.9 496.8 498.7 500.6 502.5 504.4 506.3 508.2 510.1 " 

8.0 512.0 513.9 515.8 517.8 519.7 521.7 523.6 525.6 527.5 529.5" 

8.1 531.4 533.4 535.4 537.4 539.4 541.3 543.3 545.3 547.3 549.4 «> 

8.2 551.4 553.4 555.4 557.4 559.5 561.5 563.6 565.6 567.7 569 7 =• 

8.3 571.8 573.9 575.9 578.0 580.1 582.2 584.3 586.4 588.5 590.6 »> 

8.4 592.7 594.8 596.9 599.1 601.2 603.4 605.5 607.6 609.8 612.0" 

8.5 614.1 616.3 618.5 620.7 622.8 625.0 627.2 629.4 631.6 633.8" 

8.6 636.1 638.3 640.5 642.7 645 647.2 649.5 651.7 654.0 656.2" 

8.7 658 5 660.8 663.1 665.3 667.6 669.9 672.2 674.5 676.8 679.2 *» 

8.8 681.5 683.8 686.1 688.5 690.8 693.2 695.5 697.9 700.2 702.6" 

8.9 705.0 707.3 709.7 712.1 714.6 716.9 719.3 721.7 724.2 726.6" 

9.0 729.0 731.4 733.9 736.3 738.8 741.2 743.7 746.1 748.6 751.1 " 

9.1 753.6 756.1 758.6 761.0 763.6 7661 768.6 771.1 773.6 776.2" 

9.2 778.7 781.2 783.8 786.3 788.9 791.6 794.0 796.6 799.2 801.8" 

9.3 804.4 807.0 809.6 812.2 814.8 817.4 820.0 822.7 825.3 827.9 " 

9.4 830.6 833.2 835.9 838.6 841.2 843.9 846.6 849.3 852.0 854.7 « 

9.5 857.4 8G0.1 862.8 865.6 868.3 871.0 873.7 876.5 879.2 882.0*^ 

9.6 884.7 887.5 890.3 893.1 895.8 898.6 901.4 904.2 907.0 909.9" 

9.7 912.7 916.5 918.3 921.2 924.0 926.9 929.7 932.6 935.4 938.3" 

9.8 941.2 944.1 947.0 949.9 952.8 955.7 958.6 961.5 964.4 967.4" 

9.9 970.3 973.2 976.2 979.1 982.1 985.1 988.0 991.0 994.0 997.0 "^ 

10.0 1000.0 1003.0 1006.0 1009.0 1012.0 1015.1 1018.1 1021,1 1024.21027.2 " 
10.1 1030.3 1033.4 1036.4 1039.6 1042.6 1045.7 1048.8 1051.9 1055.01058.1 •' 
10.2 1061.2 1064.3 1067.6 1070.6 1073.7 107.69 1080.0 1083.2 1086.41089.5 " 
10.3 1092.7 1095.9 1099.1 1102.3 1105.5 1108.7 1111.9 1115.2 1118.41121.6 " 
10.41124.9 1128.1 1131.4 1134.6 1137.9 1141.2 1144.4 1147.7 1151.0 1154.3 " 

10.5 1157.6 1160.9 1164.3 1167.6 1170.9 1174.2 1177.6 1180.9 1184.3 1187.6 " 
10.61191.0 1194.4 1197.8 1201.2 1204.6 1207.9 1211.4 1214.8 1218.2 1221.6 " 
10.7 1225.0 1228.5 1231.9 1235.4 1238.8 1242.3 1245.8 1249.2 1252.7 1256.2 " 

10.81259.7 1263.2 1266.7 1270.2 1273.8 1277.3 1280.8 1284.4 1287.9 1291.5 " 
10.9 1295.0 1298.6 1302.2 1305.8 1309.3 1312.9 1316.6 1320.1 1323.8 1327.4 " 



11*01331.0 13346 1338.3 134K9 1345.6 1349^2 1352!9 1356.6 1360.3 1363^9 



89 



808 



CiroumferenoM of Giroles. 



Table 3, F. 



Dim. 


OOt 


10* 


20. 


80. 


40. 


60* 


60. 


70. 


80* 


90. 


10 


3142 


3173 


3204 


3236 


3267 


3299 


3330 


3362 


3393 


3424 


11 


3456 


3487 


3519 


3550 


3681 


3613 


3644 


3676 


3707 


3738 


12 


3770 


3801 


3833 


3864 


3896 


3927 


3958 


3990 


4021 


4063 


13 


4084 


4116 


4147 


4178 


4210 


4241 


4273 


4304 


4335 


4367 


14 


4398 


4430 


4461 


4492 


4624 


4666 


4687 


461S 


4650 


4681 


15 


4712 


4744 


4776 


4807 


4838 


4869 


4901 


4932 


4964 


4995 


16 


6027 


6058 


6089 


6121 


6162 


6184 


6215 


5246 


5278 


5309 


17 


6341 


5372 


5404 


6436 


6466 


6498 


6529 


5561 


5592 


6623 


18 


6656 


6686 


5718 


5749 


6781 


5812 


6843 


6876 


5906 


6938 


19 


6969 


6000 


6032 


6063 


6095 


6126 


6158 


6189 


6220 


6262 


80 


6283 


6316 


6346 


6377 


6409 


6440 


6472 


6503 


6535 


6566 


81 


6597 


6629 


6660 


6692 


6723 


6764 


6786 


<817 


6849 


6880 


22 


6912 


6943 


6974 


7006 


7037 


7069 


7100 


7131 


7163 


7194 


28 


7226 


7257 


7288 


7320 


7351 


7383 


7414 


7446 


7477 


7508 


24 


7540 


7671 


7603 


7634 


7665 


7697 


7728 


7760 


7791 


7823 



S5 7854 7885 7917 7948 7980 8011 8042 8074 8105 8137 

26 8168 8200 8231 8262 8294 8325 8357 8388 8419 8451 

27 8482 8514 8545 8577 8608 8639 8671 8702 8734 8765 

28 8796 8828 8859 8891 8922 8954 8985 9016 9048 9079 

29 9111 9142 9173 9205 9236 9268 9299 9331 9362 9393 

80 9425 9456 9488 9519 9550 9582 9613 9645 9676 9708 

81 9739 9770 9802 9833 9865 9896 9927 9959 9990 10022 

82 10053 10085 10116 10147 10179 10210 10242 10273 10304 10336 

83 10367 10399 10430 10462 10493 10524 10556 10587 10619 10650 

84 10681 10713 10744 10776 10807 10838 10870 10901 10933 10964 

85 10996 11027 11058 11090 11121 11153 11184 11215 11247 11278 
36 11310 11341 11373 11404 11435 11467 11498 11530 11561 11592 

87 11624 11655 11687 11718 11750 11781 11812 11844 11875 11907 

88 11938 11969 12001 12032 12064 12095 12127 12158 12189 12221 

89 12252 12284 12315 12346 12378 12409 12441 12472 12504 12535 

40 12566 12598 12629 12661 12692 12723 12755 12786 12818 12849 

41 12881 12912 12943 12975 13006 13038 13069 13100 13132 13163 

42 13195 13226 13258 13289 13320 13352 13383 13415 13446 13477 

43 13509 13540 13572 13603 13635 13666 13697 13729 13760 13792 

44 13823 13854 13886 13917 13949 13980 14012 14043 14074 14106 

45 14137 14169 14200 14231 14263 14294 14326 14357 14388 14420 

46 14451 14483 14514 14546 14577 14608 14640 14671 14703 14734 

47 14765 14797 14828 14860 14891 14923 14954 14985 15017 15048 

48 15080 15111 15142 15174 15205 15237 15268 15300 15331 15362 

49 15394 15425 15457 15488 15519 15551 15582 15614 15645 15677 



60 


16708 


16739 


16771 


16802 


15834 15865 


15896 15928 


15959 


15991 


61 


16022 


16054 


16085 


16116 


16148 16179 


16211 16242 


16273 


16305 


62 


16336 


16368 


16399 


16431 


16462 16493 


16525 16556 


16588 


16619 


68 


16650 


16682 


16713 


16746 


16776 16808 


16839 16870 


16902 


16933 


64 


16966 


16996 


17027 


17059 


17090 17122 


17153 17185 


17216 


17247 


66 


17279 


17310 


17342 


17373 


17404 17436 


17467 17499 


17530 


17562 


u 


Olean) 


(I) 

• 


*? 


(J) 


(4) (5) 
18 16 


(•) CD 
i» sa 


(8) 


28 



lable3,F. 



Giroiimfereiioes of Circles. 



809 



DlaiL 00« IO4 20. 80» 40« 50. 60. 70^ 80» 90« 



55 17279 17310 17342 

56 17593 17624 17656 

57 17907 17938 17970 

58 18221 18253 18284 

59 18535 18567 18598 



17373 17404 17436 17467 

17687 17719 17750 17781 

18001 18033 18064 18096 

18315 18347 18378 18410 18441 

18630 18661 18692 18724 18755 18787 18818 



17499 17530 17562 
17813 17844 17876 
18127 18158 18190 
18473 18504 



60 18850 18881 18912 18944 18975 19007 19038 19069 19101 19132 

61 19164 19195 19227 19258 19289 19321 19352 19384 19415 19446 

62 19478 19509 19541 19572 19604 19635 19666 19698 19729 19761 

63 19792 19823 19855 19886 19918 19949 19981 20012 20043 20075 

64 20106 20138 20169 20200 20232 20263 20295 20326 20358 20389 

65 20420 20452 20483 20515 20546 20577 20609 20640 20672 20703 

66 20735 20766 20797 20829 20860 20892 20923 20954 20986 21017 

67 21049 21080 21112 21143 21174 21206 21237 21269 21300 21331 

68 21363 21394 21426 21457 21488 21520 21551 21583 21614 21646 

69 21677 21708 21740 21771 21803 21834 21865 21897 21928 21960 

70 21991 22023 22054 22085 22117 22148 22180 22211 22242 22274 

71 22305 22337 22368 22400 22431 22462 22494 22525 22557 22588 

72 22619 22651 22682 22714 22745 22777 22808 22839 22871 22902 

73 22934 22965 22996 23028 23059 23091 23122 23154 23185 23216 

74 23248 23279 23311 23342 23373 23405 23436 23468 23499 23531 

75 23562 23593 23625 23656 23688 23719 23750 23782 23813 23845 

76 23876 23908 23939 23970 24002 24033 24065 24096 24127 24159 

77 24190 24222 24253 24285 24316 24347 24379 24410 24442 24473 

78 24504 24536 24567 24599 24630 24662 24693 24724 24756 24787 

79 24819 24850 24881 24913 24944 24976 25007 25038 25070 25101 

80 25133 25164 25196 25227 25258 25290 25321 25353 25384 25415 

81 25447 25478 25510 25541 25573 25604 25635 25667 25698 25730 

82 25761 25792 25824 25855 25887 25918 25950 25981 26012 26044 

83 26075 26107 26138 26169 26201 26232 26264 26295 26327 26358 

84 26389 26421 26452 26484 26515 26546 26578 26609 26641 26672 

85 26704 26735 26766 26798 26829 26861 26892 26923 26955 26986 

86 27018 27049 27081 27112 27143 27175 27206 27238 27269 27300 

87 27332 27363 27395 27426 27458 27489 27520 27552 27583 27615 

88 27646 27677 27709 27740 27772 27803 27835 27866 27897 27929 

89 27960 27992 28023 28054 28086 28117 28149 28180 28212 28243 

90 28274 28306 28337 28369 28400 28431 28463 28494 28526 28557 

91 28588 28620 28651 28683 28714 28746 28777 28808 28840 28871 

92 28903 28934 28965 28997 29028 29060 29091 29123 29154 29185 

93 29217 29248 29280 29311 29342 29374 29405 29437 29468 29500 

94 29531 29562 29594 29625 29657 29688 29719 29751 29782 29814 



95 29845 29877 


29908 


29939 29971 30002 


30034 30065 30096 30128 


96 30159 30191 


30222 


30254 30285 30316 


30348 30379 30411 30442 


97 30473 30505 


30536 


30568 30599 30631 


30662 30693 30725 30756 


98 30788 30819 


30850 


30882 30913 30945 


30976 31008 31039 31070 


99 31102 31133 


31165 


31196 31227 31259 


31290 31322 31353 31385 


100 31416 31447 


31479 


31510 31542 31573 


31604 31636 31667 31699 


DiL (Mean) <i> 


f 


(8) (4) (5) 
• 13 If 


(«) (7) (8) (9) 
12 22 2f 28 



810 Areas of Circles. Table 3, g. 

Dim. «0 a .2 ♦S ^4 «5 «6 «7 «8 «0 on. 

10 78.5 80.1 81.7 83.3 84.9 86.6 88.2 89.9 91.6 93.3 " 

11 95.0 96.8 98.5 100.3 102.1 103.9 105.7 107.5 109.4 111.2 '' 

12 113.1 115.0 116.0 118.8 120.8 122.7 124.7 126.7 128.7 130.7 «<> 

13 132.7 134.8 136.8 138.9 141.0 143.1 145.3 147.4 149.6 151.7 '' 

14 153.9 156.1 158.4 160.6 162.9 165.1 167.4 169.7 172.0 174.4 ''' 

15 176.7 179.1 181.5 183.9 1863 188.7 191.1 193.6 196.1 198.6 ** 

16 201.1 203.6 206.1 208.7 211.2 213.8 216.4 219.0 221.7 224.3 «* 
n 227.0 229.7 232.4 235.1 237.8 240 5 243.3 246.1 248.8 251.6 '' 

18 254.5 257.3 260.2 263.0 265.9 268.8 271.7 274.6 277.6 280.6 «^ 

19 283.5 286.5 289.5 292.6 295.6 298.6 301.7 304.8 307.9 311.0 *' 



20 314.2 317.3 320.5 323.7 326.9 330.1 333.3 336.5 3398 343.1 '' 

21 346.4 349.7 353.0 356.3 359.7 363.1 366.4 369.8 373.3 376.7 ''' 
"" "'^►O.l 383.6 387.1 390.6 394.1 397.6 401.1 404.7 408.3 411.9 " 

5.5 419.1 422.7 426.4 430.1 433.7 437.4 441.2 444.9 448 6 " 

>2.4 456.2 460.0 463 8 467.6 471.4 475.3 479.2 483.1 487.0 "^ 



22 380 
28 415 

24 452 

25 490.9 494.8 498.8 502.7 506.7 510.7 514.7 518.7 522.8 526.9 *^ 
28 530.9 535.0 539.1 543.3 547.4 551.5 555.7 559.9 564.1 5683 '' 

27 572.6 576.8 581.1 585.3 589 6 594.0 598.3 602 6 607.0 611.4 '' 

28 615.8 620.2 624.6 629.0 633 5 637.9 642.4 646.9 651.4 656.0 *^ 

29 660.5 665.1 669.7 674.3 678.9 683.5 688.1 6928 697.5 702.2 "^ 

80 706.9 711.6 716.3 721.1 725.8 7306 735.4 740.2 745.1 749.9 *• 

81 754.8 759.6 764.5 769.4 774.4 779.3 784.3 789.2 794.2 799.2 *« 

82 804.2 809.3 814.3 819.4 824.5 829.6 834.7 839.8 845.0 850.1 '' 

83 855.3 860.5 865.7 870.9 876.2 881.4 886.7 892.0 897.3 902.6 '' 

84 907.9 913.3 918.6 924.0 929.4 934.8 940.2 945.7 951.1 956.6 '' 

85 962 968 973 979 984 990 995 1001 1007 1012 "" 

86 1018 1024 1029 1035 1041 1046 1052 1058 1064 1069 

87 1075 1081 1087 1093 1099 1104 1110 1116 1122 1128 

88 1134 1140 1146 1152 1158 1164 1170 1176 1182 1188 < 

89 1195 1201 1207 1213 1219 1225 1232 1238 1244 1250 

40 1257 1263 1269 1276 1282 1288 1295 1301 1307 1314 

41 1320 1327 1333 1340 1346 1353 1859 1366 1372 1379 

42 1385 1392 1399 1405 1412 1419 1425 1432 1439 1445 
48 1452 1459 1466 1473 1479 1486 1493 1500 1507 1514 

44 1521 1527 1534 1541 1548 1555 1562 1569 1576 1583 

45 1590 1598 1605 1612 1619 1626 1633 1640 1647 1655 ^ 

46 1662 1669 1676 1684 1691 1698 1706 1713 1720 1728 

47 1735 1742 1750 1757 1765 1772 1780 1787 1795 1802 

48 1810 1817 1825 1832 1840 1847 1855 1863 1870 1878 

49 1886 1893 1901 1909 1917 1924 1932 1940 1948 1956 

60 1963 1971 1979 1987 1995 2003 2011 2019 2027 2035 

51 2043 2051 2059 2067 2075 2083 2091 2099 2107 2116 • 

52 2124 2132 2140 2148 2157 2165 2173 2181 2190 2198 
58 2206 2215 2223 2231 2240 2248 2256 2265 2273 2282 

54 2290 2299 2307 2316 2324 2333 2341 2350 2359 2367 

55 2376 2384 2393 2402 2411 2419 2428 2437 2445 2454 



Table 3, Q. Areas of Circles^ 811 

Dlam. 4O A A «8 ^4 ♦& «6 J9 «8 «9 sif. 

65 2376 2384 2393 2402 2411 2419 2428 2437 2445 2454 

66 2463 2472 2481 2489 2498 2507 2516 2525 2534 2543 

67 2552 2561 2570 2579 2588 2597 2606 2615 2024 2633 • 

68 2642 2651 2660 2669 2679 2688 2697 2706 2715 2725 

59 2734 2743 2753 2762 2771 2781 2790 2799 2809 2818 

60 2827 2837 2846 2856 2865 2875 2884 2894 2903 2913 

61 2922 2932 2942 2951 2961 2971 2980 2900 3000 3009 

62 3019 3029 3039 3048 3058 3068 3078 3088 3097 3107 

63 3117 3127 3137 3147 3157 3167 3177 3187 3197 3207 
04 3217 3227 3237 3247 3257 3267 3278 3288 3298 3308 '<" 

65 3318 3329 3339 3349 3359 3370 3380 3390 3400 3411 

66 3421 3432 3442 3452 3463 3473 3484 3494 3505 3515 

67 3526 3536 3547 3557 3568 3578 3589 3G00 3610 3621 

68 3632 3642 3653 3664 3675 3685 3696 3707 3718 3728 

69 3739 3750 3761 3772 3783 3794 3805 3816 3826 3837 

70 3848 3859 3871 3882 3893 3904 3915 3926 3937 3948'' 

71 3959 3970 3982 3993 4004 4015 4026 4038 4049 4060 

72 4072 4083 4094 4106 4117 4128 4140 4151 4162 4174 

73 4185 4197 4208 4220 4231 4243 4254 4266 4278 4289 

74 4301 4312 4324 4336 4347 4359 4371 4383 4394 4406 

75 4418 4430 4441 4453 4465 4477 4489 4601 4513 4524 

76 4536 4548 4560 4572 4584 4596 4608 4620 4632 4645" 

77 4657 4669 4681 4693 4705 4717 4729 4<42 4754 4766 

78 4778 4791 4803 4815 4827 4840 4852 4865 4877 4889 

79 4902 4914 4927 4939 4051 4964 4076 4989 5001 5014 

80 5027 5039 5052 5064 5077 5090 5102 5115 5128 5140 

81 5153 5166 5178 5191 5204 5217 5230 5242 5255 5268 

82 5281 5294 5307 5320 5333 5346 5359 5372 5385 5398'' 

83 5411 5424 5437 5450 5463 5476 5489 5502 5515 5529 

84 5542 5555 5568 5581 5595 5608 5621 5635 5648 5661 

85 5675 5638 5701 5715 5728 5741 5755 5768 5782 5795 

86 5809 5822 5836 5849 5863 5877 5890 5904 5917 5931 

87 5945 5958 5972 5986 5999 6013 6027 6041 6055 6068 

88 6082 6096 6110 6124 6138 6151 6165 6179 6193 6207 

89 6221 6235 6249 6263 6277 6291 6305 6319 6333 6348'' 

90 6362 6376 6390 6404 6418 6433 6447 6461 6475 6490 

91 6504 6518 6533 6547 6561 6576 6590 6604 6619 6633 

92 6648 6662 6677 6691 6706 6720 6735 6749 6764 6778 

93 6793 6808 6822 6837 6851 6866 6881 6896 6910 6925 

94 6940 6955 6969 6984 6999 7014 7029^7044 7058 7073 

95 7088 7103 7118 7133 7148 7163 7178 7193 7208 7223"" 

96 7238 7253 7268 7284 7299 7314 7329 7344 7359 7375 

97 7390 7405 7420 7436 7451 7466 7482 7497 7512 7528 

98 7543 7558 7574 7589 7605 7620 7636 7651 7667 7682 

99 7698 7713 7729 7744 7760 7776 7791 7807 7823 7838 
100 7854 7870 7885 7901 7917 7933 7949 7964 7980 7996" 



812 Volumes of Spheres. Table s, h. 

Diam 1 f 8 4 5 6 T 8 9 Bif. 

1.0 .524 .539 .556 .572 .589 .606 .624 .641 .060 .678 " 

1.1 .697 J16 .736 .755 .776 .796 .817 .839 .860 .882 " 

1.2 .905 .928 .951 «974 .998 1.023 1.047 1.073 1.098 1.124 ^ 

1.3 1.150 1.177 1.204 1.232 1.260 1.288 1.317 1.346 1.376 1.406 •• 

1.4 1.437 1.468 1.499 1.531 1.563 1.596 1.630 1.663 1.697 1.732 '* 

1.5 1.767 1.803 1.839 1.875 1.912 1.950 1.988 2.026 2.065 2.105 »• 

1.6 2.145 2.185 2.226 2.268 2.310 2.352 2.395 2.439 2.483 2.527 '* 

1.7 2.572 2.618 2.664 2.711 2.758 2.806 2.855 2.903 2.953 3.003 ** 

1.8 3.054 3.105 3.157 3.209 3.262 3.315 3.369 3.424 3.479 3.535 '« 

1.9 3.591 3.648 3.706 3.764 3.823 3.882 3.942 4.003 4.064 4.126 '<> 

2 4.189 4.252 4.316 4.380 4.445 4.511 4 577 4.644 4.712 4 780 «• 

2.1 4.849 4.919 4.989 5.060 5.131 5.204 5.277 5 350 5.425 5.500 " 

2.2 5575 5.652 5.729 5.806 5.885 5.964 6.044 6.125 6.206 6.288 "^ 

2.3 6.371 6.451 6.538 6.623 6.709 6.795 6.882 6.970 7.059 7.148 " 

2.4 7.238 7.329 7.421 7.513 7.606 7.700 7.795 7.890 7.986 8.083 ^ 

2.5 8.18 8.28 8.38 8.48 8.58 8.68 8.78 8.89 8.99 9.10 '<> 

2.6 9.20 9.31 9.42 9.53 9.63 9.74 9.85 9.97 10.08 10.19 " 

2.7 10.31 1042 10.54 10.65 10.77 10.89 11.01 11.13 11.25 11.37 ^« 

2.8 11.49 11.62 11.74 11.87 11.99 12.12 12.25 12.38 12.51 12.64 »• 

2.9 12.77 12.90 13.04 13.17 13.31 13.44 13.58 13.72 13.86 14.00 »* 

3.0 1414 1428 14.42 14.57 14.71 14 86 15.00 15.15 15.30 15.45 »» 

3.1 15.60 15.75 15.90 16.06 16.21 16.37 16.52 16.68 16.84 17.00 '• 

3.2 17.16 17.32 17.48 17.64 17.81 17.97 18.14 18.31 18.48 18.65 " 

3.3 1882 18.99 19.16 19.33 19.51 19.68 19.86 20.04 20.22 20.40 ** 

3.4 2058 20.76 20.94 21.13 21.31 21.50 21.69 21.88 22.07 22.26 » 



3.7 26.52 26.74 26.95 27.17 27.39 27.61 27.83 28.06 28.28 28.50 *» 

3.8 28.73 28.96 29.19 29.42 29.65 29.88 30.11 30.35 30.58 30.82 " 

3.9 31.06 31.30 31.54 31.78 32.02 32.27 32.52 32.76 33.01 33.26 '« 

4.0 33.51 33.76 34.02 34.27 34.53 34.78 35.04 35.30 35.56 35.82 ** 

4.1 36.09 36.35 36.62 36 88 37.15 37.42 37.69 37.97 38.24 38.52 " 

M <k no mrk nt\ /\m n/\ ntf ma aa aa a4 ^a -a .a an m n. ba m* nm «. A. oa 




4.5 47.71 48.03 48.35 48.67 49.00 49.32 49.65 49.97 50.30 50.63 ** 

4.6 50.97 51.30 51.63 51.97 52 31 52.65 52.99 53.33 53.67 54.02 ^ 

4.7 64.36 5471 55.06 55.41 55.76 56.12 56.47 56.83 57.18 57.54 »• 

4.8 57.91 58.27 58.63 59.00 59.37 59.73 60.10 60.48 60.85 61.22 ^ 

4.9 61.60 61.98 62.36 62.74 63.12 63.51 63.89 64.28 M.67 65.06 ^ 

5.0 65.45 65 84 66 24 66.64 67.03 67.43 67.83 68 24 68.64 69.05 "• 

5.1 69.46 69.87 70 28 70.69 71.10 71.52 71.94 72.36 72.78 73.20 *• 

5.2 73.62 74.05 74.47 74.90 75.33 75.77 76.20 76 64 77.07 77.51 *• 

5.3 77.95 78.39 78.84 79.28 79.73 80.18 80.63 8108 81.54 81.99 ^ 

5.4 82.45 .S2.91 83.37 83.83 8429 84.76 85.23 85.70 86.17 86.64 *^ 

5.5 87.11 87.5»^8807 88.55 89.03 89.51 90.00 90.48 90.97 91.46 "* 



7.11 87^ 8J 



\ 



Table 3, H. Volumes of l^eres. 813 

DiaQ. i » S 4 5 6 7 S 9 

5.5 87.1 87.6 88.1 88.5 89.0 89.5 90.0 90.5 91.0 91.5 oif. 

6.6 92.0 92.4 92.9 93.4 93.9 94.4 94.9 95.4 95.9 96.5 • 

5.7 97.0 97.5 98.0 98.5 99.0 99,5 100.1 100.6 101.1 101.6 

5.8 102.2 102.7 103.2 103.8 104.3 1048 105.4 105.9 106.4 107.0 

6.9 107.5 108.1 108.6 109.2 109 J 110.3 110.9 1114 112.0 112.5 

6.0 113.1 113.7 114.2 114.8 115.4 115.9 116.5 117.1 117.7 118.3 

6.1 118.8 119.4 120.0 120.6 121.2 121.8 122.4 123.0 123.6 124.2 • 

6.2 124.8 125.4 126.0 126.6 127.2 127.8 128.4 129.1 129.7 130.3 

6.3 130.9 131.5 132.2 132.8 133.4 134.1 134.7 135.3 136.0 1C6.6 

6.4 137.3 137.9 138.5 139.2 139.8 140.5 141.2 141.8 142.5 143.1 

6.5 143.8 144.5 145.1 145.8 146.5 147.1 147.8 148.5 149 2 149.8 

6.6 150.5 151.2 151.9 152.6 153.3 154.0 154.7 155.4 156.1 156.8 ^ 

6.7 157.5 158.2 158 9 159.6 160.3 161.0 161.7 162.5 163.2 163.9 

6.8 164.6 165.4 166.1 166.8 167.6 168.3 169.0 169.8 170.5 171.3 

6.9 172.0 172.8 173.5 174.3 175.0 175.8 176.5 177.3 178.1 178.8 

7.0 179.6 180.4 181.1 181.9 182.7 183.5 184.3 185.0 185.8 186.6 

7.1 187.4 188.2 189.0 189.8 190.6 191.4 192.2 193.0 193.8 194.6 • 

7.2 195.4 196.2 197.1 197.9 198.7 199.5 200.4 201.2 202.0 202.9 

7.3 203.7 204.5 205.4 206.2 207.1 207.9 208.8 209.6 210.5 211.3 

7.4 212.2 213.0 213.9 214.8 215.6 216.5 217.4 218.3 219.1 220.0 

7.5 220.9 221.8 222.7 223.6 224.4 225.3 226.2 227.1 228.0 228.9 • 

7.6 229.8 230.8 231.7 232.6 233.5 234.4 235.3 236.3 237.2 238.1 

7.7 239.0 240.0 240.9 241.8 242.8 243.7 244.7 245.6 246.6 247.5 

7.8 248.5 249.4 250.4 251.4 252.3 253.3 254.3 255.2 256.2 257.2 

7.9 258.2 259.1 260.1 261.1 262.1 263.1 264.1 265.1 266.1 267.1 » 

ao 268.1 269.1 270.1 271.1 272.1 273.1 274.2 275.2 276.2 277.2 

8.1 278.3 279.3 280.3 281.4 282.4 283.4 284.5 285.5 286.6 287.6 

8.2 288.7 289.8 290.8 291.9 292*9 294.0 295.1 296.2 297.2 298.3 

8.3 299.4 300.5 301.6 302.6 303.7 304.8 305.9 307.0 308.1 309.2 " 

8.4 310.3 311.4 312.6 313.7 314.8 315.9 317.0 318.2 319.3 320.4 

8.5 321.6 322.7 323.8 325.0 326.1 327.3 328.4 329.6 330.7 3319 

8.6 333.0 334.2 335.4 336.5 337.7 338.9 340.1 341.2 342.4 343.6 

8.7 344.8 346.0 347.2 348.4 349.6 350.8 352.0 353.2 354.4 355.6 ^* 

8.8 356.8 358.0 359.3 360.5 361.7 362.9 364.2 365.4 366.6 367.9 

8.9 369.1 370.4 371.6 372.9 374,1 375.4 376.6 377.9 379.2 380.4 

90 381.7 383.0 384.3 385.5 386.8 388.1 3894 390.7 392.0 393.3 '* 

9.1 394.6 395.9 397.2 398.5 399.8 401.1 402.4 403.7 405.1 406.4 

9.2 407.7 409.1 410.4 411.7 413.1 414.4 415.7 417.1 418.4 419.8 

9.3 421.2 422.5 423.9 425.2 426.6 428.0 429.4 430.7 432.1 433.5 

9.4 434.9 436.3 437.7 439.1 440.5 441.9 443.3 444.7 446.1 447.5 ^ 

9.5 448.9 450.3 451.8 453.2 454.6 456.0 457J 4589 460.4 461.8 

9.6 463.2 464.7 466.1 467.6 469.1 470.5 472.0 473.5 474.9 476.4 

9.7 477.9 479.4 480.8 482.3 483.8 485.3 486.8 488.3 489.8 491.3 » 

9.8 492.8 494.3 495.8 497.3 498.9 5004 501.9 503.4 505.0 506.5 

9.9 508.0 509.6 511.1 512.7 514.2 515.8 517 3 5189 520.5 522.0 
10.0 523.6 525.2 526.7 528.3 529.9 531.5 533.1 5347 536.3 537«9 ^ 



814 Natural Sines. Tabia 4. 

in2:le «o j «2 «3 44 «5 46 «7 «8 49 Coiplement m. 

O^OOOOO 0017 0035 0052 0070 0087 0105 0122 0140 0157 0175 89* 

1 0175 0192 0209 0227 0241 0262 0279 0297 0814 0332 0349 88 

2 0349 0366 0384 0401 0419 0436 0454 0471 0488 0506 0523 87 
8 0523 0541 0558 0576 0593 0610 0628 0645 0663 0680 0698 8G 

4 0698 0715 0732 0750 0767 0785 0802 0819 0837 0854 0872 85 

5 0.0872 0889 0906 0924 0941 0958 0976 0993 1011 1028 1045 84 

6 1045 1063 1080 1097 1115 1132 1149 1167 1184 1201 1219 83 

7 1219 1236 1253 1271 1288 1305 1323 1340 1357 1374 1392 82 

8 1392 1409 1426 1444 146f 1478 1495 1513 1530 1547 1564 81 

9 1564 1582 1599 1616 1633 1650 1668 1685 1702 1719 1736 80 

10 0.1736 1754 1771 1788 1805 1822 1840 1857 1874 1891 1908 79 

11 1008 1925 1942 1959 1977 1994 2011 2028 2045 2062 2079 78 

12 2079 2096 2113 2130 2147 2164 2181 2198 2215 2233 2250 77 " 

13 2250 2267 2284 2300 2317 2334 2351 2368 2385 2402 2419 70 

14 2419 2436 2453 2470 2487 2504 2521 2538 2554 2571 2588 75 

15 0.2588 2605 2622 2639 2656 2672 2689 2706 2723 2740 2756 74 

16 2756 2773 2790 2807 2823 2840 2857 2874 2890 2907 2924 IZ 

17 2924 2940 2957 2974 2990 3007 3024 3040 3057 3074 3090 72 

18 3090 3107 3123 3140 8156 3173 3190 3206 3223 3239 3256 71 

19 3256 3272 3289 3305 3322 3338 3355 3371 3387 3404 3420 70 

20 0.3420 3437 3453 3469 3486 3502 3518 3535 3551 3567 3584 69 

21 3584 3600 3616 3633 3649 3665 3681 3697 3714 3730 3746 68 

22 3746 3762 3778 3795 3811 3827 3843 3859 3875 3891 3907 67 

23 3907 3923 3939 3955 3971 3987 4003 4019 4035 4051 4067 66 >• 

24 4067 4083 4099 4115 4131 4147 4163 4179 4195 4210 4226 66 

25 0.4226 4242 4258 4274 4289 4305 4321 4337 4352 4368 4384 64 

26 4384 4399 4415 4431 4446 4462 4478 4493 4509 4524 4540 68 

27 4540 4555 4571 4586 4602 4617 4633 4648 4664 4679 4695 62 

28 4695 4710 4726 4741 4756 4772 4787 4802 4818 4833 4848 61 

29 4848 4863 4879 4894 4909 4924 4939 4955 4970 4985 5000 60 

80 0.5000 5015 5030 5045 5060 5075 5090 5105 5120 5135 5150 59 » 
31 5150 5165 5180 5195 5210 5225 5240 5255 5270 5284 5299 58 
8*2 5299 5314 5329 5344 5358 5373 5388 5402 5417 5432 5446 57 
33 5446 5461 5476 5490 5505 5519 5534 5548 5563 5577 5592 56 

84 5592 5606 5621 5635 5650 5664 5678 5693 5707 5721 5736 55 

85 0.5736 5750 5764 5779 5793 5807 5821 5835 5850 5864 5878 54 

36 5878 5892 5906 5920 5934 5948 5962 5976 5990 6004 6018 53 ^ 

37 6018 6032 6046 6060 6074 6088 6101 6115 6129 6143 6157 52 

88 6157 6170 6184 6198 6211 6225 6239 6252 6266 6280 6293 51 

89 6293 6307 6320 6334 6347 6361 6374 6388 6401 6414 6428 50 

40 0.6428 6441 6455 6468 6481 6494 6508 6521 6534 6547 6561 49 

41 6561 6574 6587 6600 6613 6626 6639 6652 6665 6678 6691 48 ^ 

42 6691 6704 6717 6730 6743 6756 6769 6782 6794 6807 6820 47 
48 6820 6833 6845 6858 6871 6884 6896 6909 6921 6934 6947 46 
44<> 6947 6959 6972 6984 6997 7009 7022 7034 7046 7059 7071 45'^ 

Compllieil ♦A ♦& #7 46 /«5 A 48 42 4I 4O Altll 

Natural Cosines. 



Tabia 4. Natural Sines. 815 

Al|[l6 4O A ^2 48 A «5 «6 ^7 ^B ^9 Coipleient 8ii. 

45<> 0.7071 7083 7096 7108 7120 7133 7145 7157 7169 7181 7193 44o 

46 7193 7206 7218 7230 7242 7254 7266 7278 7290 7302 7314 43 » 

47 7314 7325 7337 7349 7361 7373 7385 7396 7408 7420 7431 42 

48 7431 7443 7455 7466 7478 7490 7501 7513 7524 7536 7547 41 

49 7547 7559 7570 7581 7593 7604 7615 7627 7638 7649 7660 40 

50 0.7660 7672 7683 7694 7705 7716 7727 7738 7749 7760 7771 89 

51 7771 7782 7793 7804 7815 7826 7837 7848 7859 7869 7880 38 ^^ 

52 7880 7891 7902 7912 7923 7934 7944 7955 7965 7976 7986 87 

53 7986 7997 8007 8018 8028 8039 8049 8059 8070 8080 8090 80 

54 8090 8100 8111 8121 8131 8141 8151 8161 8171 8181 8192 85 

55 0.8192 8202 8211 8221 8231 8241 8251 8261 8271 8281 8290 84 ^ 

56 8290 8300 8310 8320 8329 8339 8348 8358 8368 8377 8387 88 

57 8387 8396 8406 8415 8425 8434 8443 8453 8462 8471 8480 82 

58 8480 8490 8499 8508 8517 8526 8536 8545 8554 8563 8572 31 

59 8572 8581 8590 8599 8607 8616 8625 8634 8643 8652 8660 30 • 

60 0.8660 8669 8678 8686 8695 8704 8712 8721 8729 8738 8746 29 

61 8746 8755 8763 8771 8780 8788 8796 8805 8813 8821 8829 28 

62 8829 8838 8846 8854 8862 8870 8878 8886 8894 8902 8910 27 « 
68 8910 8918 8926 8934 8942 8949 8957 8965 8973 8980 8988 26 

64 8988 8996 9003 9011 9018 9026 9033 9041 9048 9056 9063 25 

65 0.9063 9070 9078 9085 9092 9100 9107 9114 9121 9128 9135 24 

66 9135 9143 9150 9157 9164 9171 9178 9184 9191 9198 9205 23 ^ 

67 9205 9212 9219 9225 9232 9239 9245 9252 9259 9265 9272 22 

68 9272 9278 9285 9291 9298 9304 9311 9317 9323 9330 9336 21 

69 9336 9342 9348 9354 9361 9367 9373 9379 9385 9391 9397 20 • 

70 0.9397 9403 9409 9415 9421 9426 9432 9438 9444 9449 9455 19 

71 9455 9461 9466 9472 9478 9483 9489 9494 9500 9505 9511 18 

72 9511 9516 9521 9527 9532 9537 9542 9548 9553 9558 9563 17 
78 9563 9568 9573 9578 9583 9588 9593 9598 9603 9608 9613 16 • 

74 9613 9617 9622 9627 9632 9636 9641 9646 9650 9655 9659 15 

75 0.9659 9664 9668 9673 9677 9681 9686 9690 9694 9699 9703 14 

76 9703 9707 9711 9715 9720 9724 9728 9732 9736 9740 9744 13 « 

77 9744 9748 9751 9755 9759 9763 9767 9770 9774 9778 9781 12 

78 9781 9785 9789 9792 9796 9799 9803 9806 9810 9813 9816 U 

79 9816 9820 9823 9826 9829 9833 9836 9839 9842 9845 9848 10 

80 0.9848 9851 9854 9857 9860 9863 9866 9869 9871 9874 9877 9 • 

81 9877 9880 9882 9885 9888 9890 9893 9895 9898 9900 9903 8 

82 9903 9905 9907 9910 9912 9914 9917 9919 9921 9923 9925 7 
88 9925 9928 9930 9932 9934 9936 9938 9940 9942 9943 9945 6 » 

84 9945 9947 9949 9951 9952 9954 9956 9957 9959 9960 9962 5 

85 0.9962 9963 9965 9966 9968 9969 9971 9972 9973 9974 9976 4 

86 9976 9977 9978 9979 9980 9981 9982 9983 9984 9985 9986 8 & 

87 9986 9987 9988 9989 9990 9990 9991 9992 9993 9993 9994 2 

88 9994 9995 9995 9996 9996 9997 9997 9997 9998 9998 9998 1 
89<> 9998 9999 9999 9999 9999 1.0000 1.0000 1-0000 1.0000 1.0000 LOOOO 0<> • 

Coniiinnstt ♦o ♦& «7 46 45 44 48 48 A *o imui 
Natural CoBines. 



816 Logariihmio Sines. Tabia 4. k. 

Al2le .0 .1 J .3 .4 .5 .6 .7 .8 .9 CmnlemiAtw. 

(^>-oe 7.2419 8.5439 F.7190 ?84S9 5.9408 2.0800 t.OBlO 2.1450 ?.1961 212419 89® — 

1 8li8419 2.2882 ^8210 ?.8558 7.8880 §14179 74459 2.4728 2.4971 75206 75428 88 *~ 

2 5428 5640 5842 6035 6220 6897 6567 6731 6889 7041 7188 87 ~ 

3 7188 7880 7468 7602 7781 7857 7979 8098 8218 8826 8436 86 — 

4 8486 8548 8647 8749 8849 8946 9042 , 9135 9226 9815 9408 85 — 

5 2.9408 79489 79578 79655 79786 79816 79894 £9970 7.0046 r0120 ".0192 84 ''* 

6 r.0192 r.0264 r.0884 r.0403 r.0472 7.0539 7.0605 7.0670 7.0734 7.0797 7.0859 83 ^ 

7 0859 0920 0931 1040 1099 1157 1214 1271 1326 1881 1436 82 ^ 

8 1436 1489 1542 1594 1646 1697 1747 1797 1847 1895 1948 81 *^ 

9 1948 1991 9088 2085 2181 8176 2221 2266 2310 2358 2897 80 ^ 

10 7.2897 7.2439 7.2482 7.2524 7.2565 7.2606 7.2647 r2687 72727 7.2767 7.2806 79 *^ 

11 2806 2845 2888 2921 2959 2997 8034 8070 8107 8148 8179 78 *' 

12 8179 8214 8249 8284 8819 8858 8887 8421 8455 8488 8521 77 ** 

13 8521 8554 8586 8618 8650 8682 8718 8745 8775 8806 8837 76 '^ 

14 8837 8867 8897 8927 8957 8986 4015 4044 4078 4102 4180 75 ^* 

15 7.4180 7.4158 7.4186 7.4214 7.4242 7.4269 7.4296 7.4328 7.4350 7.4877 7.4408 74 '^ 

16 4408 4430 4456 4482 4508 4533 4559 4584 4609 4634 4659 73 ** 

17 4659 4684 4709 4788 4757 4781 4806 4829 4358 4876 4900 72 ^* 

18 4900 4928 4946 4969 4992 5015 5037 5060 6082 6104 5126 71 ^ 

19 6126 5148 6170 5192 6218 6285 5256 6278 5299 6820 6841 70 ** 

20 7.5841 75861 7.5882 7.5402 7.5428 7.5448 7.5468 7.5484 7.5504 7.5528 7.5548 69 ^ 

21 5548 5568 5583 5602 5621 5641 5660 5679 5698 5717 5736 68 " 

22 5736 5754 6778 5792 5810 5828 5847 5865 5883 5901 5919 67 ^' 

23 5919 5937 5954 5972 5990 6007 6024 6042 6059 6076 6093 66 '' 

24 6098 6110 6127 6144 6161 6177 6194 6210 6227 6248 6259 65 " 

25 76259 7.6276 7.6292 76808 76834 7.6840 7.6856 7.6371 7.6387 7.6408 76418 64 '* 

26 .6418 6484 6449 6465 6480 6495 6510 6526 6541 6556 6570 63 '* 

27 6570 6585 6600 6615 6629 6644 6659 6678 6687 6702 6716 62 ^ 

28 6716 6730 6744 6759 6778 6787 6801 6314 6828 6842 6856 61 ^^ 

29 6856 6869 6888 6896 6910 6923 6937 6950 6963 6977 6990 60 

80 7.6990 7.7;,03 7.7016 7.7029 7.7042 77055 7.7068 1.7080 7.7098 7.7106 77118 59 " 

31 7118 7131 7144 7156 7168 7181 7198 7205 7218 7280 7242 58 

82 7212 7254 7266 7278 7290 7802 7814 7826 7338 7349 7361 57 ^ 

33 7861 7878 7384 7896 7407 7419 7480 7442 7453 7464 7476 56 

84 7476 7487 7498 76Ct^ 7520 7581 7542 7558 7564 7575 7586 55 ^^ 

85 7.7586 "7597 7.7607 7.7618 7.7629 7.7640 77650 "7661 7.7671 7.7682 7.7692 54 

86 7692 7703 7718 7728 7784 7744 7754 7764 7774 7785 7795 53 

87 7795 7805 7815 7825 7835 7844 7854 7864 7874 7884 7893 52 ^® 

88 7898 7903 7913 7922 7932 7941 7951 7960 7970 7979 7989 51 

89 7939 7998 8007 8017 8026 8085 8044 8058 8068 8072 8081 50 

40 7.8081 7.8090 T.8099 78108 7.8117 7.8125 "8184 7.8148 78152 7.8161 7.8169 49 * 

41 8169 8178 8187 8195 8204 8213 8221 8280 8238 8247 8255 48 

i2 8255 8264 8272 8280 8289 8297 8305 8313 8822 8330 8338 47 

^ 8888 8346 8854 8862 8370 8378 8386 8394 8402 8410 8418 46 

44*^ 8418 8426 8488 8441 8449 8457 8464 8472 8480 8487 8495 45® ' 

CU&llleflieilt J ^ .7 .6 .5 .4 .3 J 4 A iidi 
Logaxithmio Cosines. 



Table 4. A. Logarithmic Sinei. 817 

Allele .0 .1 J ^ .4 .5 •• .7 A .9 Complement w. 

45® r.8495 r.8502 "8510 r.8517 TSSSS ^8583 ".8640 ".8647 r8656 ^8562 r.85e9 44® 

46 8569 8577 8584 8591 8598 8606 8618 8620 8637 8634 8641 43 

47 8641 8648 8656 8663 8669 8676 8683 8690 8697 8^04 8711 42 ^ 

48 8711 8718 8734 8731 8738 8746 8751 8758 8765 8771 8778 41 

49 8778 8784 8791 8797 8804 8810 8317 8823 8880 8836 8848 40 

50 r.8843 r.8849 r!8855 r8868 "8868 ^8874 ^8880 "8887 ".8898 ^8899 ^8906 39 

51 8905 8911 8917 8923 8929 8935 8941 8947 8958 8959 8966 38 ' 

52 8965 8971 8977 898a 8989 8995 9000 9006 9012 9018 9023 37 

53 9028 9029 9036 9041 9046 9052 9057 9068 9069 9074 9080 36 

54 9080 9086 9091 9096 9101 9107 9112 9118 9128 9128 9184 35 

55 r.9134 ".9139 ".9144 ".9149 ".9155 ".9160 ".9165 ".9170 ".9176 r.9181 ".9186 34 

56 9186 9191 9196 9201 9206 9211 9216 9321 9336 9331 9286 33 ' 

57 9336 9341 9346 9351 9355 9360 9365 9370 9275 9279 9284 32 

58 9384 9289 9294 9298 9303 9308 9312 9317 9322 9326 9331 31 

59 9331 9336 9340 9344 9349 9353 9358 9362 9367 9371 9876 30 

60 r.9375 r.9380 ".9384 ".9883 ".9393 1.9397 ".9401 ".9406 1.9410 ".9414 1.9418 29 

61 9418 9422 9427 9431 9435 9439 9443 9447 9451 9456 9459 28 

62 9459 9468 9467 9471 9476 9479 9483 9487 9491 9495 9499 27 ^ 

63 9499 9503 95o6 9510 . 9514 9518 9522 9526 9529 9538 9637 26 

64 9537 9540 9544 9648 9551 9556 9668 9562 9666 ^569 9573 25 

65 r.9573 1 9576 ".9580 ".9583 ".9587 ".9590 ".9594 ".9597 T.9601 ".9604 ".9607 24 

66 9607 9611 9614 9617 9621 9624 9627 9631 9634 9637 9640 23 

67 9640 9643 9647 9650 9653 9656 9659 9662 9665 9669 9672 22 

68 9672 9676 9678 9681 9684 9687 9690 9693 9696 9699 9702 21 ' 

69 9702 9704 9707 9710 9713 9716 9719 9722 9724 9727 9780 20 

70 r.9730 ".9733 ".9733 ".9738 ".9741 ".9743 1.9746 ".9749 ".9761 1.9754 ".9757 19 

71 9757 9759 9762 9764 9767 9770 9772 9775 9777 9780 9782 18 

72 9782 9785 9787 9789 9792 9794 9797 9799 9801 9804 9806 17 

73 9806 9808 9811 9813 9816 9817 9820 9832 9824 9826 9828 16 

74 9828 9831 9833 9835 9837 9839 9841 9843 9845 9847 9849 15 

75 ".9849 ".9851 ".9853 ".9855 7.9857 ".9859 ".9861 ".9863 ".9865 "9867 ".9369 14 ' 

76 9869 9871 9878 9875 9876 9878 9940 9883 9884 9885 9887 13 

77 9887 9889 9891 9893 9894 9896 9897 9899 9901 9903 9904 12 

78 9904 9906 9907 9909 9910 9918 9913 9916 9S»16 9918 9919 11 

79 9919 9931 9933 9934 9926 9927 9928 9929 9931 9932 9934 10 

rO "9934 "9935 ".9936 "9937 T.9939 ".9940 1.9941 "9943 "9944 ".9945 7.9946 9 

81 9946 9947 9949 9950 9951 9953 995S 9954 9955 9956 9958 8 

82 9958 9959 9960 9961 9962 9963 9964 9966 9966 9967 9968 7 ^ 

83 9968 9968 9969 9970 9971 9972 9973 9974 9975 9975 9676 6 

84 9976 9977 9978 9978 9979 9980 9981 9981 9982 9983 9983 5 

85 ".9983 ".9984 "9935 "9985 "9986 ".9987 ".9987 "9988 "9988 ".9989 7.9989 4 

86 9989 9990 9990 9991 9991 9992 9992 9993 9993 9994 9994 3 

87 9994 9994 9995 9996 9996 9996 9996 9996 9997 9997 9997 2 

88 9997 9998 9998 9998 9998 9999 9999 9999 9999 9999 9999 1 
89® 9999 9999 0000 0000 0000 0C30 0000 0000 0000 0000 0000 0®® 

ComplGieflt .9 ^ .7 .6 .5 .4 .3 .2 4 A lu\% 

Logaxithmio CosineSi 



818 Natural Tangents. Tabia 5. 

A1I21I *0 A ♦• «3 A «5 «6 ^K «8 «9 CoipleieAt lif. 

0^" 0.0000 0017 0035 0052 0070 00S7 0105 0122 0140 0157 0175 89* 

1 0175 0192 0209 0227 0244 0262 0279 0297 0314 0332 0349 88 

2 0349 0367 0384 0402 0419 0437 0454 0472 0489 0507 0524 87 

3 0524 0542 0559 0577 0594 0612 0629 0647 0664 0682 0699 86 

4 0699 0717 0734 0752 0769 0787 0805 0822 0840 0857 0875 85 

5 0.0875 0892 0910 0928 0945 0963 0981 0998 1016 1033 1051 84 

6 1051 1069 1086 1104 1122 1139 1157 1175 1192 1210 1228 83 

7 1228 1246 1263 1281 1299 1317 1334 1352 1370 1388 1405 82 

8 1405 1423 1441 1459 1477 1495 1512 1530 1548 1566 1584 81 
1584 1602 1620 1638 1655 1673 1691 1709 1727 1745 1763 80 

10 0.1763 1781 1799 1817 1835 1853 1871 1890 1908 1926 1944 79 ^ 

11 1944 1962 1980 1998 2016 2035 2053 2071 2089 2107 2126 78 

12 2126 2144 2162 2180 2199 2217 2235 2254 2272 2290 2309 77 

13 2309 2327 2345 2364 2382 2401 2419 2438 2456 2475 2493 76 

14 8493 2512 2530 2549 2568 2586 2605 2623 2642 2661 2679 75 

15 0.2679 2698 2717 2736 2754 2773 2792 2811 2830 2849 2867 74 

16 2867 2886 2905 2924 2943 2962 2981 3000 3019 3038 3057 73 >• 

17 3057 3076 3096 3115 3134 3153 3172 3191 3211 3230 3249 72 

18 3249 3269 3288 3307 3327 3346 3365 3385 3404 3424 3443 71 

19 3443 34Ci^ 3482 3502 3522 3541 3561 3581 3600 3620 36^ 70 

20 0.3640 3659 3679 3699 3719 3739 3759 3779 3799 3819 3839 69 

21 3839 3859 3879 3899 3919 3939 3959 3979 4000 4020 4040 68 » 

22 4040 4061 4081 4101 4122 4142 4163 4183 4204 4224 4245 67 

23 4245 4265 4286 4307 4327 4348 4369 4390 4411 4431 4452 66 

24 4452 4473 4494 4515 4536 4557 4578 4599 4621 4642 4663 65 *^ 

25 04663 4684 4706 4727 4748 4770 4791 4813 4834 4856 4877 64 

26 4877 4899 4921 4942 4964 4986 5008 5029 5051 5073 5095 63 

27 5095 5117 5139 5161 5184 5206 5228 5250 5272 5295 5317 62 ** 
US 5317 5340 5362 5384 5407 5430 5452 5475 5498 5520 5543 61 

29 5543 5566 5589 5612 5635 5658 5681 5704 5727 5750 5774 60 ** 

30 0.5774 5797 5820 5844 5867 5890 5914 5938 5961 5985 6009 59 

31 6009 6032 6056 6080 6104 6128 6152 6176 6200 6224 6249 5» *^ 

32 6249 6273 6297 6322 6346 6371 6395 6420 6445 6469 6494 57 

33 6494 6519 6544 6569 6594 6619 6644 6669 6694 6720 6745 56 ^ 

34 6745 6771 6796 6822 6847 6873 6899 6924 6950 6976 7002 55 

35 7002 7028 7054 7080 7107 7133 7159 7186 7212 7239 7265 54 ^ 

36 7265 7292 7319 7346 7373 7400 7427 7454 7481 7508 7536 53 " 
87 7536 7563 7590 7618 7646 7673 7701 7729 7757 7785 7813 52 ^ 
38 7813 7841 7869 7898 7926 7954 7983 8012 8040 8069 8098 51 *• 
89 8098 8127 8156 8185 8214 8243 8273 8302 8332 8361 8391 50 ** 

40 0.8391 8421 8451 8481 8511 8541 8571 8601 8632 8662 8693 49 *• 

41 8693 8724 8754 8785 8816 8847 8878 8910 8941 8972 9004 48 >^ 

42 9004 9036 9067 9099 9131 9163 9195 9228 9260 9293 9325 47 ** 

43 9325 9358 9391 9424 9457 9490 9523 9556 9590 9623 9657 46 '• 
44'^ 9657 9691 9725 9759 9793 9827 9861 9896 9930 9965 1.0000 45<> *" 

GomleiUt «9 48 «7 «6 «5 A «3 «2 ♦! 4O AASlI 

Natural Ootangenti. 



•0 



Tabit 8. Natural Ttogeuts. 819 

AllSll «0 A *2 «3 A «5 S «T 41 «9 otf. 

45' 1.0000 1.0035 1.0070 1.0105 1.0141 1.0176 1.0212 1.0247 1.0283 1.0319 '* 

46 1.0355 1.0392 1.0428 1 0464 1.0501 1.0538 1.0576 1.0612 1.0649 1 0686 " 

47 1.0724 1.0761 1.0799 1.0837 1.0875 1.0913 1.0951 L0990 1.1028 1.1067 ** 

48 1.1106 1.1145 1.1184 1.1224 1.1263 1.1303 1.1343 1.1383 1.1423 1 1463 *^ 

49 1.1504 1.1544 1.1585 1.1626 1.1667 1.1708 1.1750 1.1792 1.1833 1.1875 «^ 

50 1.1918 1.1960 1.2002 1.2045 1.2088 1.2131 1.2174 1.2218 1.2261 12305 "^ 

51 1.2349 1.2393 1.2437 1.2482 1 2527 1 2572 1.2617 1.2662 1.2708 1.2753 "* 

52 1.2709 1.2846 1.2892 1.2938 1.2985 1.3032 1.3079 1.3127 1.3175 13222 *'' 

53 1.3270 1.3319 1.3367 1.3416 13465 1.3514 1.3564 1 3613 1.3663 1.3713 ^ 

54 1.3764 1.3814 1.3865 1.3916 L3968 1.4019 1.4071 1.4124 1.4176 1.4229 "* 

5% 1.4281 1.4335 1.4388 1 4442 1.4406 1.4550 1.4605 1.4659 1.4715 1.4770 *^ 

56 1.4826 1.4882 1.4938 1 4994 1.5051 1.5108 1.5166 1.5224 1.52V2 1.5340 f 

57 1.5399 1.5458 1.5517 1.5577 1.5637 1.5697 15757 1.5818 1.58S0 1.5941 

58 1.6003 1.6066 1.6128 1.6191 1.6255 1.6319 1.6388 1.6447 1.6512 1.6577 

59 1.6643 1.6709 1.6775 1.6842 1.6909 1.6977 1.7045 1.7113 1.7182 1.7251 

60 1.7321 1.7391 1.7461 1.7532 1.7603 1.7676 1.7747 1.7820 1.7893 1.7966 

61 1.8040 1.8115 1.8190 1.8265 1.8341 1.8418 1 8495 1.8572 1.86.';0 1 8728 

02 1.8807 1.8887 1.8967 1.9047 1.9128 1.9210 1.9292 1.9375 1.9458 1 9542 

03 1.9626 1.9711 1.9797 1.9883 1.9970 20057 20145 2 0233 2.0323 2.0413 

04 2.0503 2.0594 2.0686 2.0778 2.0872 2.0965 2.1060 2.1155 2.1251 2.1348 

05 2.145 2.154 2.164 2.174 2.184 2194 2.204 2.215 2.225 2.236 ^ 

06 2.246 2.257 2.267 2.278 2.289 2.300 2311 2.322 2.333 2.344 '' 

07 2356 2.367 2.379 2.391 2.402 2.414 2426 2.438 2.450 2.463 " 

08 2.475 2.488 2.500 2.513 2526 2539 2.552 2.565 2.578 2.592 *' 

09 2.605 2.619 2.633 2.646 2.660 2.675 2.689 2703 2.748 2.733 ^^ 

70 2.747 2.762 £.778 2.793 2.808 2.824 2.840 2.856 2.872 2.888 ^'^ 

71 2.904 2.921 2.937 2.954 2.971 2.989 3.006 3.024 3.042 3.060 '^ 

72 3.078 3.096 3.115 3.133 3.152 3.172 3.191 3.211 3.230 3 250 " 

73 3.271 3.291 3.312 3 333 3.354 S.376 8.398 3.420 3.442 3.465 *^ 

74 3.487 3.511 3.534 3.558 3 582 3606 3.630 3.655 3.681 3.700 ^ 

75 3.732 3.758 3.785 3.812 3.839 3.867 3.895 3.923 3.952 3 981 ^ 
70 4.011 4.041 4.071 4.102 4.134 4.165 4:i98 4.230 4.264 4297 »» 

77 4.331 4.366 4.402 4.437 4.474 4.511 4.548 4.586 4.G25 4.665 

78 4.705 4.745 4.787 4.829 4.872 4.915 4.959 5.005 5.050 5 097 

79 5.145 5.193 5.242 5 292 5.343 5.396 5.449 5.503 5.558 5 614 ^ 

80 5.67 5.73 5.79 5.85 5.91 5.98 6.04 6.11 6.17 6.24 '' 

81 6.31 6 39 6.46 6.54 6.61 6.69 6.77 6.85 6.94 7.03 ' 

82 7.12 7 21 7.30 7.40 7.49 7.60 7.70 7.81 7.92 8.03 ^ 

83 8.14 8 26 8.39 8.51 8.64 878 8.92 9.06 9.21 9.36 ^ 

84 9.51 9.68 9.84 10.0 10.2 10.4 10.6 10.8 11.0 112 

85 11.4 11.7 11.9 12.2 12.4 12.7 13.0 13.3 13.6 14.0 « 

86 14.3 147 15.1 15.5 15.9 16.3 16.8 17 3 17.9 18.5 « 

87 19.1 19.7 20.4 21.2 22.0 22.9 23.9 24.9 26 27.3 

88 28.6 30.1 31.8 33.7 35.8 38 2 40.9 44 1 47.7 521 
89'' 57. 64. 72. 82. 95. 115. 143. 191. 286. 573. 

AlllLll. «0 4 <8 «8 «4 «5 ^O «V ^O ♦• 

Natural Taageuto. 



87 
44 



820 Logarifhmio Tangents. Table 5. a. 

Angle .0 .1 J .3 4 «5 .6 .7 A 3 Compleient orr. 

0*^-00 T.2419 8.54W 8.7190 8.8489 8.9409 8.0200 2.0870 ?.1450 S'.19«2 2.2419 89* — 

1 2.2419 2.2888 2.8211 2.8559 2.8881 2.4181 2.4461 2.4725 2.4978 2.5208 2.5481 88 — 

2 5481 5648 5S45 6038 6228 6401 6571 6786 6894 7046 7194 87 — 

3 7194 7887 7475 7609 7789 7865 7988 8107 8228 8386 8446 86 — 

4 8446 8554 8659 8762 8862 8960 9056 9160 9241 9881 9420 85 — 

5 ?.9420 2!9506 ?9591 2.9674 2!9756 2'.9886 $'.9915 8'.9992 r.0068 7.0148 7.0216 84 ^ 

6 r.0216 r.0289 r.0360 l'.0430 T'.0499 ".0567 7.0638 7.0609 7.0764 7.0828 7.0891 83 " 

7 0891 0954 1015 1076 1185 1194 1252 1810 1867 1423 1478 82 '* 

8 1478 1583 1587 1640 1698 1745 1797 1848 1898 1948 1997 81 *' 

9 1997 2046 2094 2142 2189 2286 2282 2328 2874- 2419 2468 80 *^ 

10 7.2648 7.2507 7.2551 7.2594 7.2637 7.2680 7.2722 7.2764 7.2805 7.2846 7.2887 79 ** 

11 2887 2927 2967 8006 8046 8085 8128 8162 8200 8287 8275 78 ** 

12 8275 8812 8849 8885 8422 8458 8498 8529 8564 8599 8634 77 ** 

13 8634 8668 8702 8786 8770 8804 8837 8870 8908 8935 8968 76 ** 

14 8968 4000 4062 4064 4095 4127 4158 4189 4220 4260 4281 75 '^ 

15 74281 74811 74841 74871 7.4400 74480 74459 7.4488 74517 74546 7.4575 74 ** 

16 4575 4608 4632 4660 4688 4716 4744 4771 4799 4826 4858 73 " 

17 4858 4880 4907 4934 4961 4987 5014 5040 6066 6092 5118 72 " 

18 6118 5148 6169 5195 5220 5245 6270 6296 6820 6845 5370 71 ** 

19 6370 6394 6419 6448 6467 5491 6516 6539 6568 6587 6611 70 '^ 

20 7.5611 7.5684 7.6658 7.5681 7.5704 7.5727 7.6760 7.5778 7,5796 7.5819 7.5842 69 ** 

21 5842 5864 6887 5909 5932 5954 5976 5998 6020 ^ 6042 6064 68 ** 

22 6064 6086 6108 6129 6151 6172 6194 6215 6236 ' 6257 6279 67 '^ 

23 6279 6300 6321 6341 6362 6383 6404 6424 6445 6465 6486 66 '^ 

24 6486 6506 6527 6547 6567 6587 6607 6627 6647 6667 6687 65 ^ 

25 7.6687 7.6706 7.6726 7.6746 7.6765 7.6785 7.6804 7.6824 7.6848 7.6868 7.6882 64 

26 6882 6901 6920 6989 6958 6977 6996 7015 7084 7053 7072 63 ^ 

27 7072 7090 7109 7128 7146 7165 7188 7202 7220 7288 7257 62 

28 7257 7276 7298 7311 7380 7848 7866 7884 7408 7420 7488 61 ** 

29 7488 7455 7478 7491 7509 7626 7544 7562 7579 7597 7614 60 

30 7.7614 7.7682 7.7649 7.7667 7.7684 7.7701 7.7719 7.7786 7.7758 7.7771 77788 59 

31 7788 7805 7822 7889 7856 7878 7890 7907 7924 7941 7958 58 ^^ 

32 7958 7976 7992 8008 8026 8042 8059 8076 8092 8109 8125 57 

33 8126 8142 8158 8176 8191 8208 8224 8241 8257 8274 8290 56 

34 8890 8306 8823 8889 8356 8871 8388 8404 8420 8486 8452 55 

35 7.8452 7.8468 7.8484 7.8501 7.8517 7.8588 7.8549 7.8565 7.8581 7.8597 7.8618 54 ^ 

36 8618 8629 8644 8660 8676 8692 8708 8724 8740 8755 8771 53 

37 8771 8787 8808 8818 8834 8350 8365 8881 8897 8912 8928 52 

38 8928 8944 8959 8976 8990 9006 9022 9037 9053 9068 9084 51 

39 9084 9099 9116 9180 9146 9161 9176 9192 9207 9228 9238 50 

40 7.9288 7.9264 7.9269 7.9284 7.9300 S'.9315 7.9330 7.9346 7.936J 79876 7.9392 49 

41 9392 9407 9422 9488 9463 9468 9483 9499 9514 9529 9544 48 

42 9544 9560 9575 9590 9605 9621 9636 9651 9666 9681 9697 47 

43 9697 9718 0727 9742 9767 9778 9788 9808 9818 9833 9848 46 
44® 9848 9864 9879 9894 9909 9924 9989 ^9956 9970 9986 0000 45®^' 

CompMBit .9 .8 .7 .• J 4 J J 4 4 AniOl 
Logarlthmio Ootangenti. 



uhw 5. A. Logarithmic Tangents. 821 

Allele .0 .1 .2 .3 4 .5 .6 .7 .8 J Compliimeilt oit. 

45^ 0.0000 0.0015 0.0050 0.0045 OJXAl 0.0070 0.0091 0.0106 0.0121 0.0186 0.0158 44® '* 

40 0153 0167 0182 0197 0812 0228 0248 0258 0273 0288 0808 43 

47 0603 0319 0884 0849 0364 0879 0395 0410 0425 0440 0456 42 

48 0456 0471 0486 0501 0517 0532 0547 0562 0578 0508 0608 41 

49 0608 0634 0689 0654 0670 0685 0700 0716 0781 0716 0762 40 

50 0.0762 0.0777 0.0793 0.0808 0.0824 0.0889 0.0854 0.0870 0.0885 0.0901 0.0916 39 

51 0916 0932 0947 0963 0978 0994 lUlO 1025 1041 1056 1072 38 

52 1072 1088 1108 1119 1135 1150 1166 1182 1197 1213 1229 37 

53 1229 1245 1260 1276 1292 1308 1324 1340 1356 J371 1387 36 

54 1387 1403 1419 1435 1451 1467 1488 1499 1516 1532 1548 35 ^ 

55 0.1648 0.1564 0.1680 0.1596 0.1612 0.1629 0.1645 0.1661 0.1677 0.1694 0.1710 34 

56 1710 1726 1743 1759 177G 1792 1809 1825 1842 1858 L875 33 

57 1875 1891 1908 1925 1941 1953 1975 1992 8008 2025 2042 32 

58 2042 2059 2076 2093 2110 2127 2144 2161 2178 2195 2212 31 ^ 

59 2212 2229 2247 2264 2281 2299 8316 8338 2351 2368 2386 30 

60 0.2386 0.2103 0.2431 0.2438 0.2456 0.2474 0.2491 0.2500 0.2527 0.2545 0-2562 29 

61 2562 2580 2598 2616 2634 2652 2670 2689 2707 2725 2743 28 ^ 

62 2743 2762 2780 2798 8817 2835 2854 8372 2S91 2910 2928 27 

63 2928 2947 2966 2985 3004 3023 3045) 3061 3080 8099 3118 26 '* 

64 8118 3137 8157 8176 3196 3215 3235 3254 3274 3294 8318 25 

65 0.3813 0.3333 0.3358 0.3373 0.8398 0.8413 0.3433 0.3458 0.3 173 0.8494 0.8514 24 ^ 

66 3514 3535 8555 8576 8596 8617 3638 8659 8679 8700 8721 23 *^ 

67 8721 3743 8764 8785 8806 8828 8849 8871 8892 8914 8936 22 ^ 

68 8936 8958 3980 4002 4024 4046 4068 4091 4118 4186 4158 21 '' 

69 4158 4181 4204 4227 4250 4273 4296 4319 4842 4366 4889 20 '' 

70 0.4389 0.4413 0.4437 0.4461 0.4484 0.4509 0.4538 0.4557 0.4581 0.4606 0.4680 19 '^ 

71 4630 4655 4680 4705 4730 4755 4780 4805 4831 4S57 48S2 18 '^ 

72 4882 4903 4934 4960 4986 5013 5039 5066 5098 5120 5147 17 ^ 

73 5147 5174 5201 5329 5256 5284 5312 5340 53G8 5397 5435 16 '^ 

74 5425 5454 5483 5512 5541 5570 5600 5629 5659 5689 5719 15 ^ 

75 0.5719 0.57.50 0.5780 0.5811 0.5842 0.5878 0.5905 0.5936 0.5968 0.6000 0.6032 14 '' 

76 6032 6065 6097 6130 6163 6196 6230 6264 6298 6332 6366 13 ^ 

77 €366 6401 6436 6471 6507 6542 6578 6315 6651 6688 6725 12 ^ 

78 €725 6763 6300 6838 6877 6915 6954 6994 7038 7078 7118 11 '' 

79 7118 7154 7195 7286 7278 7820 7868 7406 7449 7498 7537 10 ^ 

80 0.7537 0.7581 0.7626 0.7672 0.7718 0.7764 0.7811 0.7858 0.7906 0.7954 0.8006 9 ^ 

81 8003 8052 8102 8152 8208 8255 8307 8360 8418 8467 8528 8 ^* 

82 8522 8577 8683 8690 8748 8806 8865 8924 8985 9046 9109 7 ^' 

83 9109 9172 9236 9301 9367 9438 9501 9570 9640 9711 9784 6 ^ 

84 9784 0857 9982 1.0008 1.0085 1.0164 1 0244 1.0326 1.0409 1.0494 1.0580 5 ^ 

85 1.0580 1.0669 1.0759 1.0850 1.0944 1.1040 1.1138 1.1238 1.1841 1.1446 1.1554 4 — 

86 1554 1664 1777 1898 8018 8135 8261 8391 8525 8668 8806 3 — 

87 3306 3954 8106 8364 8439 8599 8777 8963 4155 4357 4569 2 — 

88 4569 4793 5027 5375 5539 5819 6119 6441 6789 7167 7581 1 — 
89° 7581 8038 8550 9180 9800 8.0591 8.1561 8.8810 2.4571 2.7581 CO 0®^ 

COBplOOSlt A ^ .7 .6 .5 .4 .3 .2 .1 .0 AEKle 

Logarithmic Cotangents. 



822 Logarithms. Tables. 

10. 128 45 67 89 m 

100 00000 00043 00087 00130 00173 003217 00260 00303 00346 00389 ,^4 
lOi 00432 00475 00518 00561 00604 00647 00689 00732 00775 00817 2,! 

102 00860 00903 00945 00988 01030 01072 01115 01157 01199 01242 HI 

103 01284 01326 01368 01410 01452 01494 01536 01578 01620 01662 •)• 

104 01703 01745 01787 01828 01870 01912 01953 01995 02036 02078 {[j 

105 02119 02160 02202 02243 02284 02325 02366 02407 02449 02490 ,^\ 

106 02531 02572 02612 02653 02694 02735 02776 02816 02857 02898 i,S 

107 02938 02979 03019 03060 03100 03141 03181 03222 03262 03302 {if 

108 03342 03383 03423 03463 03503 03543 03583 03623 03663 03703 ! g 

109 03743 03782 03822 03862 03902 03941 03981 04021 04060 04100 { if 

110 04139 04179 04218 04258 04297 0433jS 04376 04415 04454 04493 t^\ 
lit 04532 04571 04610 04650 04689 04727 04766 04805 04844 04883 M 

112 04922 04961 04999 05038 05077 05115 05154 05192 05231 05269 MJ 

113 05308 05346 05385 05423 05461 05500 05538 05576 05614 05652 • g 

114 05690 05729 05767 05805 05843 05881 05918 05956 05994 06032 lU 

115 06070 06108 06145 06183 06221 06258 06296 06333 06371 06408 ,% 

116 06446 06483 06521 06558 06595 06633 06670 06707 06744 06781 f J 

117 06819 06856 06893 06930 06967 07004 07041 07078 07115 07151 t\l 

118 07188 07225 07262 07298 07335 07372 07408 07445 07482 07518 Mi 

119 07555 07591 07628 07664 07700 07737 07773 07809 07846 07882 ||; 

120 07918 07954 07990 08027 08063 08099 08135 08171 08207 08243 ,^\ 

121 08279 08314 08350 08386 08422 08458 08493 08529 08565 08600 J J 

122 08636 08672 08707 08743 08778 08814 08849 08884 08920 08955 {;« 

123 08991 09026 09061 09096 09132 09167 09202 09237 09272 09307 < Ji 

124 09342 09377 09412 09447 09482 09517 09552 09587 09621 09656 HI 

125 09691 09726 09760 09795 09830 09864 09899 09934 09968 10003 ?\ 

126 10037 10072 10106 10140 10175 10209 10243 10278 10312 10346 hi 

127 10380 10415 10449 10483 10517 10551 10585 10619 10653 10687 t\i 

128 10721 10755 10789 10823 10857 10890 10924 10958 10992 11025 ;j; 

129 11059 11093 11126 11160 11193 11227 11261 11294 11327 11361 iU 

130 11394 11428 11461 11494 11528 11561 11594 11628 11661 11694 ,^, 

131 11727 11760 11793 11826 11860 11893 11926 11959 11992 12024 hi 

132 12057 12090 12123 12156 12189 12222 12254 12287 12320 12352 ^ 

133 12385 12418 12450 12483 12516 12548 12581 12613^2646 12678 li,i 

134 12710 12743 12775 12808 12840 12872 12905 12937 12969 13001 IH 

135 13033 13066 13098 13130 13162 13194 13226 13258 13290 13322 ,'^ 

136 13354 13386 13418 13450 13481 13513 13545 13577 13609 13640 2, { 

137 13672 13704 13735 13767 13799 13830 13862 13893 13925 13956 { {} 

138 13988 14019 14051 14082 14114 14145 14176 14208 14239 14270 fS; 

139 14301 14333 14364 14395 14426 14457 14489 14520 14551 14582 iS 

140 14613 14644 14675 14706 14737 14768 14799 14829 14860 14891 ?K 

141 14922 14953 14983 15014 15045 15076 15106 15137 15168 15198 J f 

142 15229 15259 15290 15320 15351 15381 15412 15442 15473 15503 ii| 
113 15534 15564 15594 15625 15655 15685 15715 15746 15776 15806 • ij 

144 15836 15866 15897 15927 15957 15987 16017 16047 16077 16107 li 

145 16137 16167 16197 16227 16256 16286 16316 16346 16376 16406 ,^, 

146 16435 16465 16495 16524 16554 16584 16613 16643 16673 16702 | { 

147 16732 16761 16791 16820 16850 16879 16909 16938 16967 16997 !!{ 

148 17026 17056 17085 17114 17143 17173 17202 17231 17260 17289 f .*; 

149 17319 17348 17377 17406 17435 17464 17493 17522 17551 17580 i!l« 

150 17609 17638 17667 17696 17725 17754 17782 17811 17S40 17869 



Table 6. Logaritluns. 823 

10. f S 3 4 5 6 7 8 9 Dif. 

150 17609 17638 17667 17696 17725 17754 17782 17811 17840 17S69 .^, 

151 17898 17926 17955 17984 18013 18041 18070 18099 18127 18156 I { 

152 18184 18213 18241 18270 18298 18327 18355 18384 18412 18441 i\l 

153 18469 18498 18526 18554 18583 18611 18639 18667 18696 18724 M; 

154 18752 18780 18808 18837 18865 18893 18921 18949 18977 19005 I '^ 

155 19033 19061 19089 19117 19145 19173 19201 19229 19257 19285 ,^^ 
15G 19312 19340 19368 19396 19424 19451 19479 19507 19535 19562 J { 

157 19590 19618 19645 19673 19700 19728 19756 19783 19811 19838 *{[ 

158 19866 19893 19921 19948 19976 20003 20030 20058 20085 20112 •H 

159 20140 20167 20194 20222 20249 20276 20303 20330 20358 20385 Mjt 

160 20412 20439 20466 20493 20520 20548 20575 20602 20629 20656 i^^, 
101 20683 20710 20737 20763 20790 20817 20844 20871 20898 20925 I t 
162 20952 20978 21005 21032 21059 21085 21112 21139 21165 21192 i\[ 
lft3 21219 21245 21272 21299 21325 21352 21378 21405 21431 21458 f 1} 

164 21484 21511 21537 21564 21590 21617 21643 21669 21696 21722 i'H 

165 21748 21775 21801 21827 21854 21880 21906 21932 21958 21985 ,^^ 

166 22011 22037 22063 22089 22115 22141 22167 22194 22220 22246 ji i 

167 22272 22298 22324 22350 22376 22401 22427 22453 22479 22505 i ij 

168 22531 22557 22583 22608 22634 22660 22686 22712 22737 22763 ;>; 

169 22789 22814 22840 22866 22891 22917 22943 22968 22994 23019 ;:{> 

170 23045 23070 23096 23121 23147 23172 23198 23223 23249 23274 ,^. 

171 23300 23325 23350 23376 23401 23426 23452 23477 23502 23528 } f 

172 23553 23578 23603 23629 23654 23679 23704 23729 23754 23779 l\l 

173 23805 23830 23855 23880 23905 23930 23955 23980 24005 24030 ; i; 

174 24055 24080 24105 24130 24155 24180 24204 24229 24254 24279 ;^ 

175 24304 24329 24353 24378 24403 24428 24452 24477 24502 24527 
170 24551 24576 24601 24625 24650 24674 24699 24724 24748 24773 

177 24797 24822 24846 24871 24895 24920 24944 24969 24993 25018 

178 25042 25066 25091 25115 25139 25164 25188 25212 25237 25261 

179 25285 25310 25334 25358 25382 25406 25431 25455 25479 25503 

180 25527 25551 25575 25600 25624 25648 25672 25696 25720 25744 ?\ 

181 25768 25792 25816 25840 25864 25888 25912 25935 25959 25983 I ] 

182 2G007 26031 26055 26079 26102 26126 26150 26174 26198 26221 i|» 

183 26245 26269 26293 26316 26340 26364 26387 26411 26435 26458 <;« 

184 26482 26505 26529 26553 26576 26600 26623 26647 26670 26694 m 

185 26717 26741 26764 26788 26811 26834 26858 26881 26905 26928 ,^\ 

186 26951 26975 26998 27021 27045 27068 27091 27114 27138 27161 | • 

187 27184 27207 27231 27254 27277 27300 27323 27346 27370 27393 { • 

188 27416 27439 27462 27485 27508 27531 27554 27577 27600 27623 ; 15 

189 27646 27669 27692 27715 27738 27761 27784 27807 27830 27852 JIJ 

190 27875 27898 27921 27944 27967 27989 28012 28035 28058 28081 

191 28103 28126 28149 28171 28194 28217 28240 28262 28285 28307 

192 28330 28353 28375 28398 28421 28443 28466 28488 28511 28533 

193 28556 28578 28601 28623 28646 28668 28691 28713 28735 28758 

194 28780 28803 28825 28847 28870 28892 28914 28937 28959 28981 

195 29003 29026 29048 29070 29092 29115 29137 29159 29181 29203 .^^ 

196 29226 29248 29270 29292 29314 29336 29358 29380 29403 29425 ' t 

197 29447 29469 29491 29513 29535 29557 29579 29601 29623 29645 t ,\ 

198 29667 29688 29710 29732 29754 29776 29798 29820 29842 29863 ; U 

199 29885 29907 29929 29951 29973 29994 30016 30038 30060 30081 l» 

200 30103 30125 30146 30168 30190 30211 30233 30255 30276 30298 



824 Logaritluns. Tables. 

m. 01S3456T89Bit. 

800 30103 30125 30146 30168 30190 30211 30233 30255 30276 3029S » 

201 30320 30341 30363 30384 30406 30428 30449 30471 30492 30514' * 

202 30535 30557 30578 30600 30621 30643 30664 30685 30707 30728 > « 

203 30750 30771 30792 30814 30835 30856 30878 30899 30920 30942 > • 

204 30963 30984 31006 31027 31048 31069 31091 31112 31133 31154' • 

205 31175 31197 31218 31239 31260 31281 31302 31323 31345 31366>'' 

206 31387 31408 31429 31450 31471 31492 31513 31534 31555 31576 <» 

207 31597 31618 31639 31660 31681 31702 31723 31744 31765 31785 ^'^ 

208 31806 31827 31848 31869 31890 31911 31931 31952 31973 31994»'' 

209 32015 32035 32056 32077 32098 32118 32139 32160 32181 3220 P » 

210 32222 32243 32263 32284 32305 32325 32346 32366 32387 32408 ^ 

211 32428 32449 32469 32490 32510 32531 32552 32572 32593 32613 > * 

212 32634 32654 32675 32695 32715 32736 32756 32777 32797 32818' « 

213 32838 32858 32879 32899 32919 32940 32960 32980 33001 33021 > • 

214 33041 33062 33082 33102 33122 33143 33163 33183 33203 33224 « « 

215 33244 33264 33284 33304 33325 33345 33365 33385 33405 33425 "<> 

216 33445 33465 33486 33506 33526 33546 33566 33586 33606 33626 «'' 

217 33646 33666 33686 33706 33726 33746 33766 33786 33806 33826 '<« 

218 33846 33866 33885 33905 33925 33945 33965 33985 34005 34025 » '• 

219 34044 34064 34084 34104 34124 34143 34163 34183 34203 34223' » 

220 34242 34262 34282 34301 34321 34341 34361 34380 34400 34420 i9 

221 34439 34459 34479 34498 34518 34537 34557 34577 34596 34616 ' ' 

222 34635 34655 34674 34694 34713 34733 34753 34772 34792 3481 P « 

223 34830 34850 34869 34889 34908 34928 34947 34967 34986 35005 ' • 

224 35025 35044 35064 35083 35102 35122 3^141 35160 35180 35199 « • 

225 35218 35238 36257 35276 35295 35315 35334 35353 35372 35392 > '« 
236 35411 35430 35449 35468 35488 35507 35526 35545 35564 35583 «<> 

227 35603 35622 35641 35660 35679 35698 35717 35736 35755 35774 '*> 

228 35793 35813 35832 35851 35870 35889 35908 35927 35946 35965 <"'» 

229 35984 36003 36021 36040 36059 36078 36097 36116 36135 36154 »'' 

280 36173 36192 36211 36229 36248 36267 36286 36305 36324 36342 

231 36361 36380 36399 36418 36436 36455 36474 36493 36511 3653(V 

232 36549 36568 36586 36605 36624 36642 36661 36680 36698 36717 

233 36736 36754 36773 36791 36810 36829 36847 36866 36884 36903 

234 36922 36940 36959 36977 36996 37014 37033 37051 37070 37088 

285 37107 37125 37144 37162 37181 37199 37218 37236 37254 37273 

236 37291 37310 37328 37346 37365 37383 37401 37420 37438 37457 

237 37475 37493 37511 37530 37548 37566 37585 37603 37621 37639 

238 37658 37676 37694 37712 37731 37749 37767 37785 37803 37822 
289 37840 37858 37876 37894 37912 37931 37949 37967 37985 38003 

240 38021 38039 38057 38075 38093 38112 38130 38148 38166 38184 is 

241 38202 38220 38238 38256 38274 38292 38310 38328 38346 38364 > • 

242 38382 38399 38417 38435 38453 38471 38489 38507 38525 38543' « 

243 38561 38578 38596 38614 38632 38650 38668 38686 38703 38721 > » 

244 38739 38757 38775 38792 38810 38828 38846 38863 38881 38899' '' 

245 38917 38934 38952 38970 38987 39005 39023 39041 39058 39076* • 

246 39094 39111 39129 39146 39164 39182 39199 39217 39235 39252 <'^ 

247 39270 39287 39305 39322 39340 39358 39375 39393 39410 39428 t*"" 

248 39445 39463 39480 39498 39515 39533 39550 39568 39585 39602 >'« 

249 39620 39637 39655 39672 39690 39707 39724 39742 39759 39777 « "* 

250 39794 39811 39829 39846 39863 39881 39898 39915 39933 39950 



Table 6. IiOgaritluns. 825 

Ho. 1 2 3 4 5 a 7 8 9 in. 

250 39794 39811 39829 39846 39863 39881 39898 39915 39933 39950 n 

251 39967 39985 40002 40019 40037 40054 40071 40088 40106 40123 < ' 

252 40140 40157 40175 40192 40209 40226 40243 40261 40278 40295 ' " 

253 40312 40329 40346 40364 40381 40398 40415 40432 40449 40466 ' * 

254 40483 40500 40518 40535 40552 40569 40586 40603 40620 40637. ' " 

255 40654 40671 40688 40705 40722 40739 40756 40773 40790 40807 ' "" 
25G 40824 40841 40858 40875 40892 40909 40926 40943 40960 40976 * '° 

257 40993 41010 41027 41044 41061 41078 41095 41111 41128 41145 '"'' 

258 41162 41179 41196 41212 41229 41246 41263 41280 41296 41313 ""'* 

259 41330 41347 41363 41380 41397 41414 41430 41447 41464 41481 * '' 

260 41497 41514 41531 41547 41564 41581 41597 41614 41631 41647 
2G1 41664 41681 41697 41714 41731 41747 41764 41780 41797 41814 

262 41830 41847 41863 41880 41896 41913 41929 41946 41963 41979 

263 41996 42012 42029 42045 42062 42078 42095 42111 42127 42144 

264 42160 42177 42193 42210 42226 42243 42259 42275 42292 42308 

265 42325 42341 42357 42374 42390 42406 42423 42439 42455 42472 

266 42488 42504 42521 42537 42553 42570 42586 42602 42619 42635 

267 42651 42667 42684 42700 42716 42732 42749 42765 42781 42797 

268 42813 42830 42846 42862 42878 42894 42911 42927 42943 42959 

269 42975 42991 43008 43024 43040 43056 43072 43088 43104 43120 

270 43136 43152 43169 43185 43201 43217 43233 43249 43265 43281 i< 

271 43297 43313 43329 43345 43361 43377 43393 43409 43425 43441 * > 

272 43457 43473 43489 43505 43521 43537 43553 43569 43584 43600 > ' 

273 43616 43632 43648 43664 43680 43696 43712 43727 43743 43759 ' 

274 43775 43791 43807 43823 43838 43854 43870 43886 43902 43917 

275 43933 43949 43965 43981 43996 44012 44028 44044 44059 44075 ^ " 

276 44091 44107 44122 44138 44154 44170 44185 44201 44217 44232 « '^ 

277 44248 44264 44279 44295 44311 44326 44342 44358 44373 44389 ^ >* 

278 44404 44420 44436 44451 44467 44483 44498 44514 44529 44545 » " 

279 44560 44576 44592 44607 44623 44638 44654 44669 44685 44700 ^ '' 

280 44716 44731 44747 44762 44778 44793 44809 44824 44840 44855 

281 44871 44886 44902 44917 44932 44948 44963 44979 44994 45010 

282 45025 45040 45056 45071 45086 45102 45117 45133 45148 45163 

283 45179 45194 45209 45225 45240 45255 45271 45286 45301 45317 

284 45332 45347 45362 45378 45393 45408 45423 45439 45454 45469 

285 45484 45500 45515 45530 45545 45561 45576 45591 45606 45621 

286 45637 45652 45667 45682 45697 45712 45728 45743 45758 45773 

287 45788 45803 45818 45834 45849 45864 45879 45894 45909 45924 

288 45939 45954 45969 45984 46000 46015 46030 46045 46060 46075 

289 46090 46105 46120 46135 46150 46165 46180 46195 46210 46225 

290 46240 46255 46270 46285 46300 46315 46330 46345 46359 46374 » 
2»1 46389 46404 46419 46434 46449 46464 46479 46494 46509 46523 ' > 

292 46538 46553 46568 46583 46598 46613 46627 46642 46657 46672 ^ ' 

293 46687 46702 46716 46731 46746 46761 46776 46790 46805 46820 > ' 

294 46835 46850 46864 46879 46894 46909 46923 46938 46953 46967 « « 

295 46982 46997 47012 47026 47041 47056 47070 47085 47100 47114 ^ • 

296 47129 47144 47159 47173 47188 47202 47217 47232 47246 47261 • » 

297 47276 47290 47305 47319 47334 47349 47363 47378 47392 47407 ' >< 

298 47422 47436 47451 47465 47480 47494 47509 47524 47538 47553 » " 

299 47567 47582 47596 47611 47625 47640 47654 47669 47683 47698 « " 

300 47712 47727 47741 47756 47770 47784 47799 47813 47828 47842 



4 < 



826 ^ Logarithmi. Tables. 

10. \ 1 2 8 4 5 6 7 8 9 M. 

300 47712 47727 47741 47756 47770 47784 47799 47813 47828 47842 u 
SOI 47857 47871 47885 47900 47914 47929 47943 47958 47972 47986 > ^ 

802 48001 48015 48029 48044 48058 48073 48087 48101 48116,48130 * • 

803 48144 48159 48173 48187 48202 48216 48230 48244 48259 48273 » « 

804 48287 48302 48316 48330 48344 48359 48373 48387 48401 48416 « • 

805 48430 48444 48458 48473 48487 48501 48515 48530 48544 48558 • ^ 
800 48572 48586 48601 48615 48629 48643 48657 48671 48686 487U0 • • 
807 48714 48728 48742 48756 48770 48785 48799 48813 48827 48841 '"^ 
308 48855 48869 48883 48897 48911 48926 48940 48954 48968 48982 • *^ 

809 48996 49010 49024 49038 49052 49066 49080 49094 49108 49122 • '* 

810 49136 49150 49164 49178 49192 49206 49220 49234 49248 49262 

311 49276 49290 49304 49318 49332 49346 49360 49374 49388 49402 

312 49415 49429 49443 49457 49471 49485 49499 49513 49527 49541 

813 49554 49568 49582 49596 49610 49624 49638 49651 49665 49679 

814 49C93 49701 49721 49734 49748 49762 49776 49790 49803 49817 

815 49831 49845 49859 49872 49886 49900 49914 49927 49941 49956 
316 49969 49982 49996 50010 50024 50037 50051 50065 50079 50092 
817 50106 50120 50133 50147 50161 50174 50188 50202 50215 50229 
318 50243 50256 50270 50284 50297 50311 50325 50338 50352 50365 
819 50379 50393 50406 50420 50433 50447 50461 50474 50488 50501 

320 50515 50529 50542 50556 50569 50583 50596 50610 50623 50637 

821 50651 50664 50678 50691 50705 50718 50732 50745 50759 50772 

822 50786 50799 50813 50826 50840 50853 50866 50880 50893 50907 
828 50920 50934 50947 50961 50974 50987 51001 51014 51028 51041 

824 51055 51068 51081 51095 51108 51121 51135 51148 51162 51175 

825 51188 51202 51215 51228 51242 51255 51268 51282 51295 51308 
326 51322 51335 51348 51362 51375 51388 51402 51415 51428 5)!41 

827 51455 51468 51481 51495 51508 51521 51534 51548 51561 51574 

328 51587 51601 51614 51627 51640 51654 51667 51680 51693 51706 

329 51720 51733 51746 51759 51772 51786 51799 51812 51825 51838 

330 51851 51865 51878 51891 61904 51917 51930 51943 51957 51970 » 

331 51983 51996 52009 52022 52035 52048 52061 52075 52088 52101 ^ 
3H2 52114 52127 52140 52153 52166 52179 52192 52205 52218 62231 * 

833 52244 52257 52270 52284 52297 52310 52323 52336 52349 52362 * 

834 52375 52388 52401 52414 52427 62440 52453 52466 52479 52492 « 

835 52504 62517 52530 52543 52556 52569 52582 52595 52608 52621 > 
330 52634 52647 52660 52673 52686 52699 52711 52724 52737 52750 • 
337 52763 52776 52789 52802 52816 52827 52840 52853 52866 52879 * 

838 52892 52905 52917 52930 52943 62956 52969 52982 52994 53007 « »> 

839 63020 53033 53046 53058 63071 63084 63097 53110 63122 63135 •''^ 

340 63148 63161 53173 63186 53199 63212 53224 53237 53250 53263 

841 63275 53288 53301 53314 53326 63339 53352 53364 63377 53390 

842 53403 53415 53428 53441 53453 63466 53479 53491 53504 53517 

843 63529 53542 53555 53567 53580 63593 53605 63618 53631 63643 

844 63656 53668 53681 53694 63706 63719 63732 63744 63767 53769 

845 63782 63794 63807 53820 63832 63845 63867 63870 63882 53895 
346 53908 63920 53933 53945 53958 53970 63983 63996 64008 54020 
847 64033 64045 64058 64070 54083 64095 64108 64120 64133 64146 

348 64158 54170 54183 64195 64208 64220 64233 54246 64258 64270 

349 54283 54295 54307 54320 54332 64345 54357 64370 54382 64394 

350 54407 54419 64432 54444 64456 54469 64481 64494 64506 5461S 



Table 6. Logarithiiu . 827 

10. f 2 8 4 6 6 7 8 Off 

350 54407 54419 54432 54444 54456 54469 54481 54494 54506 54518 13 

351 54531 54543 54555 54568 54580 54593 54605 54617 54630 54642 ^ 

352 54654 54667 54679 54691 54704 54716 54728 54741 54753 54765 ' 

353 54777 54790 54802 54814 54827 54839 54851 54864 54876 54888 * 

354 54900 54913 54925 54937 54949 54962 54974 54986 54998 55011 « 

355 55023 55035 55047 55060 55072 55084 55096 55108 55121 55133 * 

356 55145 55157 55169 55182 55194 55206 55218 55230 55242 55255 ^ 

357 55267 55279 55291 55303 55316 55328 55340 55352 55364 55376 ^ 
3C^8 55388 55400 55413 55425 55437 55449 55461 55473 55485 55497 * '"^ 

359 65509 55522 55534 55546 55558 55570 55582 55594 55606 55618 ' " 

360 55630 55642 55654 55666 55678 55691 55703 55715 55727 55739 

361 55751 55763 55775 55787 55799 55811 55823 55835 55847 55859 

362 55871 55883 55895 55907 55919 55931 55943 55955 55967 55979 

363 55991 56003 56015 56027 56038 56050 56062 56074 56086 56098 

364 56110 56122 56134 56146 56158 56170 56182 56194 56205 56217 

865 56229 5624t 56253 56265 56277 56289 56301 56312 56324 56336 

866 56348 56360 56372 56384 66396 56407 56419 56431 56443 56455 

367 56467 56478 56490 66502 56514 56526 56538 56549 56561 56573 

368 56585 56597 56608 66620 56632 56644 56656 56667 56679 56691 

369 56703 56714 56726 56738 66750 66761 66773 66785 56797 56808 

870 56820 56832 56844 56855 56867 56879 56891 56902 56914 56926 
371 56937 66949 56961 56972 56984 56996 57008 67019 57031 57043 
872 57054 57066 57078 57089 57101 57113 67124 57136 57148 57159 
378 57171 67183 57194 57206 57217 57229 57241 57252 57264 57276 

374 57287 57299 57310 57322 57334 57345 57357 67368 57380 57392 

375 57403 57415 57426 67438 57449 57461 57473 57484 57496 57507 

376 57519 57530 57542 57553 57565 57576 57588 57C00 57611 57623 

377 57634 57646 57657 57669 57680 57692 57703 57715 57726 57738 

378 57749 57761 57772 57784 57795 57807 57818 57830 57841 57852 
879 57864 57875 57887 57898 67910 57921 57933 57944 57955 57967 

380 57978 57990 58001 58013 58024 58035 58047 68058 58070 58081 

381 58092 58104 58115 58127 58138 58149 58161 58172 58184 58195 

382 58206 68218 58229 68240 68252 68263 58274 58286 58297 58309 
883 58320 58331 68343 58354 68366 68377 58388 58399 58410 58422 

384 58433 58444 58456 68467 58478 58490 58501 58512 58524 58535 

385 68546 68557 58569 68580 68591 68602 58614 58625 58636 58647 

386 58659 58670 58681 58692 68704 58715 58726 58737 58749 58760 

387 68771 58782 58794 58805 58816 58827 58838 58850 58861 58872 

388 58883 58894 58906 58917 58928 58939 68950 58961 58973 58984 

889 58995 59006 59017 59028 59040 69051 59062 59073 59084 59095 

890 69106 59118 59129 59140 59151 69162 59173 69184 59195 59207 " 
391 69218 59229 59240 59251 59262 59273 59284 59295 59306 59318 < < 

302 69329 59340 59351 59362 59373 69384 59395 59406 59417 59428 * * 

303 59439 59450 59461 59472 59483 69494 59506 59517 59528 59539 ' ' 

394 59550 59561 59572 59583 69594 59605 59616 59627 59638 59649 * « 

395 59660 69671 69682 59693 59704 59715 59726 59737 59748 59759 * • 

396 69770 69780 59791 69802 59813 59824 59835 59846 59857 59868 • *> 
807 59879 69890 69901 59912 69923 59934 59945 59956 59966 59977 « " 
398 59988 69999 60010 60021 60032 60043 60054 60065 60076 60086 • • 
309 60097 60108 60119 60130 60141 60152 60163 60173 60184 60195 • '^^ 
400 60206 60217 60228 60239 60249 60260 60271 60282 60293 60304 



828 Logarithms. Tables. 

10. 0128 45 6V89III. 

400 60206 G0217 G0228 60239 60249 60260 60271 60282 60293 60304 u 

401 60314 60325 60336 60347 60358 60369 60379 60390 60401 60412 ^ 

402 60423 60433 60444 60455 60466 60477 60487 60498 60509 60520 ' 

403 60531 60541 60552 60563 60574 60584 60595 60606 60617 60627 • 

404 60638 60649 60660 60670 60681 60692 60703 60713 60724 60735 « 

405 60746 60756 60767 60778 60788 60799 60810 6082*1 60831 60842 • 
40» 60853 60863 60874 60S85 60895 60906 60917 60927 60938 60949 • 

407 60959 60970 60981 60991 61002 61013 61023 61034 61045 61055 ^ 

408 61066 61077 61087 61098 61109 61119 61130 61140 61151 61162 > 

409 61172 61183 61194 61204 61215 61225 61236 61247 61257 61268 • >» 

410 61278 61289 61300 61310 61321 61331 61342 61352 61363 61374 

411 61384 61395 61405 61416 61426 61437 61448 61458 61469 61479 

412 61490 61500 61511 61521 61532 61542 61553 61563 61574 61584 

413 61595 61606 61616 61627 61637 61648 61658 61669 61679 61690 

414 61700 61711 61721 61731 61742 61752 61763 61773 61784 61794 

415 61805 61815 61826 61836 61847 61857 61868 61878 61888 61899 

416 61909 61920 61939 61941 61951 61962 61972 61982 61993 62003 

417 62014 62024 62034 62045 62055 62066 62076 62086 62097 62107 

418 62118 62128 62138 62149 62159 62170 62180 62190 62201 62211 

419 62221 62232 62242 62252 62263 62273 62284 62294 62304 62315 

420 62325 62335 62346 62356 62366 62377 62387 62397 62408 62418 lo 

421 62428 62439 62449 62459 62469 62480 62490 62500 62511 62521 t i 

422 62531 62542 62552 62562 62572 62583 62593 62603 62613 62624 > • 

423 62634 62644 62655 62665 62675 62685 62696 62706 62716 62726 > • 

424 62737 62747 62757 62767 62778 62788 62798 62808 62818 62829 « « 

425 62839 62849 62859 62870 62880 62890 62900 62910 62921 62931 > • 

426 62941 62951 62961 62972 62982 62992 63002 63012 63022 63033 • • 

427 63043 63053 63063 63073 63083 63094 63104 63114 63124 63134 '' '^ 

428 63144 63155 63165 63175 63185 63195 63205 63215 63225 63236 • • 

429 63246 63256 63266 63276 63286 63296 63306 63317 63327 63337 • • 

430 63347 63357 63367 63377 63387 63397 63407 63417 63428 63438 

431 63448 63458 63468 63478 63488 63498 63508 63518 63528 63538 

482 63548 63558 63568 63579 63589 63599 63609 63619 63629 63639 

483 63649 63659 63669 63679 63689 63699 63709 63719 63729 63739 

484 63749 63759 63769 63779 63789 63799 63809 63819 63829 63839 

485 63849 63859 63869 63879 63889 63899 63909 63919 63929 63939 

486 63949 63959 63969 63979 63988 63998 64008 64018 64028 64038 

487 64048 64058 64068 64078 64088 64098 64108 64118 64128 64137 

488 64147 64157 64167 64177 64187 64197 64207 64217 64227 64237 

489 64246 64256 64266 64276 64286 64296 64306 64316 64326 64335 

440 64345 64355 64365 64375 64385 64395 64404 64414 64424 64434 

441 64444 64454 64464 64473 64483 64493 64503 64513 64523 64532 

442 64542 64552 64562 64572 64582 64591 64601 64611 64621 64631 

443 64640 64650 64660 64670 64680 64689 64699 64709 64719 64729 

444 64738 64748 64758 64768 64777 64787 64797 64807 64816 64826 

445 64836 64846 64856 64865 64875 64885 64895 64904 64914 64924 

446 64933 64943 64953 64963 64972 64982 64992 65002 65011 65021 

447 65031 65040 65050 65060 65070 65079 65089 65099 65108 65118 

448 65128 65137 65147 65157 65167 65176 65186 65196 65205 6521S 

449 65225 65234 65244 65254 65263 65273 65283 65292 65302 65312 

450 65321 65331 65341 65350 65360 65369 65379 65389 65398 65408 



Table 6. Logarithms. 829 

to. I 2 8 4 5 6 7 8 9 Off. 

450 65321 65331 65341 65350 65360 65369 65379 65389 65398 65408 

451 65418 65427 65437 65447 65456 65466 65475 65485 65495 65504 

452 65514 65523 65533 65543 i65552 65562 65571 65581 65591 65600 

453 65610 65619 65629 65639 65648 65658 65667 65677 65686 65696 

454 65706 65715 65725 65734 65744 65753 65763 65772 65782 65792 

455 65801 65811 65820 65830 65839 65849 65858 65868 66877 65887 

456 65896 65906 65916 65925 65935 65944 65954 65963 65973 65982 

457 65992 66001 66011 66020 66030 66039 66049 66058 66068 66077 

458 66087 66096 66106 66115 66124 66134 66143 66153 66162 66172 

459 66181 66191 66200 66210 66219 66229 66238 66247 66257 66266 

460 66276 66285 66295 66304 66314 66323 66332 66342 66351 66361 

461 66370 66380 66389 66398 66408 66417 66427 66436 66445 66455 
402 66464 66474 66483 66492 66502 66511 66521 66530 66539 66549 

463 66558 66567 66577 66586 66596 66605 66614 66624 66633 66642 

464 66652 66661 66671 66680 66689 66699 66708 66717 66727 66736 

465 66745 66755 66764 66773 66783 66792 66801 66811 66820 66829 

466 66839 66848 66857 66867 66876 66885 66894 66904 66913 66922 

467 66932 66941 66950 66960 66969 66978 66987 66997 67006 67015 

468 67025 67034 67043 67052 67062 67071 67080 67089 67099 67108 

469 67117 67127 67136 67145 67154 67164 67173 67182 67191 67201 

470 67210 67219 67228 67237 67247 67256 67265 67274 67284 67293 < 

471 67302 67311 67321 67330 67339 67348 67357 67367 67376 67385 ' 

472 67394 67403 67413 67422 67431 67440 67449 67459 67468 67477 > 

473 67486 67495 67504 67514 67523 67532 67541 67550 67560 67569 ' 

474 67578 67587 67596 67605 67614 67624 67633 67642 67651 67660 * 

475 67669 67679 67688 67697 67706 67715 67724 67733 67742 67752 » 

476 67761 67770 67779 67788 67797 67806 67815 67825 97834 67843 • 

477 67852 67861 67870 67879 67888 67897 67906 67916 67925 67934 ' 

478 67943 67952 67961 67970 67979 67988 67997 68006 68015 68024 » 

479 68034 68043 68052 68061 68070 68079 68088 68097 68106 68115 • 

480 68124 68133 68142 68151 68160 68169 68178 68187 68196 68205 

481 68215 68224 68233 68242 68251 68260 68269 68278 68287 68296 

482 68305 68314 68323 68332 68341 68350 68359 68368 68377 68386 

483 68395 68404 68413 68422 68431 68440 68449 68458 68467 68476 

484 68485 68494 68502 68511 68520 68529 68538 68547 68556 68565 

485 68574 68583 68592 68601 68610 68619 68628 68637 68646 68655 

486 68664 68673 68681 68690 68699 68708 68717 68726 68735 68744 

487 68753 68762 68771 68780 68789 68797 68806 68815 68824 68833 

488 68842 68851 68860 68869 68878 68886 68895 68904 68913 68922 

489 68931 68940 68949 68958 68966 68975 68984 68993 69002 69011 

490 69020 69028 69037 69046 69055 69064 69073 69082 69090 69099 

491 69108 69117 69126 69135 69144 69152 69161 69170 69179 69188 

492 69197 69205 69214 69223 69232 69241 69249 69258 69267 69276 
498 69285,69294 69302 69311 69320 69329 69338 69346 69355 69364 

494 69373 69381 69390 69399 69408 69417 69425 69434 69443 69452 

495 69461 69469 69478 69487 69496 69504 69513 69522 69531 69539 

496 69548 69557 69566 69574 69583 69592 69601 69609 69618 69627 

497 69636 69644 69653 69662 69671 69679 69688 69697 69705 69714 

498 69723 69732 69740 69749 69758 69767 69775 69784 69793 69801 

499 69810 69819 69827 69836 69845 69854 69862 69871 69880 69888 

500 69897 69906 69914 69923 69932 69940 69949 69958 69966 69976 



830 Logarithms. Tables. 

10. 1 2 8 4 5 6 7 8 9 IV. 

500 69897 69906 69914 69023 69932 69940 69949 69958 69966 69975 » 

501 69984 69992 70001 70010 70018 70027 70036 70044 70053 70062 ' ^ 

502 70070 70079 70088 70096 70105 70114 70122 70131 70140 70148 ' ' 

503 70157 70165 70174 70183 70191 70200 70209 70217 70226 70234 '• 

504 70243 70252 70260 70269 70278 70286 70295 70303 70312 70321 * « 

505 70329 70338 70346 70355 70364 70372 70381 70389 70398 70406 • • 

506 70415 70424 70432 70441 70449 70458 70467 70475 70484 70492 « » 

507 70501 70509 70518 70526 70535 70544 70552 70561 70569 70578 '• 

508 70586 70595 70603 70612 70621 70629 70638 70646 70655 70663 •'' 

509 70672 70680 70689 70697 70706 70714 70723 70731 70740 70749 »• 

510 70757 70766 70774 70783 70791 70800 70808 70817 70825 70834 

511 70842 70851 70859 70868 70876 70885 70893 70902 70910 70919 

512 70927 70935 70944 70952 70961 70969 70978 70986 70995 71003 

513 71012 71020 71029 71037 71046 71054 71063 71071 71079 71088 

514 71096 71105 71113 71122 71130 71139 71147 71155 71164 71172 

515 71181 71189 71198 71206 71214 71223 71231 71240 71248 71257 

516 71265 71273 71282 71290 71299 71307 71315 71324 71332 71341 

517 71349 71357 71366 71374 71383 71391 71399 71408 71416 71425 

518 71433 71441 71450 71458 71466 71475 71483 71492 71500 71508 

519 71517 71525 71533 71542 71550 71559 71567 71575 71584 71592 

520 71600 71609 71617 71625 71634 71642 71650 716S9 71667 71675 8 

521 71684 71692 71700 71709 71717 71725 71734 71742 71750 71759 ^ 
622 71767 71775 71784 71792 71800 71809 71817 71825 71834 71842 « 
528 71850 71858 71867 71875 71883 71892 71900 71908 71917 71925 * 

524 71933 71941 71950 71958 71966 71975 71983 71991 71999 72008 « 

525 72016 72024 72032 72041 72049 72057 72066 72074 72082 72090 » 

526 72099 72107 72115 72123 72132 72140 72148 72156 72165 72173 ' 

527 72181 72189 72198 72206 72214 72222 72230 72239 72247 72255 » 

528 72263 72272 72280 72288 72296 72304 72313 72321 72329 72337 ^ 

529 72346 72354 72362 72370 72378 72387 72395 72403 72411 72419 • 

530 72428 72436 72444 72452 72460 72469 72477 72485 72493 72501 

531 72509 72518 72526 72534 72542 72550 72558 72567 72575 72583 

532 72591 72599 72607 72616 72624 72632 72640 72648 72656 72665 
538 72673 72681 72689 72697 72705 72713 72722 72730 72738 72746 
584 72754 72762 72770 72779 72787 72795 72803 72811 72819 72827 

535 72835 72843 72852 72860 72868 72876 72884 72892 72900 72908 

536 72916 72925 72933 72941 72949 72057 72965 72973 72981 72989 

537 72997 73006 73014 73022 73030 73038 73046 73054 73062 73070 

538 73078 73086 73094 73102 73111 73119 73127 73135 73143 73151 

539 73159 73167 73175 73183 73191 73199 73207 73215^3223 73231 

540 73239 73247 73255 73263 73272 73280 73288 73296 73304 73312 

541 73320 73328 73336 73344 73352 73360 73368 73376 73384 73392 

542 73400 73408 73416 73424 73432 73440 73448 73456 73464 73472 

543 73480 73488 73496 73504 73512 73520 73528 73536 73544 73552 

544 73560 73568 73576 73584 7359^ 73600 73608 73616 73624 73632 

545 73640 73648 73656 73664 73672 73679 73687 73695 73703 73711 

546 73719 73727 73735 73743 73751 73759 73767 73775 73783 73791 

547 73799 73807 73815 73823 73830 73838 73846 73854 73862 73870 

548 73878 73886 73894 73902 73910 73918 73926 73933 73941 73949 

549 73957 73965 73973 73981 73989 73997 74005 74013 74020 74028 
650 74036 74044 74052 74060 74068 74076 74084 74092 74099 74107 



Table 6. Logarithms. 831 

10. 1 2 8 4 6 6 7 8 9 BV 

550 74036 74044 74052 740G0 74068 74076 740S4 74092 74099 741G7 » 

551 74115 74123 74131 74139 74147 74155 74102 74170 74178 74186 '« 

552 74194 74202 74210 74218 74225 74233 74241 74249 74257 74265 '* 
658 74273 74280 74288 74296 74304 74312 74320 74327 74335 74343 » • 

554 74351 74359 74367 74374 74382 74390 74398 74406 74414 74421 « " 

555 74429 74437 74445 74453 74461 74468 74476 74484 74492 74500 * « 

556 74507 74515 74523 74531 74539 74547 74554 74562 74570 74578 • » 

557 74586 74593 74C01 74609 74617 74624 74632 74640 74648 74656 ^ « 

558 74663 74671 74679 74087 74695 74702 74710 74718 74726 74733 » * 

559 74741 74749 74757 74764 74772 74780 74788 74796 74803 74811 •'' 

560 74819 74827 74834 74842 74850 74858 74865 74873 74881 74889 

561 74896 74904 74912 74920 74927 74935 74943 74950 74958 74966 

562 74974 74981 749S9 74997 75005 75012 75020 75028 75035 75043 

563 75051 75059 75066 75074 75082 75089 75097 75105 75113 75120 

564 75128 75136 75143 75151 75159 75166 75174 75182 75189 75197 

565 75205 75213 75220 75228 75236 75243 75251 75259 75266 75274 

566 75282 75289 75297 75305 75312 75320 75328 75335 75343 75351 

567 75358 75366 75374 75381 75389 75397 75404 75412 75420 75427 

568 75435 75442 75450 75458 75465 75473 75481 75488 75496 75504 

569 75511 75519 75526 75534 75542 75549 75557 75565 75572 75580 

570 75587 75595 75603 75610 75618 75626 75633 75641 75648 75656 

571 75664 75671 75679 75086 75694 75702 75709 75717 75724 75732 

572 75740 75747 75755 75762 75770 75778 75785 75793 75800 75808 

573 75815 75823 75831 75838 75846 75853 75861 75868 75876 75884 

574 75891 75899 75906 75914 75921 75929 75937 75944 75952 75959 

575 75967 75974 75982 75989 75997 76005 76012 76020 76027 76035 

576 76042 76050 76057 76065 76072 76080 76087 76095 76103 76110 

577 76118 76125 76133 76140 76148 76155 76163 76170 76178 76185 

578 76193 76200 76208 76215 76223 76230 76238 76245 76253 76260 

579 76268 76275 76283 76290 76298 76305 76313 76320 76328 76335 

580 76343 76350 76358 76365 76373 76380 76388 76395 76403 76410 

581 76418 76425 76433 76440 76448 76455 76462 76470 76477 76485 

582 76492 76500 76507 76515 76522 76530 76537 76545 76552 76559 

583 76567 76574 76582 76589 76597 76604 76612 76619 76626 76634 

584 76641 76649 76656 76664 76671 76678 76686 76693 76701 76708 

585 76716 76723 76730 76738 76745 76753 76760 76768 76775 76782 

586 76790 76797 76805 76812 76819 76827 76834 76842 76849 76856 

587 76864 76871 76879 76886 76893 76901 76908 76916 76923 76930 

588 76938 76945 76953 76960 76967 76975 76982 76989 76997 77004 

589 77012 77019 77026 77034 77041 77048 77056 77063 77070 77078 

590 77086 77093 77100 77107 77115 77122 77129 77137 77144 77151 T 

591 77159 77166 77173 77181 77188 77195 77203 77210 77217 77225 ^ » 

592 77232 77240 77247 77254 77262 77269 77276 77283 77291 77298 « » 
598 77305 77313 77320 77327 77335 77342 77349 77357 77364 77371 » • 

594 77379 77386 77393 77401 77408 77415 77422 77430 77437 77444 « * 

595 77452 77459 77466 77474 77481 77488 77495 77503 77510 77517 » * 

596 77525 77532 77539 77546 77554 77561 77568 77576 77583 77590 6 * 

597 77597 77605 77612 77619 77627 77634 77641 77648 77656 77663 ' » 

598 77670 77677 77685 77692 77699 77706 77714 77721 77728 77735 « • 

599 77743 77750 77757 77764 77772 77779 77786 77793 77801 77808 » • 

600 77815 77822 77830 77837 77844 77861 77859 77866 77873 77880 



882 Logarithms. Tables. 

lO. • 1 2 3 4 5 6 7 8 (T DR 

600 W815 77822 7V830 77837 77844 77851 77859 77866 77873 77880 

«01 77887 77895 77902 77909 77916 77924 77931 77938 77945 77952 

002 77960 77967 77974 77981 77988 77996 78003 78010 78017 78025 

603 78032 78039 78046 78053 78061 78068 78075 78082 780S9 78097 

604 78104 78111 78118 78125 78132 78140 78147 78154 78161 78168 

605 78176 78183 78190 78197 78204 78211 78219 78226 78233 78240 

606 78247 78254 78262 78269 78276 78283 78290 78297 78305 78312 

607 78319 78326 78333 78340 78347 78355 78362 78309 78376 78383 

608 78390 78398 78405 78412 78419 78426 78433 78440 78447 78455 

609 78462 78469 78476 78483 78490 78497 78504 78512 78519 78526 

6t0 78533 78540 78547 78554 78561 78569 78576 78583 78590 78597 

611 78604 78611 78618 78625 78633 78640 78647 78654 78661 78668 

612 78675 78682 78689 78696 78704 78711 78718 78725 78732 78739 

613 78746 78753 78760 78767 78774 78781 78789 78796 78803 78810 

614 78817 78824 78831 78838 78845 78852 78859 78866 78873 78880 

615 78888 78895 78902 78909 78916 78923 78930 78937 78944 78951 

616 78958 78965 78972 78979 78986 78993 79000 79007 79014 79021 

617 79029 79036 79043 79050 79057 79064 79071 79078 79085 79092 

618 79099 79106 79113 79120 79127 79134 79141 79148 79155 79162 

619 79169 79176 79183 79190 79197 79204 79211 79218 79225 79232 

620 79239 79246 79253 7&260 79267 79274 79281 79288 79295 79302 f 

621 79309 79316 79323 79330 79337 79344 79351 79358 79365 79372^ * 

622 79379 79386 79393 79400 79407 79414 79421 79428 79435 79442" ^ 

623 79449 79456 79463 79470 79477 79484 79491 79498 79505 79511 »» 

624 79518 79525 79532 79539 79546 79553 79560 79567 79574 79581 « ' 

625 79588 79595 79602 79609 79616 79623 79630 79637 79644 79650 '« 

626 79657 79964 79671 79678 79685 79692 79699 79706 79713 79720 •* 

627 79727 79734 79741 79748 79754 79761 79768 79775 79782 79789'* 

628 79796 79803 79810 79817 79824 79831 79837 79844 79851 79858 «• 

629 79865 79872 79879 79886 79893 79900 79906 79913 79920 79927** 

630 79934 79941 79948 79955 79962 79969 79975 79982 79989 79996 

631 80003 80010 80017 80024 80030 80037 80044 80051 80058 80065 

632 80072 80079 80085 80092 80099 80106 80113 80120 80127 80134 

633 80140 80147 80154 80161 80168 80175 80182 80188 80195 80202 

634 80209 80216 80223 80229 80236 80243 80250 80257 80264 80271 

635 80277 80284 80291 80298 80305 80312 80318 80325 80332 80339 

636 80346 80353 80359 80366 80373 80380 80387 80393 80400 80407 

637 80414 80421 80428 80434 80441 80448 80455 80462 80468 80475 

638 80482 80489 80496 80502 80509 80516 80523 80530 H0536 80543 

639 80550 80557 80561 80570 80577 80584 80591 80598 80604 80611 

640 80618 80625 80632 80638 80645 80652 80659 80665 80672 80679 

641 80686 80693 80($99 80706 80713 80720 80726 80733 80740 80747 

642 80754 80760 80767 80774 80781 80787 80794 80801 80808 80814 

643 80821 80828 80835 80841 80848 80855 80862 80868 80875 80882 

644 80889 80895 80902 80909 80916 80922 80929 80936 80943 80949 

645 80956 80963 80969 80976 80983 80990 80996 81003 81010 81017 

646 81023 81030 81037 81043 81050 81057 81064 81070 81077 81084 

647 81090 81097 81104 81111 81117 81124 81131 81137 81144 81151 

648 81158 81164 81171 81178 81184 81191 81198 81204 81211 81218 
U9 81224 81231 81238 81245 81251 81258 81265 81271 81278 81285 

'50 81291 81298 81305 81311 81318 81325 81331 81338 81345 81351 



Table 6. Logarithms. 838 

fO. f 9 8 4 5 6 7 8 9 Dif. 

650 S12D] 81298 81305 81811 81318 81325 81331 81338 81345 81351 7 

651 S1358 81365 81371 81378 81385 81391 81398 81405 81411 81418 > 

652 81425 81431 81438 81445 81451 81458 81465 81471 81478 81485 » 

653 81491 81498 81505 81511 81518 81525 81531 81538 81544 81551 > 

654 8155S 81564 81571 81578 81584 81591 81598 81604 81611 81617 « 

655 81624 81631 81637 81644 81651 81657 81664 81671 81677 81684 ' 

656 81690 81697 81704 81710 81717 81723 81730 81737 81743 81750 « 

657 81757 81763 81770 81776 81783 81790 81796 81803 81809 81816 ' 

658 81823 81829 81836 81842 81849 81856 81862 81869 81875 81882 ' 

659 81889 81895 81902 81908 81915 81921 81928 81935 81941 81948 > 

660 81954 81961 81968 81974 81981 81987 81994 82000 82007 82014 

661 82020 82027 82033 82040 82046 82053 82060 82066 82073 82079 
602 82086 82092 82099 82105 82112 82119 82125 82132 82138 82145 

663 82151 82158 82164 82171 82178 82184 82191 82197 82204 82210 

664 82217 82223 82230 82236 82243 82249 82256 82263 82269 82276 

665 822S2 82289 82295 82302 82308 82315 82321 82328 82334 82341 

666 82317 82354 82360 82307 82373 82380 82387 82393 82400 82406 

667 82413 82419 82426 82432 82439 82445 82452 82458 82465 82471 

668 82478 82484 82491 82497 82504 82510 82517 82523 82530 82536 

669 82543 82549 82556 82562 82569 82575 82582 82588 82595 82601 

670 82607 82614 82620 82627 82633 82640 82646 82653 82659 82666 

671 82G72 82G79 82685 82G92 82698 82705 82711 82718 82724 82730 

672 82737 82743 82750 82756 82763 82769 82776 82782 82789 82795 

673 82S02 82808 82814 82821 82827 8^834 82840 82847 82853 82860 

674 82866 82872 82879 82885 82892 82898 82905 82911 82918 82924 

675 82930 82937 82943 82950 82956 82963 82969 82975 82982 82988 

676 82005 88001 88008 83014 83020 83027 83033 83040 83046 88052 

677 83059 880G5 88072 83078 83085 83091 88097 88104 88110 88117 

678 88123 88129 88186 88142 83149 83155 83161 83168 88174 83181 

679 83187 88193 88200 88206 83213 83219 83225 83232 88288 83245 

680 83251 83257 832G4 83270 83276 83283 83289 83296 83302 83308 « 

681 83315 88821 88327 83884 88340 83347 83353 83359 83366 88872 

682 83378 83:!85 88891 83898 83404 83410 83417 83423 83429 83436 

683 88442 88448 88455 88461 83467 83474 83480 83487 83493 83499 

684 83506 83512 88518 88525 88581 83537 83544 83550 83556 83563 

685 83569 83575 88582 88588 88594 83601 83607 83613 83620 83626 

686 88682 88639 88G45 83G51 88658 88664 83670 83677 83683 83689 

687 83696 83702 83708 88715 83721 83727 83734 83740 88746 83753 

688 88759 83765 83771 83778 88784 83790 88797 83803 88809 83816 » * 

689 88822 83828 83885 83841 88847 83853 83860 83866 88872 83879 ^ * 

690 88885 83891 83897 83904 83910 83916 83923 83929 83935 83942 

601 83948 88954 88960 83967 83973 83979 83985 83992 83998 84004 

602 84011 84017 84023 84029 84036 84042 84048 84055 84061 84067 

603 84073 84080 84086 84092 84098 84105 84111 84117 84123 84180 

694 84136 84142 84148 84155 84161 84167 84173 84180 84186 84192 

695 84198 84205 84211 84217 84223 84280 84236 84242 84248 84255 
606 84261 84267 84273 84280 84286 84292 84298 84305 84311 84317 
697 84323 84330 84836 84342 84348 84354 84361 84367 84873 84379 

608 81386 S4392 84308 84404 84410 84417 84423 84429 84435 84442 

609 84148 84454 84460 844C0 84473 84479 84485 84491 84497 84504 
700 84510 S4516 84522 84528 84535 84541 84547 84553 84559 84566 



1 1 

2 1 

3 8 

4 8 



7 4 



834 Logarithms. Tables. 

10. 1 t 8 4 5 « 7 8 9 Dif. 

700 84510 84516 84522 84528 84535 84541 84547 84553 84559 84566 

701 84572 84578 84584 84590 84597 84603 84609 84615 84621 84628 

702 84634 84640 84646 84652 84658 84665 84671 84677 84683 84689 

703 84696 84702 84708 84714 84720 84726 84733 84739 84745 84751 

704 84757 84763 84770 84776 84782 84788 84794 84800 84807 84813 

705 84819 84825 84831 84837 84844 84850 84856 84862 84868 84874 

706 84880 84887 84893 84899 84905 84911 84917 84924 84930 84936 

707 84942 84948 84954 84960 84967 84973 84979 84985 84991 84997 

708 85003 85009 85016 85022 85028 85034 85040 85046 85052 85058 

709 85065 85071 85077 85083 85089 85095 85101 85107 85114 85120 

710 85126 85132 85138 85144 85150 85156 85163 85169 85175 85181 

711 85187 85193 85199 85205 85211 85217 85224 85230 85236 85242 

712 85248 85254 85260 8526& 85272 85278 85285 85291 85297 85303 
718 85309 85315 85321 85327 85333 85339 85345 85352 85358 85364 

714 85370 85376 85382^85388 85394 85400 85406 85412 85418 85425 

715 85431 85437 85443 85449 85455 85461 85467 85473 85479 85485 

717 85491 85497 85503 85509 85516 85522 85528 85534 85540 85546 

716 85552 85558 85564 85570 85576 85582 85588 85594 85600 85606 

718 85612 85618 85625 85631 85637 85643 85649 85655 85661 85667 

719 85673 85679 85685 85691 85697 85703 85709 85715 85721 85727 

720 85733 85739 85745 85751 85757 85763 85769 85775 85781 85788 « 

721 85794 85800 85806 85812 85818 8i^824 85830 85836 85842 85848 ^ ^ 

722 85854 85860 85866 85872 85878 85884 85890 85896 85902 85908 ' ^ 

723 85914 85920 85926 85932 85938 85944 85950 85956 85962 85968 >' 

724 85974 85980 85986 85992 85998 86004 86010 86016 86022 86028 « > 

725 860H4 86040 86046 86052 86058 86064 86070 86076 860^2 86088 > • 

726 86094 86100 86106 86112 86118 86124 86130 86136 86141 86147 •« 

727 86153 86159 86165 86171 86177 86183 86189 86195 86201 86207 t« 

728 86213 86219 86225 86231 86237 86243 86249 86255 86261 86267 • • 

729 86273 86279 86285 86291 86297 86303 86308 86314 86320 86326 > • 

730 86332 86338 86344 86350 86356 86362 86368 86374 86380 86386 

731 86392 86398 86404 86410 86415 86421 86427 86433 86439 86445 

732 86451 86457 86463 86469 86475 86481 86487 86493 86499 86504 

733 86510 86516 86522 86528 86534 86540 86546 86552 86558 86564 

734 86570 86576 86581 86587 86593 86599 86605 86611 86617 86623 

735 86629 86635 86641 86646 86652 86658 86664 86670 86676 86682 

736 86688 86694 86700 86705 86711 86717 86723 86729 86735 86741 

737 86747 86753 86759 86764 86770 86776 86782 86788 86794 86800 

738 86806 86812 86817 86823 86829 86835 86841 86847 86853 86859 

739 86864 86870 86876 86882 86888 86894 86900 86906 86911 86917 

740 86923 86929 86935 86941 86947 86953 86958 86964 86970 86976 

741 86982 86988 86994 86999 87005 87011 87017 87023 87029 87035 

742 87040 87046 87052 87058 87064 87070 87075 87081 87087 87093 

743 87099 87105 87111 87116 87122 87128 87134 87140 87146 87151 

744 87157 87163 87169 87175 87181 87186 87192 87198 87204 87210 

745 87216 87221 87227 87233 87239 87245 87251 87256 87262 87268 

746 87274 87280 87286 87291 87297 87303 87309 87315 87320 87326 

747 87332 87338 87344 87349 87355 87361 87367 87373 87379 87384 

748 87390 87396 87402 87408 87413 87419 87425 87431 87437 87442 

749 87448 87454 87460 87466 87471 87477 87483 87489 87495 87500 

750 87506 87512 87518 87523 87529 87535 87541 87547 87552 87558 



Tables. Logarithiiu. 835 

n. 01 28 45 6709 DK. 

750 87506 87512 87518 87523 87529 87535 87541 87547 87552 87558 

751 87564 87570 87576 87581 87587 87593 87599 87604 87610 87616 

752 87622 87628 87633 87639 87645 87651 87656 87662 87668 87674 

753 87679 87685 87691 87697 87703 87708 87714 87720 87726 87731 

754 87737 87743 87749 87754 87760 87766 87772 87777 87783 87789 

755 87795 87800 87806 87812 87818 87823 87829 87835 87841 87846 

756 87852 87858 87864 87869 87875 87881 87887 87892 87898 87904 

757 87910 87915 87921 87927 87933 87938 87944 87950 87955 87961 

758 87967 87973 87978 87984 87990 87996 88001 88007 88013 88018 

759 88024 88030 88036 88041 §8047 88053 88058 88064 88070 88076 

760 88081 88087 88093 88098 88104 88110 88116 88121 88127 88133 

761 88138 88144 88150 88156 88161 88167 88173 88178 88184 88190 

762 88195 88201 88207 88213 88218 88224 88230 88235 88241 88247 

763 88252 88258 88264 88270 88275 88281 88287 88292 88298 88304 

764 88309 88315 88321 88326 88332 88338 88343 88349 88355 88360 

765 88366 88372 88377 88383 88389 88395 88400 88406 88412 88417 

766 88423 88429 88434 88440 88446 88451 88457 88463 88468 88474 

767 88480 88485 88491 88497 88502 88508 88513 88519 88525 88530 

768 88536 88542 88547 88553 88559 88564 88570 88576 88581 88587 

769 88593 88598 88604 88610 88615 88621 88627 88632 88638 88643 

770 88649 88655 88660 88666 88672 88677 88683 88689 88694 88700 « 

771 88705 88711 88717 88722 88728 88734 88739 88745 88750 88756 ' > 

772 88762 88767 88773 88779 88784 88790 88795 88801 88807 88812 « » 

773 88818 88824 88829 88835 88840 88846 88852 88857 88863 88868 ' * 

774 88874 88880 88885 88891 88897 88902 88908 88913 88919 88925 ' * 

775 88930 88936 88941 88947 88953 88958 88964 88969 88975 88981 ^ * 

776 88986 88992 88997 89003 89009 89014 89020 89025 89031 89037 * « 

777 89042 89048 89053 89059 89064 89070 89076 89081 89087 89092 "* « 

778 89098 89104 89109 89115 89120 89126 89131 89137 80143 89148 * • 

779 89154 89159 89165 89170 89176 89182 89187 89193 89198 89204 * • 

780 89209 89215 89221 89226 89232 89237 89243 89248 89254 89260 

781 89265 89271 89276 89282 89287 89293 89298 89304 89310 89315 

782 89321 89326 89332 89337 89343 89348 89354 89360 89365 89371 

783 89376 89382 89387 89393 89398 89404 89409 89415 89421 89426 

784 89432 89437 89443 89448 89454 89459 89465 89470 89476 89481 

785 89487 89492 89498 89504 89509 89515 89520 89526 89531 89537 

786 89542 89548 89553 89559 89564 89570 89575 89581 89586 89592 

787 89597 89603 89609 89614 89620 89625 89631 89636 89642 89647 

788 89653 89658 89664 89669 89675 89680 89686 89691 89697 89702 

789 89708 89713 89719 89724 89730 89735 89741 89746 89752 89757 

790 89763 89768 89774 89779 89785 89790 89796 89801 89807 89812 

791 89818 89823 89829 89834 89840 89845 89851 89856 89862 89867 

792 89873 89878 89883 89889 89894 89900 89905 89911 89916 89922 

793 89927 89933 89938 89944 89949 89955 89960 89966 89971 89977 

794 89982 89988 89993 89998 90004 90009 90015 90020 90026 90031 

795 90037 90042 90048 90053 90059 90064 90069 90075 90080 90086 

796 90091 90097 90102 90108 90113 90119 90124 90129 90135 90140 

797 90146 90151 90157 90162 90168 90173 90179 90184 90189 90195 

798 90200 90206 90211 90217 90222 90227 90233 90238 90244 90249 

799 90255 90260 90266 90271 90276 90282 90287^90293 90298 90304 

800 90309 90314 90320 90325 90331 90336 90342 90347 90352 90358 



886 Logarithms. Tables. 

10. Of S« 46 6789 on. 

800 90309 00314 90320 90325 90331 90336 90342 90347 90352 90358 

801 90363 90369 90374 90380 90385 90390 90396 90401 90407 90412 

802 90417 90423 90428 90434 90439 90445 90450 90455 90461 90466 

803 90472 90477 90482 90488 90493 90499 90504 90509 90515 90520 

804 90526 90531 90536 90542 90547 90553 90558 90563 90569 90574 

805 90580 90585 90590 90596 90601 90607 90612 90617 90623 90628 

806 90634 90639 90644 90650 90655 90660 90666 90671 90677 90682 

807 90687 90693 90698 90703 90709 90714 90720 90725 90730 90736 

808 90741 90747 90752 90757 90763 90768 90773 90779 90784 90789 

809 90795 90800 90806 90811 90816 90822 90827 90832 90838 90843 

810 90849 90854 90859 90865 90870 90875 90881 90886 90891 90897 

811 90902 90907 90913 90918 90924 90929 90934 90940 90945 90950 

812 90956 90961 90966 90972 90977 90:)82 90988 90993 90998 91004 

813 91009 91014 91020 91025 91030 91036 91041 91046 91052 91057 

814 91062 91068 91073 91078 91084 91089 91094 91100 91105 91110 

815 91116 91121 91126 91132 91137 91142 91148 91153 91158 91164 

816 91169 91174 91180 91185 91190 91196 91201 91206 91212 91217 

817 91222 91228 91233 91238 91243 91249 91254 91259 91265 91270 

818 91275 91281 91286 91291 91297 91302 91307 91312 91318 91323 

819 9132^ 91334 91339 91344 91350 91355 91360 91365 91371 91376 

820 91381 91387 91392 91397 91403 91408 91413 91418 91424 91429 s 

821 91434 91440 91445 91450 91455 91461 91466 91471 91477 91482' 

822 91487 91492 91498 91503 91508 91514 91519 91524 91529 91535 » 

823 91540 91545 91551 91556 91561 91566 91572 91577 91582 91587' 

824 91593 91598 91603 91609 91614 91619 91624 91630 91635 91640 « 

825 91645 91651 91656 91661 91666 91672 91677 91682 91687 91693' 

826 91698 91703 91709 91714 91719 91724 91730 91735 91740 91745' 

827 91751 91756 91761 91766 91772 91777 91782 91787 91793 91798^ 

828 91803 91808 91814 91819 91824 91829 91834 91840 91845 91850' 

829 91855 91861 91866 91871 91876 91882 91887 91892 91897 91903' 

630 91908 91913 91918 91924 91929 91934 91939 91944 91950 91955 

831 91960 919o5 91971 91976 91981 91986 91991 91997 92002 92007 

832 92012 92018 92023 92028 92033 92038 92044 92049 92054 92059 

833 92065 92070 92075 92080 92085 92091 92096 92101 92106 92111 
034 92117 92122 92127 92132 92137 92143 92148 92153 92158 92163 

835 92169 92174 92179 92184 92189 92195 92200 92205 92210 92215 

836 92221 92226 92231 92236 92241 92247 92252 92257 92262 92267 

837 92273 92278 92283 92288 92293 92298 92304 92309 92314 92319 

838 92324 92330 92335 92340 92345 92350 92355 92361 92366 92371 

839 92376 92381 92387 92392 92397 92402 92407 92412 92418 92423 

840 92428 92433 92438 92443 92449 92454 92459 92464 92469 92474 

841 92480 92485 92490 92495 92500 92505 92511 92516 92521 92526 

842 92531 92536 92542 92547 92552 92557 92562 92567 92572 92578 

843 92583 92588 92593 92598 92603 92609 92614 92619 92624 92629 

844 92634 92639 92645 92650 92655 92660 92665 92670 92675 92681 

845 92686 92691 92696 92701 92706 92711 92716 92722 92727 92732 

846 92737 92742 92747 92752 92758 92763 92768 92773 92778 92783 

847 92788 92793 92799 92804 92809 92814 92819 92824 92829 92834 

848 92840 92845 92850 92855 92860 92865 92870 92875 92881 92886 
ft49 92891 92896 92901 92906 92911 92916 92921 92927 92932 92937 
J50 92942 92947 92952 92957 92962 92967 92973 92978 92983 92988 



Table 6. Logarithms. 837 

10. Of 28 45 6739 01. 

B50 92942 92947 92952 92957 92962 92967 92973 92978 92983 92988 

851 92993 92998 93003 93008 93013 93018 93024 93029 93034 93039 

852 93044 93049 93054 93059 93064 93069 93075 93080 93085 93090 

853 93035 93100 93105 93110 93115 93120 93125 93131 93136 93141 

854 93146 93151 93156 93161 93166 93171 93176 93181 93186 93192 

855 93197 93202 93207 93212 93217 93222 93227 93232 93237 93242 

856 93247 93252 93258 93263 93268 93273 93278 93283 93288 93293 

857 93298 93303 93308 93313 93318 93323 93328 93334 93339 93344 

858 93349 93354 93359 93364 93369 93374 93379 93384 93389 93394 

859 93399 93404 93409 93414 93420 93426 93430 93435 93440 93445 

860 93450 93455 93460 93465 93470 93475 93480 93485 93490 93495 

861 93500 93505 93510 93515 93520 93526 93531 93536 93541 93546 

862 93551 93556 93561 93566 93571 93576 93581 93586 93591 93596 

863 93601 93606 93611 93616 93621 93626 93631 93636 93641 93646 

864 93651 93656 93661 93666 93671 93676 93682 93687 93692 93697 

865 93702 93707 93712 93717 93722 93727 93732 93737 93742 93747 

866 93752 93757 93762 93767 93772 93777 93782 93787 93792 93797 

867 93802 93807 93812 93817 93822 93827 93832 93837 93842 93847 

868 93852 93857 93862 93867 93872 93877 93882 93887 93892 93897 

869 93902 93907 93912 93917 93922 93927 93932 93937 93942 93947 

870 93952 93957 93962 93967 93972 93977 93982 93987 93992 93997 » 

871 94002 94007 94012 94017 94022 94027 94032 94037 94042 94047' 

872 94052 94057 94062 94067 94072 94077 94082 94086 94091 94096' 

873 94101 94106 94111 94116 94121 94126 94131 94136 94141 94146' 

874 94151 94156 94161 94166 94171 94176 94181 94186 94191 94196' 

875 94201 94206 94211 94216 94221 94226 94231 94236 94240 94245 <^ 

876 94250 94255 94260 94265 94270 94275 94280 94285 94290 94295 * 

877 94300 94305 94310 94315 94320 94325 94330 94335 94340 94345^ 

878 94349 94354 94359 94364 94369 94374 94379 94384 94389 94394' 

879 94399 94404 94409 94414 94419 94424 94429 94433 94438 94443' 

880 94448 94453 94458 94463 94468 94473 94478 94483 94488 94493 

881 94498 94503 94507 94512 94517 94522 94527 94532 94537 94542 

882 94547 94552 94557 94562 94567 94571 94576 94581 94586 94591 

883 94596 94601 94606 94611 94616 94621 94626 94630 94635 94640 

884 94645 94650 94655 94660 94665 94670 94675 94680 94685 94689 

885 94694 94699 94704 94709 94714 94719 94724 94729 94734 94738 

886 94743 94748 94753 94758 94763 94768 94773 94778 94783 94787 

887 94792 94797 94802 94807 94812 94817 94822 94827 94832 94836 

888 94841 94846 94851 94856 94861 94866 94871 94876 94880 94885 

889 94890 94895 94900 94905 94910 94915 94919 94924 94929 94934 

890 94939 94944 94949 94954 94959 94963 94968 94973 94978 94983 

891 94988 94993 94998 95002 95007 95012 95017 95022 95027 95032 

892 95036 95041 95046 95051 95056 95061 95066 95071 95075 95080 

893 95085 95090 95095 95100 95105 95109 95114 95119 95124 95129 

894 95134 95139 95143 95148 95153 95158 95163 95168 95173 95177 

895 95182 95187 95192 95197 95202 95207 95211 95216 95221 95226 

896 95231 95236 95240 95245 95250 95255 95260 95265 95270 95274 

897 95279 95284 95289 95294 95299 95303 95308 95313 95318 95323 

898 95328 95332 95337 95342 95347 95352 95357 95361 95366 95371 

899 95376 95381 95386 95390 95395 95400 95405 95410 95415 95419 

900 95424 95429 95434 95439 95444 95448 95453 95458 95463 95468 



888 Logarithms. Tables. 

10. 01 t3 4 56 7 8 9oit. 

900 95424 95429 95434 95439 95444 95448 95453 95158 95463 95468 

901 95472 95477 95482 95487 95492 95497 95501 95506 95511 95516 

902 95521 95525 95530 95535 95540 95545 95550 95554 95559 95564 
908 95569 95574 95578 95583 95588 95593 95598 95602 95607 95612 

904 95617 95622 95626 95631 95636 95641 95646 95650 95655 95660 

905 95666 95670 95674 95679 95684 95689 95694 95698 95703 95708 

906 95713 95718 95722 95727 95732 95737 95742 96746 95751 95756 

907 95761 95766 95770 95775 95780 95785 95789 95794 95799 95804 

908 95809 95813 95818 95823 95828 95832 95837 95842 95847 95852 

909 95856 95861 95866 95871 95875 95880 95885 95890 95895 95899 

910 95904 95909 95914 95918 95923 95928 95933 95938 95942 95947 

911 95952 95957 95961 95966 95971 95976 95980 95985 95990 95995 

912 95999 96004 96009 96014 96019 96023 96028 96033 96038 96042 
918 96047 96052 96057 96061 96066 96071 96076 96080 96085 96090 

914 96095 96099 96104 96109 96114 96118 96123 96128 96133 96137 

915 96142 96147 96152 96156 96161 96166 96171 96175 96180 96185 

916 96190 96194 96199 96204 96209 96213 96218 96223 96227 96232 

917 96237 96242 96246 96251 96256 96261 96265 96270 96275 96280 

918 96284 96289 96294 96298 96303 96308 96313 96317 96322 96327 

919 96332 96336 96341 96346 96350 96355 96360 96365 96369 96374 

920 96379 96384 96388 96393 96398 96402 96407 96412 96417 96421 s 

921 96426 96431 96435 96440 96445 96450 96454 96459 96464 96468 > 

922 96473 96478 96483 96487 96492 96497 96501 96506 96511 96515 * 
928 96520 96525 96530 96534 96539 96544 96548 96553 96558 96562 ' 

924 96567 96572 96577 96581 96586 96591 96595 96600 96605 96609 * 

925 96614 96619 96624 96628 96633 96638 96642 96647 96652 96656 ' 

926 96661 96666 96670 96675 96680 96685 96689 96694 96699 96703 • 

927 96708 96713 96717 96722 96727 96731 96736 96741 96745 96750 ^ 

928 96755 96759 96764 96769 96774 96778 96783 96788 96792 96797 > 

929 96802 96806 96811 96816 96820 96825 96830 96834 96839 96844 > 

930 96848 96853 96858 96862 96867 96872 96876 96881 96886 96890 
9-U 96895 96900 96904 96909 96914 96918 96923 96928 96932 96937 
932 96942 96946 96951 96956 96960 96965 96970 96974 96979 96984 
938 96988 96993 96997 97002 97007 97011 97016 97021 97025 97030 

934 97035 97039 97044 97049 97053 97058 97063 97067 97072 97077 

935 97081 97086 97090 97095 97100 97104 97109 97114 97118 97123 

936 97128 97132 97137 97142 97146 97151 97155 97160 97165 97169 

937 97174 97179 97183 97188 97192 97197 97202 97206 97211 97216 

938 97220 97225 97230 97234 97239 97243 97248 97253 97257 97262 
989 97267 97271 97276 97280 97285 97290 97294 97299 97304 97308 

940 97313 97317 97322 97327 97331 97336 97340 97345 97350 97354 

941 97359 97364 97368 97373 97377 97382 97387 97391 97396 97400 

942 97405 97410 97414 97419 97424 97428 97433 97437 97442 97447 

943 97451 97456 97460 97465 97470 97474 97479 97483 97488 97493 

944 97497 97502 97506 97511 97516 97520 97525 97529 97534 97539 

946 97543 97548 97552 97557 97562 97566 97571 97575 97580 97585 

946 97589 97594 97598 97603 97607 97612 97617 97621 97626 97630 

947 97635 97640 97644 97649 97653 97658 97663 97667 97672 97676 

948 97681 97685 97690 97695 97699 97704 97708 97713 97717 97722 

949 97727 97731 97736 97740 97745 97749 97754 97759 97763 97768 

950 97772 97777 97782 97786 97791 97795 97800 97804 97809 97813 



• 4 

• 6 



Table 6. Logarithms. 839 

10. 1 S 8 4 5 « V 8 9 oif. 

950 97772 97777 97782 97786 97791 97795 97800 97804 97809 97813 » 

951 97818 97823 97827 97882 97836 97841 97845 97850 97855 97859 ^ ^ 

952 97864 97868 97873 97877 97882 97886 97891 97896 97900 97905 "> ' 

953 97909 97914 97918 97923 97928 97932 97937 97941 97946 97950 * * 

954 97955 97959 97964 97968 97973 97978 97982 97987 97991 97996 * ' 

955 98000 98005 98009 98014 98019 98023 98028 98032 98037 98041 ^ * 

956 98046 98050 98055 98059 98064 98068 98073 98078 98082 98087 « * 

957 98091 98096 98100 98105 98109 98114 98118 98123 98127 98132 ^« 

958 98137 98141 98146 98150 98155 98159 98164 98168 98173 98177 

959 98182 98186 98191 98195 98200 98204 98209 98214 98218 98223 

960 98227 98232 98236 98241 98245 98250 98254 98259 98263 98268 

961 98272 98277 98281 98286 98290 98295 98299 98304 98308 98313 

962 98318 98322 98327 98331 98336 98340 98345 98349 98354 98358 

963 98363 98367 98372 98376 98381 98385 98390 98394 98399 98403 

964 98408 98412 98417 98421 98426 98430 98435 98439 98444 98448 

965 98453 98457 98462 98466 98471 98475 98480 98484 98489 98493 

966 98498 98502 98507 98511 98516 98520 98525 98529 98534 98538 

967 98543 98547 98552 98556 9856ri 98565 98570 98574 98579 98583 

968 98588 98592 98597 98601 98605 98610 98614 98619 98623 98628 

969 98632 98637 98641 98646 98650 98655 98659 98664 98668 98673 

970 98677 98682 98686 98691 98695 98700 98704 98709 98713 98717 

971 98722 98726 98731 98735 98740 98744 98749 98753 98758 98762 

972 98767 98771 98776 98780 98784 98789 98793 98798 98802 98807 
978 98811 98816 98820 98825 98829 98834 98838 98843 98847 98851 

974 98856 98860 98865 98869 98874 98878 98883 98887 98892 98896 

975 98900 98905 98909 98914 98918 98923 98927 98932 98936 98941 

976 98945 98949 98954 98958 98963 98967 98972 98976 98981 98985 

977 98989 98994 98998 99003 99007 99012 99016 99021 99025 99029 

978 99034 99038 99043 99047 99052 99056 99061 99065 99069 99074 

979 99078 99083 99087 99092 99096 99100 99105 99109 99114 99118 

980 99123 99127 99131 99136 99140 99145 99149 99154 99158 99162 

981 99167 99171 99176 99180 99185 99189 99193 99198 99202 99207 

982 99211 99216 99220 99224 99229 99233 99238 99242 99247 99251 

983 99255 99260 99264 99269 99273 99277 99282 99286 99291 99295 

984 99300 99304 99308 99313 99317 99322 99326 99330 99335 99339 

985 99344 99348 99352 99357 99361 99366 99370 99374 99379 99383 

986 99388 99392 99396 99401 99405 99410 99414 99419 99423 99427 

987 99432 99436 99441 99445 99449 99454 99458 99463 99467 99471 

988 99476 99480 99484 99489 99493 99498 99502 99506 99511 99515 

989 99520 99524 99528 99533 99537 99542 99546 99550 99555 99559 

990 99564 99568 99572 99577 99581 99585 99590 99594 99599 99603 « 

991 99607 99612 99616 99621 99625 99629 99634 99638 99642 99647 ^^ 

992 99651 99656 99660 99664 99669 99673 99677 99682 99686 99691 ' > 

993 99695 99699 99704 99708 99712 99717 99721 99726 99730 99734 * ^ 

994 99739 99743 99747 99752 99756 99760 99765 99769 99774 99778 « * 

995 99782 99787 99791 99795 99800 99804 99808 99813 99817 99822 * « 

996 99826 99830 99835 99839 99843 99848 99852 99856 99861 99865 «> 

997 99870 99874 99878 99883 99887 99891 99896 99900 99904 99909 ^ « 

998 99913 99917 99922 99926 99930 9^935 99939 99944 99948 99952 • ' 

999 99957 99961 99965 99970 99974 99978 99983 99987 99991 99996 • « 

1000 00000 00004 00009 00013 00017 00022 00026 00030 00035 00039 



840 Logarithms. Table 6. 

12 8^^--<^ 5 « 7 8 9 Oil. 

1000 00000 00004 00009 00013 00017 00022 00026 00030 00035 00039 

1001 00043 00048 00052 00056 00061 00065 00069 00074 00078 00082 

1002 00087 00091 00095 00100 00104 00108 00113 00117 00121 00126 

1003 00130 00134 00139 00143 00147 00152 00156 00160 00165 00169 

1004 00173 00178 00182 00186 00191 00195 00199 00204 00208 00212 

1005 00217 00221 00225 00230 00234 00238 00243 00247 00251 00255 
lOOG 00260 00264 00268 00273 00277 00281 00286 00290 00294 00299 

1007 00303 00307 00312 00316 00320 00325 00329 00333 00337 00342 

1008 00346 00350 00355 00359 00363 00368 00372 00376 00381 00385 

1009 00389 00393 00398 00402 00406 00411 00415 00419 00424 00428 

1010 00432 00436 00441 00445 00449 00454 00458 00462 00467 00471 

1011 00475 00479 00484 00488 00492 00497 00501 00505 00509 00514 

1012 00518 00522 00527 00531 00535 00540 00544 00548 00552 00557 

1013 00561 00565 00570 00574 00578 00582 00587 00591 00595 00600 

1014 00604 00608 00612 00617 00621 00625 00629 00634 00638 00642 

1015 00647 00651 00655 00659 00664 00668 00672 00677 00681 00C85 

1016 00689 00694 00698 00702 00706 00711 00715 00719 00724 00728 

1017 00732 00736 00741 00745 00749 00753 00758 00762 00766 00771 

1018 00775 00779 00783 00788 00792 00796 00800 00805 00809 00813 

1019 00817 00822 00826 00830 00834 00839 00843 00847 00852 00856 

1020 00860 00864 00869 00873 00877 00881 00886 00890 00894 00898 « 

1021 00903 00907 00911 00915 00920 00924 00928 00932 00937 00941 ' <> 

1022 00945 00949 00954 00958 00962 00966 00971 00975 00979 00983' > 

1023 00988 00992 00996 01000 01005 01009 01013 01017 01022 01026' ' 

1024 01030 01034 01038 01043 01047 01051 01055 01060 01064 01068' > 

1025 01072 01077 01081 01085 01089 01094 01098 01102 01106 OlllP' 

1026 01115 01119 01123 01127 01132 01136 01140 01144 01149 01153« ' 

1027 01157 01161 01166 01170 01174 01178 01182 01187 01191 01195" 

1028 01199 01204 01208 01212 01216 01220 01225 01229 01233 01237' ' 

1029 01242 01246 01250 01254 01258 01263 01267 01271 01275 01280' ' 

1030 01284 01288 01292 01296 01301 01305 01309 01313 01317 01322 

1031 01326 01330 01334 01339 01343 01347 01351 01355 01360 013C4 

1032 01368 01372 01376 01381 01385 01389 01393 01397 01402 01406 

1033 01410 01414 01418 01423 01427 01431 01435 01439 01444 01448 

1034 01452 01456 01460 01465 01469 01473 01477 01481 01486 014C9 

1035 01494 01498 01502 01507 01511 01515 01519 01523 01528 01532 

1036 01536 01540 01544 01549 01553 01557 01561 01565 01569 01574 

1037 01578 01582 01586 01590 01595 01599 01603 01607 01611 01616 

1038 01620 01624 01628 01632 01636 01641 01645 01649 01653 01657 

1039 01662 01666 01670 01674 01678 01682 01687 01691 01695 01699 

1040 01703 01708 01712 01716 01720 01724 01728 01733 01737 01741 

1041 01745 01749 01753 01758 01762 01766 01770 01774 01778 01783 

1042 01787 01791 01795 01799 01803 01808 01812 01816 01820 01824 

1043 01828 01833 01837 01841 01845 01849 01853 01858 01862 01866 

1044 01870 01874 01878 01883 01887 01891 01895 01899 01903 01907 

1045 01912 01916 01920 01924 01928 01932 01937 01941 01945 01949 

1046 01953 01957 01961 01966 01970 01974 01978 01982 01986 01991 

1047 01995 01999 02003 02007 02011 02015 02020 02024 02028 02032 

1048 02036 02040 02044 02049 02053 02057 02061 02065 02069 02073 

1049 02078 02082 02086 02090 02094 02098 02102 02107 02111 02115 

1050 02119 02123 02127 02131 02135 02140 02144 02148 02152 02156 



Table 6. LogarithniA. 841 

01t846«789oif. 

1050 02119 02123 02127 02131 02135 02140 02144 02148 02152 02156 

1051 021uO 02164 02169 02173 02177 02181 02185 02189 02193 02197 

1052 02202 02206 02210 02214 02218 02222 02226 02230 02235 02239 

1053 02243 02247 02251 02255 02259 02263 02268 02272 02276 02280 

1054 02284 02288 02292 02296 02301 02305 02309 02313 02317 02321 

1055 02325 02329 02333 02338 02342 02346 02350 02354 02358 02362 

1056 02366 02371 02375 02379 02383 02387 02391 02395 02399 02403 

1057 02407 02412 02416 02420 02424 02428 02432 02436 02440 02444 

1058 02449 02453 02457 02461 02465 02469 02473 02477 02481 02485 

1059 02490 02494 02498 02502 02506 02510 02514 02518 02522 02526 

1060 02531 02535 02539 02543 02547 02551 02555 02559 02563 02567 

1061 02572 02576 02580 02584 02588 02592 02596 02600 02604 02608 

1062 02612 02617 02621 02625 02629 02633 02637 02641 02645 02649 

1063 02653 02657 02661 02666 02670 02674 02678 02682 02686 02690 

1064 02694 02698 02702 02706 02710 02715 02719 02723 02727 02731 

1065 02735 02739 02743 02747 02751 02755 02759 02763 02768 02772 

1066 02776 02780 02784 02788 02792 02796 02800 02804 02808 02812 

1067 02816 02821 02825 02829 02833 02837 02841 02845 02849 02853 

1068 02857 02861 02865 02869 02873 02877 02882 02886 02890 02894 

1069 02898 02902 02906 02910 02914 0291S 02922 02926 02930 02934 

1070 02938 02942 02946 02951 02955 02959 02963 02967 02971 02975 4 

1071 02979 02983 02987 02991 02995 02999 03003 03007 03011 03015 lo 

1072 03019 03024 03028 03032 03036 03040 03044 03048 03052 03056 > ^ 

1073 03060 03064 03068 03072 03076 03080 03084 03088 03092 03096 > > 

1074 03100 03104 03109 03113 03117 03121 03125 03129 03133 03137 «> 

1075 03141 03145 03149 03153 03157 03161 03165 03169 03173 03177 • > 
107G 03181 03185 03189 03193 03197 03201 03205 03209 03214 03218 « > 

1077 03222 03226 03230 03234 03238 03242 03246 03250 03254 03258 i > 

1078 03262 03266 03270 03274 03278 03282 03286 03290 03294 03298 » > 

1079 03302 03306 03310 03314 03318 03322 03326 03330 03334 03338 • « 

1080 03342 03346 03350 03354 03358 03362 03366 03371 03375 03379 

1081 03383 03387 03391 03395 03399 03403 03407 03411 03415 03419 

1082 03423 03427 03431 03435 03439 03443 03447 03451 03455 03459 

1083 03463 03467 03471 03475 03479 03483 03487 03491 03495 03499 

1084 03503 03507 03511 03515 03519 03523 03527 03531 03535 03539 

1085 03543 03547 03551 03555 03559 03563 03567 03571 03575 03579 

1086 03583 03587 03591 03595 03599 03603 03607 03611 03615 03619 

1087 03623 03627 03631 03635 03639 03643 03647 03651 03655 03659 
10»8 03663 03667 03671 03675 03679 03683 03687 03691 03695 03699 

1089 03703 03707 03711 03715 03719 03723 03727 03731 03735 03739 

1090 03743 03747 03751 03755 03759 03763 03767 03771 03775 03778 

1091 03782 03786 03790 03794 03798 03802 03806 03810 03814 03818 

1092 03822 03826 03830 03834 03838 03842 03846 03850 03854 03858 
1003 03862 03866 03870 03874 03878 03882 03886 03890 03894 03898 

1094 03902 03906 03910 03914 03918 03922 03926 03930 03933 03937 

1095 03941 03945 03949 03953 03957 03961 03965 03969 03973 03977 

1096 03981 03985 03989 03993 03997 04001 04005 04009 04013 04017 

1097 04021 04025 04029 04033 04036 04040 04044 04048 04052 04056 

1098 04060 04064 04068 04072 04076 04080 04084 04088 04092 04096 

1099 04100 04104 04108 04112 04116 04120 04123 04127 04131 04135 

1100 04139 04143 04147 04151 04155 04159 04163 04167 04171 04175 



842 



Probability of Errori. 



Table 7. 



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Table 8. Properties of Elementary Substances. 843 



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844 Properties of Elementary Substances Tables. 



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Tables. Properties of Elementary Substances. 845 



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ogpacfs 



Suniog 






013 gL Vooi • 

a© 'iBOiqnf) ; 

uoisucdxg * 

ia3iogj903 : 






sninpoj^ 9 






8S9apje(| 



uia gL puB ^ ; 
IB XiisudQ I 



^loioiv 



loquiXs ^ 



6 •? 



oo^Of^ ^ oo o oo q oq 



>0 '^^ 'O • o 



I 
I 
I 



S8"8-88^82-i S.S>:8. :88 ! 8 I 88 | 




q ■ 

Ck^ ^. fr- 

\n • fo tn« ,^ 

d —6 o' ' ^ 

— — (rf 

t^ ,^ c>, 

••••••d •••••••o •• ••••o • 

• •■■••^■••••••I ••••••! • 

:! 1^ ^ 

+ , . . I .J. . .+ .+ 1 . ,.+ ....+ . I 

00 I »n oorou-»OMr»0 ' ^00 "^ w I f »^ ' I [ Q , ** 
d '. ii^«^«ddc«c«doi-*wt^» •.odtn'. • t^^ <f 

pm, *^ ^ ^* 

O"^. « . . ^. *>? ^*S^ ...... ^ . **. ^ . ^_. 

O^ ^ »A> ^ OsOO l> f^ l>H ^ N 00 ^ NOO O »p O •- PO O ^ O 
fOOOOOl>Hf«OWOO «*»00 f* O fO ^ \r.QO ^ ^r^QO 0^«0 Ov 






•6 •SSgc 



•iiiJisH 



i 5^H;5'« o a « S « § o'S a a 



846 



DUIDD13 6 






XifAiionp 



Properties of Solids. 



N< SO • Q\ • • 

i+-r- • 



Table 9. 









:-H-H : 

o o 



in: 



*o« 



In 



8 :«£888g-88 -8: 



IB3H 



,001— ,o 
IBaH 






U13 9^ 
JUIOJ 

guqiog 



Wioj 



83. 



;88 



.o. 






6 



,001— ,0 

IBoiqn') 
noisaBaxg 
laajogjdo^ 



>r 



aranpA jo ^ 






sninpop^ S 
s^Sunoj^ 2 



'3^ a: rJ*!- 



NOVO 












ftiiqsnjf) oj « 



o 



qiSaajis « 
SuiiiEdjg I 



O * r9 I O tofo I 



"?. •T) 



i 



S 



ssdupjcH ! 



I + :+++ :+ 



+ 



X)isa>a *3 



+* 



ti sd d OiOO 00 t-^ 00 00 00 6 



+ 1 + 

»r>ir»*r»5^ t>. 

00 ri ro m ci 



•5 

» s 

i -a 



I 

•3 



.•^ 



.9- 
'3 
S 



Is 



•I 



•«5 

CO c^ 



C n 3* <^ N V ^ 



« G 

^ o S^ 






"I 






Table 9. Building Materials, eta 




848 



Properties of Solids. 



Table 9a. 



NanM 



Symbol 









S' 



n.S 



CO 












Acid 

„ Acetic • . . 

„ Oxalic . . . 

„ Phosphoric . 

„ Phosphorous 

„ „ Hypo- 

„ Sulphuric (hyd.) 

„ Tartaric . . 
Arsenate of 

„ Lead. . • . 

„ Potassium . 

„ „ (anhyd.) 
Borate of 

„ Lead . . . 

ff Potassium . 

n ^ »» . ^i- • • 
„ Sodium . . 

n »» ^i"» • 

., (Borax) . . 
Bromide of 

„ Lead . . . 

„ Potassium . 

„ Silver . . . 
Carbonate of 

„ Barium . . 

„ Calcium . . 

„ Iron . . . 

„ Lead . . . 

„ Potassium . 

„ Sodium . . 

„ „ (acid) . 

„ Strontium . 
Chloral .... 
Chlorate of 

„ Barium . . 

„ Potassium . 

»' o ".. ^^^' • 
„ Sodium . . 

Chloride of 

„ Ammonium. 
„ Barium . . 
„ „ (crystals) 
„ Calcium . . 
11 ^i» (crystals) 
„ Carbon . . 
„ Copper • 
n Iron • • 



HCalhOa. . 
H^C->Oi,3H20 

H.POa - . 
HtPOy . . 
H.S0j,H30 
H3C4H4O6 



PbsAssOg 
KHo' ^ 



l2As04 , 

KAsOs . 



PbBsOi . 
PbB407 , 
KBO2. , 
K2B4O7 , 
NaBOa 



Na2B407 
Na2B407. 

PbBra. 
KBr . 
AgBr. 



10H2O 



BaCOs . . 
CaCOs . . 
FeCOj . . 
PbCOs . . 
KaCOj , . 
Na2G0j. . 
NaHCOs . 
SrCO. . . 
C2H3GI2O2 



BaCl206.H20 
KCIO3 . . 
KCIO4 . . 
NaClOs . . 



H4NCI . . 
BaCl2 . . . 
BaClj.iHsO 
CaCla . • • 
GaCl2.6H20 
C2CL , . , 
CuaClj • . 
FeCl,. . . 



2.9 



16 

40 
72 
17 
10 

135 



117 



600 



,000126 
.000104 



500 
700 
430 



1+ 
4 



850 
850 



46 



.32 
.29 

.073 
.^7S 
.156 

.090 

.114 
.205 
.220 
.257 
.233 
.38s 

.0S3 
"3 
.074 

.110 
,210 

.'93 
.079 
.211 
.26 



.000188 



50 

400 
350 
600 
300 

sub 



97 



400 



.0006 



720 

29 
187 
434 
300 



187 
1000 



148 



.157 

.20 

.19 

.38 

.ogO 

.171 
,164 

4+ 
,2 

.138 



33 



40 



100 
12 



100 
58 



70 

o 

sol 
sol 
47 

4 

7 

o? 

39 
o 

o 
o 
o 
o 

51 

20 

9 

o 

sol 

29 

7 

26 

31 
42 

0+ 

o 

47 



100 

90? 
100 
100 
100 
100 

77 



sol 

36 
67 



SI 
o 

o 
o 
o 
o 

62 

33 



59 
38 

70 

42 
37 
44 
60 
100 



Table 9a. 



Ohemical Materials. 



849 



Name 



Chloride of 

„ Lead . 
„ Lithium 
„ Magnesiu 
„ Mercury . . 
„ ,, (calomel) 
„ Potassium . 
,, & platinum . 
,, Rubidium . 
„ Silver . . . 
,, Sodium . . 
n Strontium . 
„ Tin .... 
i» jj (crystals) 
,, Zmc . • • 
Cliromate of 
„ Lead . • . 
„ Potassium . 

" ^" ,^^" ' • ' 
„ Sodium . • 

Cyanide of 

„ Mercury . . 

„ Potassium . 

„ „ Ferri-. . 

,, „Ferro. . 
Fluoride of 

n, Calcium . . 
Hyposulphite of 

„ Barium . . 

9, Lead . . . 

„ Potassium . 

„ Sodium . . 

,, ,, (crystals) 
Iodide of 

,, Copper • • 

9, Lead . • . 

„ Mercury . . 

„ „(mercurous) 

„ Potassium . 

„ Silver . . . 

„ Sodium . . 
Naphthalene 
Nitrate of 

„ Ammonium. 

^ Barium • • 

y, Lead • • • 

„ Potassium . 



Symb ol 



V O I 

*^ 5 o 
i? S 

V D.-2 



.5 c 



.E'R 






X B 



^« 



i=.S 

I! 



sium 



PbCla . 
LiCl . 
MgCla . 
HgCla . 
H^aCl, 
KCl. . 
KgPtCle 
RbCl . 
AgCl . 
NaCi . 
SrClj . 
SnClj 



500 
600 
700 
290 



900 



300 



.000114 



730 



,00010 
.000121 



SnCla.2H,0 
ZnCla . . • 



PbCr04 
KaCrO* 



450 

77S 
850 
250 

260 



625 
700 



_ ^4 • 

KaCrjOj . 
Na,Cr04. 
NajCrjOt 



.067 
.282 
.194 
067 
.052 
.172 

"3 
.112 
091 
.214 
.120 

.102 

.136 

.090 
.187 
.188 



I 
45 

70? 

7 

o 

26 

I 



27 
35 

67? 
80? 
80 

o 

39 
II 



KSf"* : : ; 

KeFeaCijNia . 
K4FeCeN5.3H20 

CaFj . . . . 



1.9 



.100 

.233 
.280 



60? 

12 

30 
25 



BaSaOs.HjO . 
PbSaOs . . . 
K2S2O3 . . . 
NaaSaOs . . . 
NajSaOs.sHaO 



Cu«Tt 
Pbl, 
Hgl, 
Hf... 

Agl. 
Nal. 
CioHg 



.00004? 



900 



1.7 



.00013 



,000101 
,000072 



,000128 
— jx)ooo4 



30? 

600 
380 
250 
290 
640 
530 
630 
80 



.212 
anh 
.163 
.092 

.'97 
.221 

.445 



5 

57 

35 

36 
5 

o 
28 
50 



45 

50 



35 
55 
44 
50 



.38 



sol 

41 
64 



100 



760 ,069 



900 

350 
310 



.043 

.042 

.039 
,082 
.062 

.088 



215 



36 



H4NN0, 

BaNjOe 
PbNaOe 
KNOj. 



150 
600 



340 



.455 

.150 

114 

.235 



49 



0.1 

5T 


59 

o 
64 

o 

67? 

8 
36 

24 



0.5 



67T 
o 

76 
o-t- 

26 

5« 

71 



850 



Properties of Solids. 



Table 9a. 



Name 



Symbol 



Mi 

O X.O 



si 



"I 



'"if 



S3 



I! 



Nitrate of 

„ Silver . . . 

„ Sodium . . 

„ Strontium . 
Oxalate of 

^, Potassium . 

*, «Tetr-. . 
Oxide of 

^ Aluminum . 

„ Antimony . 

„ Arsenic . . 

,, Bismuth . . 

„ Boron . . . 

„ Calcium . . 

,, ,, (hydrate) 

„ Chromium 

„ Copper . . 

« ,« (cuprous) 

,, Iron . . . 

,, Lead . • . 

„ Magnesium . 

„ (hydrate) 

„ Manganese . 

», Mercury . . 
„ Molybdenum 
„ Nitrogen . . 
„ Potassium . 

„ Silicon ^ . • 

,, Sodium . . 

•1 ^? (hydrate) 

„ Tin .... 

„ Titanium. . 

„ Tungsten. . 

„ Zinc . . . 
Phosphate of 

„ Calcium . . 

„ Lead . . . 

„ Potassium . 

„ „ (acid) . 

f, Sodium . . 

,, ,, (acid) . . 
Silicate of 

„ Al etc. (ciay) 

f, Calcium . • 

^ Zirconium . 



AgNO, 
NaNo.. 
SrNsCSe 



K,Cs04.H20 
KH3C4O8.2H2O 



AljOs . 
SbaOs . 
AsaOs. 
BiaOj . 
B2O3 . 
CaO . 
CaOaHa 
CraOj . 
CuO . 
CuaO . 
FeaOs. 
PbO . 
MgO . 
MgOaH, 
MnO . 
MnOa . 
HgO . 
M0O3 . 
NaOs . 
KaO . 
KOH . 
SiOa . 
NajO . 
NaOH. 
SnOa . 
TiOa . 
WO3 . 
ZnO . 



2P2O7 



CaPj 

Pi 

K4P2O7 

KHaPOi 

Na4P207 

NaaHPO* 



2 — 



210 
310 
650 



3.9 



S-S 



580 



4— 
5+ 



.00004 



3-3 



4.4 



2.2 



.00004 



<4- 



.00002 
.00003 



5-7 



2.3 



30 



i2HaO 



AlaSi307.2HaO,etc 
CaSiOs . . . 
ZrSiOi • . • 



900 
36 



,00002? 



7+ 



,144 
,270 
181 

236 
.283 



65 



198 

093 
,128 
.061 
.237 



.177 
US 
.III 
.16 
.051 

244 
.312 

.157 
.159 
.052 
.154 



47 



19 



.091 
.172 
.085 
125 



.199 
.082 
.191 
,208 
,228 
454 

.2+ 
.178 
.132 



77 



70 
47 
4* 

*5 
5 



90 
64 
50 

40 





0+ 

2 
o 

2 

O 

O 
O 

o 

of 
o 
o 
o 

0.2 
80? 

SO 

67 
o 

40 
60 

o 
o 
o 
o 

(?) 

0+ 

o 

sol 
sol 
5 

20 



o 

0+ 

8? 



.1 — 

0.1 

o 
o 
o . 



0+ 

0.1 



o 

70 



as 





o 





Table 9a 



Chemical Materials. 



851 



Kame 



Symbol 



If] Ti 

1= i; 

it n 

C X 



u X I 



2 



E 



.1 



1^ 

If 









Snip hate of 

,, Ammonium. 
J, Lkiriuni 
,, Calcium . , 
,, „ (hydrat) 
„ Cobalt , , , 
,1 Copper - . 
„ „ (crystals) 
„ !ron . . , 
,, Lead . , , 
„ Magnesium . 
^ „ (hydrat,) 
„ Man^nnese . 
„ „ (hydrat.) 
„ Nickel . * . 
'I „ » (hydrat.) 
,j Potassium » 
» >j (acid) , 
f) „ & Al (alum] 

I, Sodium * . 
n » (cry.staJs) 
„ Strontium . 
I Zinc , . ^ 
11 „ (hydrat*) 
Hiilpliicle of 
J, Antimony . 
J, Bismuih . , 
ff Copper , , 
„ „ (cuprous) 
Tt „ & iron . 
„ Iron 

" t '' P^" 
,1 Lead , 

„ Mercury 

t. Nickel , 

II Po[assiurn 

,T Silver . 

., Tin - , 

Ti „." ^^' 

n Zinc * 
Talc. , . 
Tartrate of . . 

„ Potass, (acid) 
„ ^ ^sodium 



(H<N)3S04 . 

BaSOi , . . 
CaSO^ . , . 

CoS04,7HaO 

CuSOa . , . 

CuiSOi.sHaO 

PeSOj.yHaU 

Pby04 . . . 

MgSO* 

MgSOi 

MnSOi 
N1SO4 . 
NLSO4 
K2SO4 
KHSO4 

K3Al2S40ig.34H30 

KHjCr2S40ie.24H20 
Na2S04 .... 
NasSOi^ioHaO 
SriS04 - . . 
ZnSU4 , . . 
ZnSO4.7H30 

SbS, 



.7HaO 



a— 



Bis S3 . 
CuS , 
Cu^S . 
CuFeSa 
FeS. . 
FeSa , 
PbS • 
H^S , 
NiS. , 
KaS, , 
Ag^S . 
bnS - 
SnSa . 
ZnS , 
3MgO,4SiO 



KHCiHiOfl 



.HoO, 



4HaO 



^3 
■9 

2.32+ 
T.92 

6-3 
2.7 
'■7 

2.0 
2.6 

^'3 

T-724- 

i,S 

27 

1-5 

37 

lo 

2.0|2+ 

74 
4-0 
5.63- 

4.24— 
4.8 

7-5 
7.9 
4.6 
2.1 

5.0 
+^S 
4-', 
2*7, 



.00C06 



T40 



5 



2+ 



4- 
I 



.00^06 



.0000 J 
.00c 07 



POD0036 



* 



205 



900 



SO 



^350 
^rto 

.19 
.36 

'34J, 

'JO 

■3S^ 

.225 

.38 

ag 

'33 
.216 

.341 
•'93 
,244 

^37 ^ 

.530 

J40 
.T74 
■34 

.084 
,o6ci 

,121 
.131 

L136 
,128 
.050 

.128 

^07S 
,084 

JI9 

.122 



'33 



43 

o 

0.2 
0.2 
48 
19 
30 
SO 

o 

26 

31 
SO 
28 
52 
10 

f4 

30? 

60? 

o 

35 
63 

o 
o 
o 
o 
o 
o 
o 
o 
o 
o 

50? 

o 
o 



o 
o 

0.6 

50T 



50 

o 

oa 
0.2 

43 
67 
So 
o 
40 
87 



II 

S3 

* 

o 
87 



852 



Properties of Solids. 



Table 10« 



H-rV 

uoisjddsiQ JO 
xapui 


: 


00 

9 • 








: : : 


!vo !fo! ! !o«^!r) 
•q -q • • -riO -G 




oinuiixBp^ 

(a) UOlJDBJja^ 


. 


. . 








to ^ VO !-• 

. U^ . ON .00 . »^ . , . . 

: »c ^ «j ' ^. * *?• • • • • 




JO xdpui 
















XaBuipjo JO 


o 


VO — M 


1/ 


^n 





^^< ^inr4 c« r4 o>oo o^tnyo 




^>£ 


u^oo « 


oc 


oc 


or 


m ^ rooo ^OO \0 vO !/-> ir> "^ . »* 


■> 


((1) uinjpaj^ 


tn f* 


■>'«i-«*T 


■■^ 


r ^ 


■ T 


■ »i^ u^ u->QO vOvOfO^^nwO ,u 


■» 


•iaBjj3>I xapui 


H* m 










^- ^- c*- ^ ^ J «• ^ ^ cJ ►: M 




uinuiimpv 


. 










NO t^OO O O vo 

; I ', T^x) vo u-> ; Tj- • ; ^ ; 




JO xapui 












c» - »^ •-• •-• ** 




XaiDCdB3 9M% 
















-anpui ogpadg 


• 














XjiAii^npuo^ 


. 














l«5H 


• 














Suiippv JO 


. 














IB3H laaiB'i 


• 














IB9H ^m^^JS 




l-l ^ 








:?:::^:^^f ::: 


•i 


imoj 3aii!og 


• 










:::::::::::§'; 


1 


luioj gqippi 


• 










. ::::::::8^:S: 


_ ft 


fOl-^ 












tn c^ oo 
^ rr>0 •-• 


a 


*\eoiqno 














1 


uoisuedx^ 


• 










S**8SS'' 


Ia3p9j903 












q o o 8 






+ . 








+ . . +1 . . . . 


^ 


8Sdaw»H 








w t^ • 'k© 4 "^ •••«©• • 




A\]SU9(J 


vq + t>. . 




Ch 


+ r^oo <^w o^ot^t^*/^i^l C 


% 




N«-«i-i • • ' hm "^M roO rOC» ^i-<S(VO«iM«N 








i>^0>>'».a*.'^ •••••<-•• 


^ 


'aooc 






k 




:^sii ^^s : : : :^: : :6 : : : 




k 


o 


• SOOO^XO-' • • 'li • • • • • 




k 




S • 


« 


CA 


2 ^^Sffff^sf sS^xi .?« c «■« oca s 


•I 






. . .g . 








1 






• . . 3 • 






► • . g . . . . 




6 
2 


* c 


• : : -I-- 

' • • -eg 

:j :i|e 


•tt 


q -5^ 


, . ..2 . . . , 

• -is • • ' 


1 






• 




« 


< 








<<< 


<« 


aQfflcQ U 


U'J 





Table 10. 



Optical Materials, eto. 



858 



aoisjddsiQ }0 

' xapu'i 



? 



(a) UOliDBJJ3"a 

JO xdpui 



o vo 



!^ 



XjBaipJO ^o 
(a) i"n!P»W 






■j'jc: 



(a) UOJ10IUJ3H 
jO X9pUI 



O M 

QO >0 



■onpui agpadg 



it 



XiiAipnp 

-UQO 

JB»H 



IB3H laaJBl 



IB9H 35P3ds 






2 

a 

8 



•3 

•a 

o 



laioj Snniog 



luioj Suiipi^ 



uoisiiedx^ 
juapgjao^ 



8S9UpjBH 



:± 



+ 1 + 

00 r^ c» 






XllSUSQ 



vO ^o ri »n 



rofn 



c4 C« roro 









o 

HI 

S3 ^ 



I 






£cfe2< 



CQc/3Uc/D CO 



2 
2 

a 




•t, 



o o -a 
UUUU 



•5 



G «2 cSco 



11 ill 




I 



854 



Properties of Soilds. 



Table 10. 



UOISJSdSTQ 

JO xspiij 



uinuiixvpii 

(a) uoiioBjja-a 

JO xspai 



XjreaipjQ ^^ 
(a) wnip^w 



(a) noiabiij^-a 
JO xspuf 



XiioedB^ 9An 
-oripai 3y!39cls 



XiiAq^np 
-003 ;b3H 






;b9h 3g!33<IS 



mo gL 
*Hiioj Saniog 



luioj SnnidK 



IBaiqnp 
noisaBoxg 
juspgjao^ 



fisaopoBH 



iUisaoQ 



6 

CO 



:a 



•o 



• -CL 



■5 



00 M 
VO fO 



► ^8 • • o 



00 



a 
9 



•5 8 






' Ov 



:8 



+-0 



^« 



00 r«« 



«i^ iM d fo ^ 






O 

c« to <0 



M ro t^ • O 00 



:::8 






O 



8S? 



.+->! 



:«• 



q r« O'oo vc 
M ri d 1^ « 



•+ 
9 









ua,c;3< 



CO - 



..2 I . 



'I'sll 

• « « 5 o 

•S 1-g.J 






■g 

m 
S 

s 

1 






o 



•o 



.S 

I 






o-l- 






•5 









I 



Table 10. 



Optioal Materials, eto. 



855 



H — V 

UOISJddSIQ JO 

• xapui 


9- • • • 

eO • • • 


0^ 0^ 

^1 : : : : : : : -^s i : : : i 


uinuiTXBp^ 

(a)aoiiDBJj3^ 

JO xapui 


• • • • t>. ^0 0^ 0^00 ^.'OQ •« . . • .t** 

• • • '^^^^^^(^ *^^ *^ . • . . o^ 


XjBUipjQ JO 

(a) uinipaw 




uinojiuii^ 

(a)uonbBJia'a 
JO X9pui 


i>H lit N r>H ro t>. M Q %rt d NO 

• • • • r*i ¥•* t*^\o O" \^ \ri • r^ 0^ • p^ r^ . ■ • • 

• • • •»^»^^^^^CJ» '^ ^ '^^^ • . • • 


onpuxoyi^dds 


: ^ : 


^ 


..... .-f- ..t.IC! •«! 




iliAiiDnp 

-003 IB9H 


• • • g . 


...::.::.::::::::::§: 


JB3H laajBT 


• •••a. •...^••. •.•••(«)• 




JB9H ogpadS 




naioj Suiijoa 


(^ 6 


>o t^ 


inioj Jmipw 


sr .5 -8, S- •«§. . . . .5* • 


IBDiqnp 
aoTsuBOxg 
W3iogj903 


.0002 

• • • 

• • • 

• • • 

• • • 
.0002 

.000 [ 

• • • 

• • • 

.001 

• • • 


ssdupjBH 


• • '00 • A 00 r^ • • 'lis 


ilimaa 


NO "T q^oq inr»i>.qNoqqq^ , t>.oq in q "^ "V q* ^ 




e 

to 


" • • 


• •......••..•..... 


. . . , 


....... •^5* ••••..•• 


... 1 


.OOn .0 . . , . . S " . . . . 


■ • • t 

• • '6 

• • • e- 

. . .< 




S 




Selenium (crystals) , 
Shellac ...... 

Spermaceti . . . . 

Sninel 


feg.^ 


Sugar (crystals). 

Sulphate* of Copp 
„ Magnesiu 
„ Nickel. 
„ Potassiun 
„ Zinc . 

Sulphur. . . . 

Tallow .... 

Tartar Emetic . 

Tartaric Acid . 

Thallium (prisms) 

Topaz .... 

Tourmaline . . 

Varnishes . . . , 

Vulcanite • . . 

Wax 

Zircon 



4 



856 



Properties of Liquids. 



TablBll. 






•H-y 

uoisjddsiQ 
JO xapn'i 



(a) 

noiioejjd-^ 



XlIOBdB^ 



XiiAipnp 
-U03 iB»H 



-ijo'dB^ 

)B3H tU3lBq 



bOOi— ,o 

'JB3H 






9jn} 
-Bjaduiax 



QQ 



q ? 5. Q 5 Q O Q 



' q S • q q q" o o o S 



ro fO fO fO fO fO (^ fO 



, ^ ^ u^ ro eo fO CO 



.^O O 






« o • o o 



eo « 0\ « ro 



, \n\0 fO M po N 
^ CfOOO POf* 

w d w rt w « 



M - .00 

S. •• =8 



• O ( 



' « 



9 s ••• 



'3 



I 

t 



Suiitog 



O ti^OO 00 . VO t>.00 O ^xTi .MO ^ r*^\0 !>>. »r» *1 
tr> t>* O ti^ • in I- POVO tn O * ^ fOOO r^ PO O ^00*0 



^ ^ ti ^ C^O On 



lUIOd 



-f 



oO JB 
UOISUBdx;} 

ju9jogj303 



8.8.188.8.8.1 'gp.!! • 'I •^§.' 



8uinio^jooj 

93U9I1!S9'S 2 



(,pe)uoisu9X 
dOBjing 



X}lS03SI^ 



(fi) itiSUSQ 






:8 



, O ^ 



o On t>*>o ^ o r** >o o^ o «^vo o ts. 

a^O •- mO •-'00 O^oo vo ro>0 »-• ^»r>»r>rovO r» i-* ►- n< 
00 O'OO On OnOO OO OvO^CimOOOOs 0>QO O 00 00 OO 00 

d o' d d d d 1^ «^' o* d M »-.' i4 1^* d d 6 ■-* o* o* o' d 






•o 

.3? . • 



•9 

-J 






2 



n-S o o 2 



ja o 



^■^l 



O ft K ft 



.•;3 

« 



^3 



u • § § o >> ^ ..t 

O:^ u 3 e« 

fe;2;a,c/3> .^ 

K r^ :: s s s So 



6§ 



^ 



Table It Properties of Liquids. 



857 



(,P2) JOdBA% 

JO dJnssdJj ^ 



H-y 



(a) 

uoipBjja^ 



XipBdB3 



XjiAipnp 



UUBZIJOd 

-b'a IBSH 

__1U31B1_ 

nB^H 






9jn)Bj 
-aduiax 



Suiiiog 



^aaioj 
SuizddJj 



uoisuBdx^ ; 
juaiDyj303 • 






(^c)uoisa3j. 



XjlSODSJA 



(fi) Xireu^a 



6 



S 



a. 
2 

S 






rooo o o H< ^ 

m tr> tr) vr> »r> «^ 



S? 






• q q q q q 






mo ■ 

00 li^ 



»^m' 



r4oooo ^ o N r^^POH* c^r^ «no ir> i-i o , r» S% 

00 ^ ri O ^ t^ *^0 

• •••On*^^***i^ ••••• ts«t/) 

• I ••• • •••11 •••••! 

(s 00 m o^ %i^oo ^ c^ n Th w^ ^ onc« o^r^i/^ 
00 •-• o^oo o xnm o^ n-oo ,OnO ,»-C«Om-^On« 

8 §. 8. § 1. 8 1 8 § 8 § -is. •88.§.8.8. ' 

• •^M •••••••••••••••••• 

« « q 

• •••••••.••••• r^ ••••••• 

S '■ ^ O eo O 

• • li^ » » • • rr) » * m » » m 0\ u^ » |>» • • • 

: :8 : : : :8 : : : : : :8 8 :8 : : : 

O <^ o^o O^ t^i->-vor»oo»-c^p«0»-' •0**o« 

o5 q 00 q "^ <*; ^ '^ •". ^ ^ '?'*. •". *^ ^ ^ ' ^ *0 • 

o •-* 6 •-* "^ «-^ fo »-' rJ N** c« i-I f« fo •-" d d • >-' nJ • 

o 

• • -o^ OqO • •:£ 

. • • "9'^ ••••••••••^ t^-1. 

O « • t-X • • ' :*il -ihO -O 

^xxX:^cQxx££xJ^ y^Xx^^^^^ 
u Of u u u u CO 'Jcj^u cu u 3 c5 Of t3'u u y^u u 



' • • ^ • « o • ' 



jC C 0) O 



5< 



G oi Qi 



s 






:: :: s: r P r 



4> ^ V 

•Hof 



•a 



G-g 

CQ.G 



858 



Properties of Liquids. Tableit 



'H~V 
uofSJddsiQ 



O tr% 

O O 



o • • O O 



• o 



?? 



(a) 

nonoBijd'^ 
JO xapui 






: ^^ 



XllDBdB3 



XliAipnp 
-003 IBSH 



■■I 



aonez 
-uodB^ 



Ov -vo 



X01-.0 

nB3H 



(fiZ) JOdBA » 

jodjnssaj^ m 



|B3ll{J3 2 



s 



9jni 
-Bjadui^x 



00 00 t/> 
•-I w w 



:S 



oiog^'iaioj 
Sufiiog 



^O M fOt-i t>.w >-oo invo t>. u^ in ^ »-t fo < 

N« CS N* M »^ »^ M I 



^lUlOJ 

9aiz99Jj 



T 



,PJB 

aoisuBdx^ 
luabgjaoQ 



^ ONr>.ooo 



8 8 8 • 8 • 8 8 8 : 



9ainioAJo9 



OjOC)UOlSU3X 



XlisoosiA 



88 -8 



ip)liisa9a 



»^* d d « r« 1-4 J N*' 6 •-.* tM* o »-' d N? H? cf hJ i-T J d d 



o 
.0 

a 
>* 

CD 




» I I 
as O 






6 



'♦3 

s 




o5^6SeoS 



7::i:::;:::prp;i<:;::: 



tJ-tsts o^ ti^'B.ati GEO 

O OT3 
•• " ft o s * 



uuu 



Table It 



JO JJpui 



Properties of Liquids. 



859 



|6 xspai 



pnpui-jaJg 



AiTAiijnp 



rfOOl— 1^ 

(^t)JoJDA ^ 



aim 









•o IB 
Doisu^dTcg 












(^) X^isasQ 



B 









:i^: 



r^ 0-. »"Ti Q ^o 
ri >— ri r<T 



p4 fc- iv^ f^oO 
Q q q q q 






O « 

8 



:++ 



^3- r*T Q O O *^ CTH 






§g 



is: 



ss; 



§■■ 



ss 



8 :s 



38S 









o 
pa 



o 
u 



6 



X 






iio z^u u u 3: u u d-u u ^ 



^s 















o 



§ 



O 



&< *- r^'^ ri rJ n 
ir. rl \0 O C'' CJ^ <> 






^ ^ p; ^ 

xuua 



c3 



O 

c6 fe S 



•. *, , n<'U3^Sa.'75 c*l: ti ^ ^ S 

*:; *j *^ r^u jC^ ^"^^ t/j 

fc O cu JJ 3^ ty^>4- 



860 



Properties of Liquids. 



Table 11. 



}6 I3pUJ *j 


q ' O^ -. ■ ■ ■ ^ q ' q q ' q q o^ q q q q 


to) 

JO sjpuj 




'ijnpu[j5J5 * 


MM MM 


.CiiAJT.inp . 


__^-4-P*l,*-vCi^,*-NO 






-■--*»- ^OO » . so • * ■ • O 




s • • -g ■ ■ • s 




"£ 


ajTii^J3dui3X 


gss'S' ..{:-. -s r 


5^. ^ e- UU 




■ o * '« ^ - • • ' * ■ - ' 2^ ■ 1 - - ■ ' 


tiorsuEd£3 ' 


g 8 • § § ■ 8 i 1 i § § 1 8 • • 8 g 8 • S"^ 


»uin[OAJO ES 


■ • • q q - • ■ q • ■ ■ ■ S. 




■ • -a 2, • ■ -^ ■ • • -S 




8g8.3 • ■ -8 -8 • - ■? '5 


Cp) ijreaia 1 


O so "rw gD pi Oh r* o O iJHi CT p* •« co r? t^oo O Q 
H- ^ i-^m O r- c?o3 Ch o^ M o f*Qo , oo is «a » x^ cT' 

-" -: d6'^'^6666^*^'-^6 6 6 6 6 6 6 ^ 




o ' 

€d:,--q8oOo5 5J=^^- 

^.jas ■ -x££i£i ■ • ■ - -^^x- ■ 

mmnM 


1 t 

P. 

1 


o " a ,^rt <y «u «. .5 iJ rr 



•> eg 
CO* 



^1 



i+i 






o 
o 

«^ 

«o 



"Table 18. Properties of Oases and Vapors. 861 






— ddsi 



noisJddsiQ ' 
JO xapai g 



JO xapaj 



Xipcd63 
ogiDdds 



XjiAjianp 

-003 JBSH 



uoues 

•UOpQO^ 



10A nsuop 
lBaH'JP^°S 



9JnSS9J(l 
JUBlSUOf) 






9jm 
•ijddaidx 

IBDIIU^ 



•UID gL uoii 
•B8a3puo3 
JO 'dm ax 



aouBogipn 
-03 JO 9jm 
-BwdwjL 



*ui9 gL 

•^1—0 

noisuBdxg 



rf>i=d*^ . 
*aaiDioA JO o 
9DO»njwH_^ 



•TOO 'bB 

J9d SduXp 

OOO'OOO*! 

> *iCl!SU9a 



98ojpXH 
iwjndG 



092 

oi-j9JJods 



o 
.0 

a 



•a 

s 



8 









C« ! O vO <^ O^ 

N« * cs C4 M r^ 



#-vO e-.*- 'oO 



hfO fO U^ fO 



I 



;8 : 



O Q 






r?5 

O M ^ ^ 



so NO 

00 fO 



5^8^: 



.s: 



00 00 00 

« N. « 



^ fe *1 OP 00 ok 

■j^t«^^oo>ONfOO»/^00«-»^opiKOoo an-tn>* 



IT) » 

• CM 



:+ :- It : 



, rxvo 

:88 



8: 



r» ON 

OxOv 






l^-rt 



•§8 8;^^^ 



S 3 2^2 



$'^\=i-^ 



§88S ^S^'SS 



T3 

* 



:l 






h P^ C^ 4 rONO d 06 \nc6 6 • 
Km r4 »« M >-^ W »* CO ^ 



• tOM O^ON^^MOt^ 






•^.-SP 



s:fx^-££2 -^^-fj ^"o^Ef d^":?d 






•s 












If iyi|2|i§ 'si 



■i .til§§i •1^ '^ 



a 



•^1 






4 

II 

u 



It 
ll 

I- 



O » K 1: S « 



862 Properties of Gases and Vapors. Table 12. 






aoisjddsTQ 1 
JO* »3po'i S 



*at9 gL 'jo 

JO X8pUI 



i;pBdB3 

9Aipnp 

-oio'gpddS 



XliAipnp ^ 
-no3 JB3H O 



noi) 
-Bsudpuo3 



•lOAlsuop 



lOBlSUOf) 

^»»HJPgds 



3JIISS9JJ 



% 



ajiu 
-Bjadoiax 



*ai3 gL uoi) 
-Bsuapuof) 
JO -daiajL 



uouBogipq 
-OS JO ajni 



UJ3 9^ 
aoiSQBdxg 



^i=d ',0 « 
aiunio^ JO o 



•mo 'bs 

J3d 89UXp 

oocyooo'i 



098 

-ojpiCH oi 






■"ST 

;8 



6 N« • 



:| :8 



m 



■s: 



• NO • • 



• eo*n • 

• 0^ I • 



O O N. 



' .3 I ^ I o o I I I 



I 



. - . ^ i^o 
• • • \rtco « 

:::| IT 



::8 









'888 



8? 



tnd ^o ,00 f?oo »A .NO 
r» t>*oo vo "^ ON»o 



*■ d^QO 



• CO • • ••g • 



Scu*S3c»PP 



I 

Q 



K ft ft ft fi 



-I. 






« S 03 8 8 




•I 
cS 



o 

t 



I 



fO ^ 10 eo i-i O 







Table 12. Properties of Gases and Vapors, 

' - ■ . > . « 



■U13 91^ 'pCK 



:8 

O 






863 



I 












a 3"n 

■U13 9i *^ 
jd K3pU| 






o 5 o O q 









-uspuo^ 

1H3HJ P^'JS 
3Jti'^s9Jt[ 



Sin: 






Ul f^ i#i ui 

Q O O O 



^ :2 



^ M ; t^ :^0 "Ci h>VD 



8 ::^ 



2" ^a 



T-i 



I I 



s ■ s § ■ i 



:S 



:| 






|iSiiftirllfl^-^?^tr^§ 



U01JT?3tJ!pt[ 
-BJ3dlUX3j^^ 



irt ^ bj O c^ 

I 1^" ! 






r*^ fn r*Ti 

000 

q o q 



5Pt=d V> , 
3:iu3ilts3Tl 



CO o^ 















0__ tfl 






M5 



^ 



q o q o^^. i^ 

i/^ fil ^ ir^ (A r-loo 
N- n -M tMf J. »^ F^ 



^0^0 



■ \^ r\ 






< 

Sen 
1-^ 



Uli* ff, * trX ^O en « en ^ "'/JrT^Tl^OO S 






(U 



1^ 



< 

§ g a '#■* c?.0 

3 



C/r 



- C 






t/^:^ 



''^^^ 



ut: 5b - Cc^ 



tie - 



^ J3. 



0004l3tiSFi'j=3Ey[;tJr:^ ^"2 ^ "r^ _e _2 ^ h 



t3 _ _ 






CJ'J 



864: 



Pressure of Vapors 



Table 18, A. 



+ § 


* • 

• • 




» • . . 


















: :5. 


+ 2 


• • 




► • , • . 




















+ § 


:-S : 




... 


















: :? 


+\ 


• in \ 




• *. • 




















+1 


• 5- ** 




tn • • • 


















• 00 


+\ 


:5? : 




^% : : 


















s? 


+ li 


I'S : 




^-^:- 




















+% 


:8 : 




Jr^£ : 












00 ■ 




5^5 


+\. 


.• m • 


« •-; « r>. 


» 00 








•S : 


^R5 


+\ 


S 2 - 




« 








»j-.r?'5. 


+1 


>J5 00 • 


s:-q"8':t5? . 




^ ^ kO i-« fo 


+1 


«°-2 !»J?SI?';.5S : 




»o fo ^ ►* fl 


% 


aa°^"s.:«^ = ^?^ ■ 




. q 1^ »n vo w »^ 


11 


8^-J?S :'g'5S• 


» 


: ^ 


; S s- ^ ff- S 


1^ 


3 S^'^S « t^-s 2 


.^ . 




1^ 




> M 4 


• :2 s'S ♦J 


\\ 


• • « o 

«• m • • • • • 




: : + : s "^ : 


\% 


• • *C t> 


M M • • • 


\l 


: :f* : : : : : : 


r>oo :<*>:»« 2" • 


\\ 


• ..^^••..« 

• ••<^.*... 


' •« tn ; « r '^ 2 • 


o 

a 


CaH, 

HsN 

HsAs 

GO 




a 
'0 


■i 


• . lA • • • 

: :K : : : 
: rcJ : : : 


o 
8 

« 


Acetylene .• 

Ammonia 

Arsen'd Hvdroffen 


4 

.a 

c 


Chloride Boron., 
t. Phosphorus 
„ Silicon ,..• 

Chlorine 


1 


• 

J 




1 


Hydriodtc Acid . . 
Hydrochloric,, .. 

Methylether 

Nitrous Oxide ... 
Sulph'd Hydrogen 
Sulphurous Anh. 



Table IBB. 



Pressure of Vapors. 



866 





8. 


• • • 

• • • 


r 




. ^. . 


. : :^? : ■ 








:S 




<s 


* * 00 * 


► ^ . < 


. : :S= : : 






■ "S 




g. 


I r *? : 


« : 


1 tr> • sq q» ^o * 
* 00 * r» 00 ^ 










5 


: :"§> 






:«. : 


: q : - «^ • 






:? 


i 


a 


: :^ 




??1.-t^'J5- 


o> 




: 2« 


t 




ro O u^ 




8 ^No 00 t> *"?« q ^ 
^ t"^ ON ^ ^ M m 


= 5 • 




. 2 5. 


t 








K *8 00 xq 00 o & q j; 

• 11^ l^ ro « ^ 


vd 




. J'^ 


8 


5?^ 




^^.2t?.S«'^^3." 


5.2 


: S^ 


S 


O 


«^8 






'%S 




8 


5;S-t 


s. 








. J- 


1 


% 


^^2 


• • * M cl M * * >-• d\ 


•S-a 




i 


1 


S^? 




r4 % 00 Q u^ ON rt 
Mqqt>.q roMMt^ 

M N« »« N« f>. 


>q 


» 00 

r> q 


H 


"a 




' ^ ' ' ' ' \f% 


'55. 


' 3^ 


1 


oo 

n n o. *^. 


i^eor^t^n OOO o 

q q 'T t> *o «. q ^. k; 


^ q ^3 




sq q 


a 


CO q q 


t>. 


q q n V? *? ^. q ^ 1- * 


d ON 
f> ^ f 


:tS 


s 


% 


O ON 


i-< 


vo vo r^ 
wi-ioo^omMfoON 

q q -. ^9 <* -. q n ^. 


^: 


^ 


^.s! 


«8 


° 


00 RJ 
ro q q 


f 


^ ** A o so *^ c? ON 

q q -. « n q q -. 5;^ 


fo u^ ro 

fOOO d 


%i 


8 


u^ O 


1 


q q q -. -. q q ". ^^ J 


^ 00 ti% 
N w vp -. 


%% 


a 




. o. s. 


8. 


8 8 0% q q q q q^ »< 




^1 


?, 




• 


: -.sf ^8. §|v?« 


' :^? 


£r8. 


flci 

CO 


1 

a 




5^5 






2: 


i • • • • S fl -J 

• • • • o h5 " "^^ • ,^ a> O --< ^ 




* 

leg 


• • 

• • 

• • 

H 



866 



Pressure of Vapors Tables isc-P. 






^ O 



«l 



O » 



2 § 



o S 



§1 



M 



M 



3 ac 



o o 
o 00 



00 tn 



O r-* 



5 « 



ti-» 00 

eo 00 



O d 



o w 



2 8 



8 ? 



s>s 



II 



C4 i-« 



5 



m O 



O OO 



§ is 



\5 vo 



vq 
r4 



8< 






5: * t 



S 2^ 






% 



|2 



2 iP 
I* 

si 



w 



S 



.S 



s> 



s 



5 f ? 



? r 



<j q 



1 3. 



• o O O S • 

• ^ ^ * 4. 

. o o o o . 

, s ^ ^^ ^ . 
g ffi a: X a: 

j3 -P -^ ^ -P O 
^ o o o o «^ 

5 OO h* »n fo 
^ jodv^snoanby^ 




»f N" 






in 



O O^ M 

p o s S. :f 



Tables 14—14 A. Aqueous Vapor. 867 

14. Boiling; Points of Water at Different Pressures (g = 980.61). 

cm. .0 .1 .2 .3 4 .5 .6 .7 .8 .9 
j5 68 96.92 96.96 97.00 97.05 97.09 97.13 97.17 97.21 97.25 97.29 
g 69 97.33 97.36 97.40 97.44 9748 97.52 97.56 97.60 97.64 97.68 
g 70 97.72 97.76 97.80 97.84 97.88 97.92 97.96 98.00 98.03 98.07 
^ 71 98.11 98.15 98.19 98.23 98.27 98.31 98.34 98.38 98.42 98.46 
^ 72 98.50 98.54 98.58 98.61 98.65 98.69 98.73 98.77 98.80 98.84 
*B 73 <)8.88 98.92 98.96 98.99 99.03 99.07 99.11 99.14 99.18 99.22 
g 74 99 26 99.30 99.33 99.37 9941 99.44 99.48 99.52 99.56 99.59 
§ 75 99.63 99.67 99^71 99.74 99.78 99.82 99.85 99.89 99.93 99.96 
g 76 f 00.00 100.04 100.07 100.11 100.15 100.18 100.22 100.26 100.29 100.33 
PQ 77 100.36 10040 100.44 10047 100.51 100.55 100.58 100.62 100.65 100.69 

14A* Dew Points eorresponding to Different Deicrees of Tem peratore 
and Aelative Humidity. 

Relative Humidity of the Air. 

loo/o 20O/0 30P/0 40% 50^/0 60O/0 70»/o 80O/0 90% looO/o 

o® .. .. —16 —12 —9 —7—5 —3 —I 0° 

1 •• .. 15 II 8 6 4 2 o -f < 

2 .. —19 14 10 7 5 3 —I +1 2 

3 .. 18 13 9 6 4 3 o 2 3 
4** .. —17 —12 —8 —6 -3 —I +1 +3 +4 
50 ^^ _,6 _ji _7 _5 _2 o +2 +3 +5** 
6 .. 15 10 7 4 —I +1346 

7.. 15 96302457 
8 ,. 14 —9 52+13568 

9» .. —13 —8 —4 —I +2 +4 +6 +7 +9« 

iqo .. —12 -7—3 o +3 +5 +7 +8 +T00 

.^ II .. II 6 2 +1 3 6 8 9 II 

< 12 —19 10 5—1 2 4 7 9 10 12 

^13 18 10 4 o 3 5 8 10 II 13 

js 140 —17 —9—3+1 +4 +6 +9 +11 +12 +140 

^ 150 _,7 —8 —3 +2 +5 +7 +10 +12 +13 +15** 

o 16 16 7 2 2 6 8 II 13 14 t6 

<o 17 15 6—1 3 6 9 II 14 15 17 

iH 18 14 5 o 4 7 10 12 14 16 18 

2 190 —13 -5 +1 +5 +8 +11 +13 +15 +17 +190 

2 20© —13 —4 +2 +6 +9 +12 +14 +16 +18 +200 

« 21 12 3 3 7 10 13 15 17 19 21 

^22 II 3 4 8 IX 14 16 18 20 22 

6 23 10 —I 4 9 la 15 17 19 21 23 

^ 24® —10 o +5 +10 +13 +16 +18 +20 +22 +240 



H 



250—9 0+6 +10 +14 +17 +19 +21 +23 +25* 

26 8+1 7 II 15 18 20 22 24 26 

27 7 a 8 12 16 19 21 23 25 27 

28 7 3 9 13 17 20 22 24 26 28 
290 —6 +4 +10 +14 +18 +20 +23 +25 +27 .+290 

3o» —5 +5 +" +»5 +18 +21 +24 +26 +28 +300 

31 4 5 II 16 19 22 25 27 29 31 

32 3 6 12 17 2d 23 26 28 30 32 

33 3 7 13 18 21 24 27 29 31 33 

34 2 8 14 18 22 25 28 30 32 34 
3-0 _x +9 +15 +19 +23 +26 +29 +31 +33^+35^ 

A 



868 



Aqaearis Vapor. 



Tables 15, 15 A, 



and a dry bulb thermometer must be multiplied to find the difference between 
the dew-point and the temperature (T) of the air. 



T 


P 


D 


F 


T 


P 


D 


F 


-10* 


0.22 


.0000023 


8.8 


+I00 


0.91 


.0000093 


2.1 


=1 


.23 


2S 


8.5 


II 


0.98 


.0000100 


2.0 


.2S 


27 


8.2 


12 


1.04 


106 


2.0 


-7 


.27 


29 


7.9 


13 


i.ii 


112 


2.0 


—6 


.29 


32 


7.6 


14 


1.19 


120 


1.9 


-5^ 


0.32 


.0000034 


I'l 


+If 


1.27 


.OOOOJ28 


1.9 


—4 


.34 


37 


6.8 


16 


1.35 


ns 


1.9 


—3 


.37 


40 


6.0 


\l 


t.44 


144 


'•2 


— 2 


•39 


42 


5.0 


I.S3 


152 


1.8 


—X 


.42 


45 


4.1 


19 


1.63 


162 


1.8 


OO 


046 


.0000049 


3.3 


+20° 


'•Z^ 


.0000172 


1.8 


4 


-I 


.49 


52 


2.9 


21 


1.85 


182 


1.8 


- 


-2 


.53 


56 


2.6 


22 


1.96 


193 


1-7 


- 


-3 


.57 


60 


2.S 


23 


2.09 


204 


1.7 


- 


h4 


.61 


64 


24 


24 


2.22 


216 


17 


- 


hr 


0.65 


.0000068 


2.3 


+2f 


2.35 


.0000229 


1.7 


- 


-6 


.70 


73 


2.2 


26 


a.50 


242 


1.7 


- 


-7 


.75 


77 


2.2 


27 


2.6s 


2S6 


1.7 


- 


-8 


.80 


82 


2.1 


28 


2.81 


27Q 


1.7 


- 


■^-. 


.85 


87 


2.1 


29 


2.97 


285 


H 




i(fi 


.91 


.0000093 


2.1 


+30« 


3.15 


.0000301 


1.6 



IB A« Speeille Heat of Moist Air nnder Constuit Presrare (76 em.) 



Dew- 


Sjecmc 


Dew- 


^m:t 


Dew- 


%^ei'^ 


Point 


Point 


Point 


— ooo 


.2383 


—1 10 


.2387 


+120 


.2404 


—33 


.2383 


—10 


.2387 


13 


.2405 


—32 


.2384 


zt 


.2388 


H 


,2407 


-31 


.2384 


.2388 


15 


.2408 


—30 


.2384 


-7 


.2388 


16 


.2410 


—29 


.2384 


— 6 


.2389 


17 


.2412 


—28 


.2384 


— 5 


.2389 


18 


.2414 


-27 


.2384 


- 4 


.2390 


19 


.2416 


—26 


.2384 


— 3 


.2390 


20 


.2418 


—25 


.2384 


— > 


.2391 


21 


.2420 


-24 


.2384 


— I 


.2392 


22 


.2423 


—23 


.2384 





.2392 


23 


12428 


—22 


.2385 


+ 1 


.2393 


24 


— 21 


.238s 


2 


.2394 


25 


.2430 


—20 


.2385 


3 


.2394 


26 


.2433 


z\l 


.228s 


4 


.239s 


27 


.2436 


.2385 


5 


.2396 


28 


.2440 


^^l 


•^38S 


6 


.2397 


29 


.2443 


—16 


.2386 


7 


.2398 


30 


.2447 


-15 


.2386 


8 


.2399 


31 


.2451 


-14 


.2386 


9 


.2400 


32 


.245s 


—13 


.2386 


10 


.2401 


33 


351 


—la* 


.2387 


no 


.2403 


I30« 



Tnbles 16. B— C. Aqueoiis Vapor. 



869 



16, B Velocity of Sound In centimetres per second throvgli Atniospberie 
Air at DilTerent Temperatnres and under Differsnt Conditions of Rdatiya 

Humidity. 



=ii^ 














'IT 


0% 


20O/0 


40% 


60O/0 


80O/0 


looo/o 


midity 














0© 


83,220 


33,225 


33,231 


33,236 


33,242 


'33,247 


lO 


33,281 


33,286 


33,292 


33,298 


33,304 


33,310 


3° 


33,34^ 


33,347 


33,353 


33,360 


33,367 


33,373 


^l 


33,402 


33,408 


33,415 


33,422 


33,429 


33,436 


4** 


33,462 


33,469 


33,476 


33,484 


33,491 


33,499 


5* 


33,523 


33,530 


33,538 


33,546 


33,554 


33,562 


6o 


33,583 


33,591 


33,600 


33,608 


33,617 


33,625 


t 


33,643 


33,652 


33,661 


33,670 


33,679 


33,689 


8° 


33,703 


33,713 


33,722 


33,732 


33,742 


33,752 


C ^' 


33,763 


33,773 


33,784 


33,794 


33,805 


33,815 


.-too 


33,823 


33,834 


33,845 


33,856 


33,867 


33,879 


< IlO 


33,882 


33,894 


33,906 


33,918 


33^30 


33,942 


« 12^ 


33,942 


33,955 


33,967 


33,980 


33,993 


34,006 


•S'3^ 


34,001 


34,015 


34,029 


34,043 


34,056 


34,070 


-,40 


34,060 


34,075 


34,090 


34,105 


34,119 


34,134 


I50 


34,120 


34,^36 


34,151 


34,167 


34,183 


34,198 


0,160 


34,179 
34,23» 


34,196 


34,213 


34,229 


34,246 


34,263 


u 170 


34,256 


34,274 


34,292 


34,310 


34,328 


2i8o 


34,297 


34,316 


34,335 


34,354 


34,374 


34,393 
34,458 


«I9° 


34,356 


34i376 


34,397 


34,417 


34,438 


V 20° 


34,41s 


34,436 


34,458 


34,480 


34,502 


34,524 


CU2I<> 


34,474 


34,496 


34,520 


aisi 


34,566 


34,589 


6 22° 


34,532 


34,557 • 


34,581 


34,630 


34,655 


«230 


34,590 
34,649 


34,617 


34,643 


34,669 


34,695 


34,722 


H24O 


34,677 


34,705 


34,732 


34,761 


34,789 


250 


34,707 


34,737 


34,766 


34,796 


34,826 


34,856 


.26O 


34,765 


34,797 


34,828 


34,860 


34,892 


34,924 


27O 


34,823 


34,857 


34,890 


34,924 


34,958 


34,992 


,28° 


34,881 


34,917 


34,953 


34,988 


35,025 


35,061 


290 


34,939 


34,977 


35,015 


35,053 


35,092 


35,130 


30*» 


34,997 


35,037 


35.077 


35,"8 


35,158 


35,«99 


.3^: 


35,055 


35,097 


35,139 


35,182 


35,225 


35,269 


^^! 


35,ti3 


35,157 


35,202 


35,247 


35,293 


35,340 


33** 


35,170 


35t2i8 


35,265 


35,313 


35,362 


35,412 



16, 0« CoeOcienti of InterdiAision of Gases. (0. 6. S.)* 



Air 



Car- 
bonic 
Oxide 

CO 



Hy- 
drogen 
H. 



Meth- 
ane 
CH4 



Nitrous 
Oxide 
N,0 



Oxygen 
O, 



Sulphur- 
ous An- 
h^'dride 
SO, 



Carbonic 

Dioxide CO, 
Hydrogen H s 
Oxygen Oj . . 



.1423 



.6422 
.1802 



.5614 
.7214 



.1586 



.0982 



.1409 
.7214 



•4800 



• See Maxwell's Theory of Heat. 4th Ed. page 392* CErerett Ait 1314 



S70 Barometric Tables. Table 16. 

Reduction op Inches to Centimetres. 



Inches. 





z 


a 


3 


4 


5 


6 


7 


8 

.323 


9 


28.0 


7X.119 


.145 


.170 


.196 


.2:11 


.246 


.272 


.297 


.348 


28.1 


71373 


.399 


•424 


.450 


•475 


.500 


.526 


.551 


.577 


.602 


28.2 


71.627 


.653 


.678 


i^ 


.7^9 


.754 


.780 


.805 


.831 


.856 


28.3 


71.881 


.907 


'^% 


.9S3 


♦008 


*034 


•^59 


*to85 


*IIO 


28.4 


72.135 


.161 


.186 


.212 


.^37 


.262 


.288 


•313 


•339 


:^ 


28.5 


72.389 


.415 


.440 


.466 


.491 


.516 


•542 


.567 


•593 


28.6 


72643 


.669 


.694 


.720 


.745 


.770 


•796 


.821 


.847 


.872 


28.7 


72.897 


.923 


.948 


.974 


.999 


*b24 


*oso 


*o75 


♦lOI 


*I26 


28.8 


73.151 


.177 


.202 


.228 


•253 


.278 


.304 


.329 


.355 


.380 


28.9 


73-405 
73-659 


'W 


.456 


.482 


.5C7 


.532 


.558 


.583 


.609 


:^ 


29.0 


.685 


.710 


.736 


.761 


.786 


.812 


.837 


.863 


29.1 


73.913 


.939 


tt 


.990 


*t>J5 


♦040 


''b66 


^i 


♦117 


•143 


29.2 


74.167 


.193 


.244 


.^69 


.294 


.320 


.345 


.371 


.396 


293 


74.421 


.447 


.472 


.498 


5^i 


.548 


.574 


.599 


.625 


.650 


29.4 


74.675 


.701 


.726 


.752 


.777 


.802 


.828 


.853 


.879 


•904 


29.5 


74.929 


.955 


.980 


♦006 


*o3i 


♦056 


♦082 


♦107 


*i33 


•158 


29.6 


75.183 


.209 


.234 


.260 


.2S5 


.310 


.336 


.361 




:^ 


29.7 


75-437 


.463 


.488 


.514 


•539 


.564 




.615 


.641 


29.8 


75.691 


.717 


•74; 


.768 


.793 


.818 


.844 


.^ 


.895 


.920 


29.9 


75.945 


.971 


.996 


*022 
.276 


♦047 
.30?. 


*op 


*iO^ 


♦123 


♦149 


*i^. 


30.0 


76.199 


.225 


.25b. 


.326 


.352 




.403 


.li 


30.1' 


76.453 


.479 


.504 


.530 


.555 


..5?o 


.606 


.657 


30.2 


76.707 


.733 


.758 




.809 


'^ 


.860 


.885 


.911 


.936 


30.3 


76:961 


.987 


•012 


*o38 


•063 


-1 14 


*i39 


*i65 


♦190 


30.4 


77.215 


.241 


.266 


.292 


.317 


.342 


,368 


.393 


.419 


:« 


30.5 


77.469 


.49s 


.520 


.546 


•571 


.596 


.622 


.647 


.673 


30.6 


77.723 


.749 


■zit 


.800 


.8^5 


.850 


.876 


.901 


.927 


.952 


30.7 
30.8 


77.977 


♦003 


*o53 


*o79 


♦104 


♦130 


*i55 


*i8o 


♦206 


75-^^' 


•257 


.282 


.307 


.333 


.358 


.384 
.638 


.409 


■m 


,460 


30.9- 


78.485 


.511 


.536 


.561 


.5^7 


.612 


.663 


.714 


31.0 


78.739 


.765 


.790 


.815 


.S41 


.866 


.892 


.917 


.942 


.968 


31.1 


78.993 


'*oi9 


*o^ 


•^ 


*09S 


♦120 


♦146 


*i7i 


♦196 


♦222 


31.2 


79.247 


.273 


.298 


.323 


.349 


:U 


.400 


.425 


.450 


.476 


31.3 


79-501 


.527 


.552 


.577 


.603 


:S 


.679 


.704 




314 


79-755 
80.009 


.781 


.806 


.831 


.857 


.882 


•933 


.958 


31.5 


.035 


.060 


.085 


.III 


.136 


.162 


.187 


.212 


.238 


S5> 


.001 


.002 


.003 


.004 


.005 


.006 


.007 


.008 


.009 


.010 


5 Cm. 


.003 


.005 


.008 


.010 


.013 


.015 


.018 


.020 


.023 


.025 



* The star indicates that the namber of whole centimetres is to be read from 
the line underneath it. 



Tables 16 A-~17B . Barometric Tabled. 

16 A. Redaotioii «f Mercurial Oentiinetrefl to Heffadynes per 
cm .0 .1 .2 .3 ,4 A .6 .7 

70 0.9327 0.9340 0.9354 0.9367 0.9380 0.9393 0.9407 0.9420 o, 

71 .9460 .9473 .9487 .9500 .9513 .9527 .9S40 .9553 . 

72 .9593 .9607 .9620 .9633 .9647 .9660 .9673 .9687 . 

73 .9727 .9740 .9753 .9767 .9780 .9793 .9807 .9820 , 

74 .9860 .9873 .9886 .9900 .9913 .9926 .9940 .99«>3 . 
75' .9993 1.0006 1.0020 1.0033 1.0046 1.0060 1.0073 1.0086 1. 
76 1.0126 1.0140 1.0153 1.0166 T.0180 1.0193 1.0206 1.0220 1, 
77 1.0260 1.0273 1.0286 1.0300 1.0313 1.0326 1.0339 1-0353 I 



871 

sq.em. ^=980. 
.8 .9 Dif. 

.94330.944715a 
.9567 .9580 1 I 

.9700 .9713 I 3 

9833 .9847 4 5 
,9966 
.01001.0113 - Q 

,0246 S ,f 



16R Redaction 
cm .0 .1 

70 0.9336 0.9350 

71 .9470 .9483 

72 .9603 .9616 

73 .9737 .9750 

74 .9870 .9883 
76 1.0003 1. 001 7 
761.0137 1.0150 
771.02701.0283 



of Heronrial Centimetres to 

.2 .3 .4 .5 

0.9363 0.9376 0.9390 0.9403 
.9496 .9510 .9523 .9536 
.9630 .9643 .9656 .9670 

.9763 .9777 .9790 .9803 

.9897 .9910 .9923 .9937 

1.0030 1.0043 1.0057 1.0070 

1.0163 1.0177 1.0190 1.0203 

1.0297 1.0310 1.0323 1.0337 



Hegadynes per 

.6 .7 

0.94160.94300. 
.9SS0 .9563 , 
.9683 .9696 . 
.9817 .9830 , 
.9950 .9963 < 

1.0083 1.0097 I, 

1. 021 7 1.0230 I. 

1.0350 1.0363 I. 



; .9980 5 2 

►I.0II3 S 5 

0233 1.< 

,0366 1.0379 9 " 

sq. cm. g=i981. 

.8 .9 Dif, 
.9443 0.9456 135 
.9576 .9590 
.9710 
.9843 

•9977 .9990 5 2 
.01101.0123 2 J 
.0243 1.0257 i iT 
0377 1.0390 9 la 



I 

•9723 5 I 
.9857 4 5 



17. ElcTfttioii in Metres above the Sea Leyel corresponding to Different 
Barometric Pressures at lOR Centigrade (g=980.6). 

cm .0 .1 .2 .3 .4 .5 .0 .7 .8 .9 

00 1959 1945 I931 191 8 1904 1890 1876 1863 1849 1836 

61 1822 1808 1795 1782 1768 1754 1741 1727 1714 1701 

62 1687 1674 1660 1647 1634 1621 1607 1594 1581 1568 

63 1555 ^S4i 1528 1515 1502 1489 1476 1463 1450 1437 

64 1424 1411 1398 1385 1372 1360 1347 1334 1321 1308 

65 1295 1283 1270 1257 1245 1232 1219 1207 1194 1182 

66 1169 1157 1144 1131 1119 1107 1094 1082 1069 1057 

67 1044 1032 1020 1007 995 983 971 958 946 034 

68 922 910 897 885 873 861 849 837 825 813 

e» 801 789 777 765 753 741 729 717 705 693 

70 681 670 658 646 634 623 611 599 587 576 

71 564 552 541 529 517 506 494 483 471 460 

72 448 437 425 4H 402 39' 379 368 356 345 

73 334 322 311 300 288 277 266 255 243 232 

74 221 210 199 187 176 165 154 143 132 121 

75 no 99 88 77 66 55 44 33 22 11 

76 —II —22 —33 —43 — 54 —65 —76 —87 —98 

77 —108 —119 —130 —141 —151 —162 —173 —183 —194 —205 

78 — 215 — 226 — 236 — 247 — 258 — 268 — 279 — 289 — 300 — 310 



17 A. Correction for Temperature in 17. 



Mean Subtc 
Temp. % 

0° 3.5 



I 

2 

3 
4 
5 
6 

7 
8 

9** 



3.2 
2.8 

2.5 
2.1 
1.8 
1.4 
I.I 

0.7 
0.4 



Mean Add 
Temp. •/• 

10 0.0 

II 

12 

13 

14 

IS 

16 

17 
18 

19 



0.4 

0.7 
I.I 

11 

2.1 

2.5 
2.8 

3.2 



Mean Add 

Temp. •/• 

20 



21 
22 
23 
24 
25 
26 

27 
28 
29 



3.5 
3.9 

4.6 
5.0 
5.3 
5.7 

6.0 

64 
6.7 



17 B. Correction for Humidity in 17. 



Dew 
Point 



Add 
% 
0.0 
— 20 0.0 

—15 0.1 
— 10 O.I 



—5 
o 

-2 



0.2 
0.2 
0.3 



--4 0.3 
--6 0.3 
+8 04 



Dew- 
Point 

+10 
II 
12 
13 
14 
IS 
16 

17 
18 

19 



Add 
•/o 

0.5 
0.5 
0.S 
0.6 
0.6 
0.6 
0.7 
0.7 
0.8 
0.8. 



Dew- 
Point 

+20 

21 
22 

23 
24 

26 

27 
28 

29 



Add 
•/. 
0.9 
0.9 
1.0 
I.I 
I.I 
1.2 
1.3 
1.3 
14 
1.5 



872 



Baromotric Tables. Tables 18, a— e. 



to be subtracted. 


Tempe- 


Length in centimetres of the Mercurial Column mcaiurcd 
by a Brass Scale. 


Correction 
for glass 


rature 


70 71 72 73 74 75 76 77 78 


scale 




cm cm cm cm cm cm cm cm cm 




OO 


0.000 0.000 0.000 0.000 0.000 6.000 0.000 0.000 0.000 


0.000 


I 


on oil 012 012 012 012 012 012 OI3 


001 


2 


023 023 023 024 024 024 024 025 025 


002 


3 


034 034 035 035 036 036 037 037 038 

045 046 046 047 048 04^ 049 oso 050 


002 


4 


003 


S 


0.056 0.057 0.058 0.059 0.060 0.060 0.061 0.062 0.063 


0.004 


6 


068 069 069 071 072 072 073 074 075 


005 


7 


079 080 081 082 083 085 086 087 088 


006 


8 


090 092 093 094 095 097 098 099 101 


006 


9 


102 103 104 106 107 109 110 112 113 


007 


10 


O.I 13 0.1 14 O.II6 0.1 18 O.I 19 O.I 21 0.122 0.124 0.126 


0.008 


II 


124 126 128 129 131 133 135 137 138 


009 


12 


135 137 139 141 143 145 147 149 is» 


009 


13 


147 149 151 153 155 157 159 161 164 


010 


14 


158 160 163 165 167 169 172 174 176 


on 


15 


ai69 0.172 0.174 0.177 0.179 0.181 0.184 0.186 0.189 


0.012 


i6 


181 183 186 188 191 194 196 199 201 


013 


;i 


192 195 197 200 203 206 208 211 214 


013 


203 206 209 212 215 218 221 224 227 


014 


19 


215 218 221 224 227 230 233 236 239 


015 


20 


0.226 0.229 0.232 0.236 0.239 0.242 0.245 0.248 0.252 


0.016 


21 


237 241 244 247 251 254 258 261 264 


017 


22 


249 252 256 259 263 266 270 273 277 


017 


23 


260 264 267 271 275 278 282 286 290 


018 


«4 


271 275 279 283 287 291 294 298 302 


019 


25 


0.283 0.287 0.291 0.295 0.299 0.303 0.307 0.311 0.315 


a02o 


26 


294 298 302 306 311 315 319 323 327 


021 


27 


305 310 3H 318 323 327 331 336 340 


021 


28 


317 321 326 330 335 339 344 348 353 


022 


29 


328 333 337 342 347 35^ 35^ 361 365 


023 


30 


0.339 0.344 0.349 0.354 0.359 0.363 0.368 0.373 378 


024 



18b. Correetlon for the Capillarity of Mercurial Colnmna 


to bet 


kdded. 


Internal 


Height of 




Diameter 
of Tube 


Meniscus 
unknown 

.9? 


Height of Meniscus in Centimetres 


0.1 cm 


















0.2 
0.3 
04 


46 
.29 
.26 


0.04 


0.06 


0.08 


0.10 


0.12 


0.14 


0.16 


0.18 


0.083 


0.122 


0.154 


0.198 


0.237 


... 


. . • 


• . • 


0.5 


.15 


.047 


.065 


.086 


.119 


.145 


0.180 


• • • 


• • • 


0.6 


.11 


.027 


.041 


.056 


.078 


.098 


.121 


0.143 


... 


0.7 


.09 


.018 


.028 


.040 


.053 


.067 


.082 


.097 


0.113 


0.8 


.07 




.020 


.029 


.038 


.046 


.056 


.065 


.077 


0.9 


.05 




.015 


.021 


.028 


.033 


.040 


.046 


.052 


1.0 


.04 






.015 


.020 


.02| 
.018 


.029 


.033 


.037 


I.I 


^3 






.010 


/)14 


.021 


.024 


.027 


1.2 


^3 






.007 


^10 


.013 


.015 


.018 


.019 


1.3 


.02 






.004 


.007 


.010 


.012 


.013 


.014 



18 e. Correetlon for the Preasnra of Herenrial Vapor to be added. 

Temperature o<> s** 10^ 15** 2o<» 25** 30** 35"^ 40* 
Add cm. 0.001? /»!? .002? .002? .002? .002? .003? .003? .004? 



Tables 18, d-g. AednotloiiB to <r aud 76 om. 



878 



pres- 
sure 
cm 

70 
71 
72 
73 
74 
75 
76 
77 



18, d. 
.0 

1.0857 
1.0704 
1.0556 
1.041 1 
1.0270 
1.0133 
1. 0000 
.9870 



Faeton for ibe Rednettoit of tbe DensUy of a 6m to 76 em,^ 
.1 .2 \3 4 ^ .6 .7 .8 .9 Oif. 



1.0842 1 
1.0689 I 
1.0541 1, 
1.0397 I 
1.0256 I 
1. 01 20 1 
.9987 
.9857 



.08261/18111/795 
.0674 ijoOsg 1.0644 
.05261.05121.0497 
.0383 1.0368 1.0354 
,0243 1.0229 1.0215 
>oio6 1.0093 1.0080 

9975 .9961 .9948 
9845 .9832 .9819 



1.0780 1.0765 
1.0629 1.0615 
1.0483 1.0468 
1.0340 1.0326 
1. 020 1 1. 01 88 
1.0066 1.0053 

.9935 .9922 
.9806 .9794 



r.0750 1, 
1.0600 I. 
1.0454 I, 
1.0312 I 
1.0174 I, 
1.0040 I 
.9909 
.9781 , 



►0734 1-0719 
.0585 1.0570 
.0440 1 /H^S 
.0298 1.0284 
,01601.0147 
.00261.0013 
,9896 .9883 
9769 .9756 



18, e. Factors for tlie BednetlQB of the Pensity of a 

*^am?e'+^ +1** +2° +3*» +4° +5*» +6*» 

0° I 0000 1.0037 1.0073 1.0110 1,0147 1.0184 1.0220 1 

10 1.0367 1.0404 1.0440 1.0477 1.0514 1.0551 1.0587 I. 

20 1.0734 1.0771 1.0807 1*0844 i>o88i 1.0918 1.0954 I. 

30 i.iioi I.I 138 1.1174 I.I2II 1.1248 1.1285 1.1321 1, 

40 1.1468 1. 1505 1.1541 M578 1,1615 1.165I 1.1688 1. 

50 1.1835 1.1872 1. 1908 1.1945 1.1982 1.2019 1.2055 1 

60 1.2202 1.2239 1*2275 1.2312 1.2349 1.2386 1.2422 I 

70 1.2569 1.2606 1.2642 1.2679 1.2716 1.2753 1.2789 1, 

80 1.2936 1.2973 1.3009 1.3046 1.3083 1.3120 1.3156 1 

90 1.3303 1.3340 1.3376 1.3413 1.3450 1.3487 1.3523 I 

100® 1.3670 1.3707 1.3743 1.3780 1.3817 1.3854 1.3890 1 



Gas to 0^ Centigrade. 
+70 +go ^90 D,^, 

.0257 1.0294 1.0330 atf. 7 
.0624 1.0661 1.0697 1 4 

.0991 1.1028 I.IO64 f T 

.13581.13951.1431 S 11 

.1725 1.1762 1.1798 4 1ft 

.2092 1.2129 1.2165 ft M 

.24591.24961.2532 e tt 

28261.28631.2899 T M 
.3193 1.3230 1.3266 • M 

3560 1.3597 1.3633 • 1^ 
.3927 1.3964 1.4000 



18, f. 

Pres- A 
sure •" 
cm 

70 0.9211 

71 .9342 



Factors for the BedactioB of the Volnme of a Gas to 76 en^ 
.1 .2 ^ 4 ^ .6 .7 ^ 4) 



Dif. 



72 
73 
74 
75 



.9474 
.9605 

.9737 
.9868 



0.9224 0.9237 
.9355 .9368 
.9487 
.9618 
.9750 



.9500 
,9632 
.9763 



76 i.oooo 

77 1.0132 



.9882 .9895 
1. 001 3 1.0026 
1.0145 1.0158 



0.92500.9263 
.9382 .9395 
.9513 .9526 
.9645 .9658 
.9776 .9789 
.9908 *992i 
1.0039 1.0053 
1.0171 i/)i84 



0.9276 0.9289 a 
•9408 .9421 . 

.9539 .9553 . 
.9671 .9684 . 
.9803 .9816 , 

.9934 .9947 . 
1.0066 1.0079 1 
1.0197 1.0211 1 



■93030.93160.9329 t9,n 
.9434 .9447 .9461 
.9566 .9579 .9592 
•9697 .97" .9724 
,9829 .9842 .9855 
.9961 .9974 .9987 
.00921.01051.0118 

.0224 1.0237 1.0250 • IS 



18, g. Factors for the Bednetloii of the Volune of a Gas to 0^ Centliiprade. 



0° 1.0000 5® 0.9820 to9 0.9646 1 5 0.9478 20® 0.9316 25® 0.9160 30^ 0.9008 



V' 



0.9963 
.9927 
.9891 
.9855 
36 



.9785 
.9750 
.9715 
•9680 

35 



.9612 
.9518 
.9545 
.95" 
34 



•9445 
.9413 
.9380 
-9348 
33 



.9285 

.9253 
.9222 
*9i90 

3a 



26 

27 
28 

29 



.9129 
.9098 
.9068 
.9038 

31 



.8978 
.8949 
.8920 
.8891 

29 



574 Atmospheric Density. Table 19 -aoA. 

19. Wcinrlit in ippamg of 1 cable ccntjinctre of dry air. 

Barometric pressure (g = 98aQ 





72 cm 


78 cm 


74 cm 


75 cm 


76 cm 


77 cm 


Diff. 
per cm. 


o® 


.001225 


.001242 


.001259 


.001276 


.001^93 


.001310 


17 


I 


1220 


1237 


1254 


1271 


1288 


1305 


.1 fl 


2 


1216 


1233 


1249 


1267 


1283 


1300 


J fl 


3 


1212 


1228 


1245 


1262 


1279 


1296 


.• 


4 


1207 


1224 


1241 


1257 


1274 


1290 


.4 f 


5* 


.001203 


.001219 


•001236 


.001253 


.001270 


.001286 


.5 • 


6 


1 198 


1215 


1232 


1248 


1265 


1282 


.6 10 


I 


1194 


1211 


1227 


1244 


1260 


1277 


.7 Ifl 


1 190 


1206 


1223 


1239 


1256 


1272 


4 14 


i 9 


1 186 


1202 


1219 


«3S 


1251 


1268 


.9 1ft 


:-^ 


.001181 


.001198 


.001214 


.001231 


.001247 


.001263 


16 


X II 


1 177 


"94 


1210 


1226 


1243 


I2S9 


.1 t 


Z " 


"73 


1189 


1206 


1222 


1238 


I2S5 


j» i 


o 13 


1 169 


1185 


1202 


1218 


1234 


1250 


.8 ft 


V^ 


1 165 


I181 


"97 


1214 


1230 


1246 


.4 • 


S 150 


.001161 


.001177 


.001193 


.001209 


.001225 


.001242 


.ft • 


i 16 


IIS7 


"73 


1189 


1205 


1221 


1237 


.6 10 


^'l 


"53 


1 169 


1185 


1201 


1217 


1233 


.T II 


B 18 


1149 


1165 


1181 


"97 


1213 


1229 


.8 IS 


h'9 


1 145 


1161 


"77 


"93 


1209 


1224 


.9 U 


200 


.001141 


.001157 


.00T173 


.001189 


.001204 


.001220 


15 


21 


"37 


"53 


1 169 


1185 


1200 


1216 


.1 fl 


22 


"33 


"49 


1165 


1181 


1 196 


I2I2 


.8 i 


*3 


1130 


"45 


Ii6[ 


"77 


1192 


1208 


.8 4 


H 


1126 


1141 


"57 


"73 


1188 


1204 


A • 


25« 


•001122 


.001138 


.001153 


.001169 


.001184 


.001200 


.ft f 


26 


1118 


"34 


"49 


1165 


1 180 


1196 


.6 • 


^Z 


1114 


1 130 


"45 


1161 


1176 


1192 


.7 10 


28 


IIIO 


1126 


1142 


"57 


1172 


II88 


.8 18 


^9^ 


1107 


1 122 


1 138 


"53 


1169 


II84 


.9 18 


- 300 


.001 103 


.001119 


.001134 


.001149 


.001165 


.001180 





20. Correctioii for Moisture in Table 19. 



Dew- 

Point 

—8 
—6 

—4 

— 2 



Subtract 

.000,001 
.000,002 
/>00,002 
.000,002 
.000,003 



Dew- 
Point 






Subtract 

.000,003 
.000,003 
•000,004 
.000,004 
.000,005 



Dew- 
Point 

--I0« 

--I2 

--14 

-ti6 
--18 



Subtract 

.000,006 
.000,006 
.000,007 
.000,008 
.000,009 



Dcw- 
Point 
-j-200 
--22 
--24 
--26 
--28 



Subtract 
.000,010 

.000,012 
.000,013 
.000,015 
.000,016 



20 A. Weight in yramg of air digplaced by 1 gram of brans of density 8.4. 



Density of Air 
Weight Displaced 



.00110 
.000131 



.00112 
.000133 



.00114 
,000136 



.00116 
.000138 



.00118 
.000140 



.00120 
.000143 



Density of Air 
Weight Displaced 



.00120 
.000143 



.00122 
.000145 



.00124 
,000148 



.00126 
,000150 



.00128 
,000152 



.00130 
.000155 



Tables 21,22. Beduotion of Apparent Weights. £75 



21. Factors for the Bedaotion of Apparent Weigliiiiga in Air witli Brass 
Weights to Vacuo. 


Density of the Air. 


Density of the Air. 




.00115 


.00120 


.00125 




.00115 


.00120 


.00125 


"2 ^•7<> 


1.00151 


I. 001 57 


1. 001 64 


73 


2.0 


1.00044 


1.00046 


1.00048 


.So.75 


„ 140 


„ 146 


V ^52 


Jc 


2.5 


ff 32 


>i 34 


>i 35 


.g^O.80 


n 130 


» 136 


i> MI 


.^ 


3.0 


>i 25 


n 26 


n ^7 


i 0.85 
•^ 0.90 


„ 122 


» «7 


» 132 


^ 


3.5 


»i 19 


„ 20 


** ^1 


„ I '4 


1) "9 


» 124 


4.0 


„ 15 


„ 16 


„ 16 


8 0-95 


» 107 


„ 112 


H "7 





4.5 


« " 


V " 


•„ u 


« 1.0 


I.OOIOI 


1.00106 


1.00110 




5.0 


1.00009 


I.OOOIO 


I.OOOIO 


.ooog. 


1.00095 
„ 86 


1.00099 


6.0 
7.0 


1.00005 
»> 3 


1.00006 
» 3 


1.00006 
w 3 


^J'-3 


" ^§ 


n 78 


;; 8? 


t/J 


8.0 


« I 


,, I 


„ I 


£ ^4 


„ 68 


It 71 


f, 74 


^ 


9.0 


0.99999 


0.99999 


0.99999 


c; 1.5 


1.00063 


1.00066 


1.00068 


<4^ 


10 


0.99998 


0.99998 


0.99998 


1.6 


„ 58 


„ 61 


„ 63 





12 


„ 6 


„ 6 


>» 5 


fri.7 


n S4 


n 56 


„ 59 


b 


4 


» 5 


i> 4 


n 4 


•g 1.8 
g 1.9 


n 50 


» 52 


yf 55 




16 


ff 3 


>i 3 


n 3 


» 47 


» 49 


»' 5' 


u 


18 


„ 3 


» 2 


« 2 


Q 2.0 


1.00044 


1.00046 


1.00048 


U 


20 


0.99992 


0.99992 


0.99991 



Apparent Specific Volume of Water. 

22. Space in cubic centimetres occupied by a quantity of Water weighing 
apparently 1 gram when counterpoised in Air with Brass Weights of the 
Density 8>4. 







Density of the 


Air 






.00110 


.00115 


.00120 


.00125 


.00130 . 


&> 


1.00109 


1.00113 


1.00117 


I.00122 


1.00126 


I 


1) 103 


» 107 


„ 112 


„il6 


n 121 


2 


1.00099 


,y 103 


„io8 


n 112 


n "6 


3 


V 97 


n lOI 


„ 106 


n IIO 


1, "4 


4 


» 96 


„ 100 


» 105 


1, 109 


1, 114 


5* 


1.00097 


I.OOIOI 


1. 00106 


I.OOIIO 


i.ooit4 


6 


V 99 


» 103 


„ 108 


,1 112 


II 117 


i I 


I.OOI03 


» 107 


„ III 


„ 116 


1, 120 


„ 108 


„ 112 


„ 116 


yy 121 


» 125 


- 9 


» "4 


„ 118 


n 123 


II 127 


II 131 


^ lOO 


1. 00 1 22 


I.COI26 


I.OOI3I 


I.OOI35 


1.00T39 


« II 


,, '31 


V 135 


y, 140 


II M4 


«I48 


pC 12 


V 141 


„ 146 


II 150 


II 155 


n 159 


•^ 13 


» 153 


« 158 


1,162 


n 166 


» 171 


^ M 


„ 166 


» 171 


1, 175 


1, 179 


II 184 


« i5« 


1.00180 


I.OOI85 


I.OOI89 


I.OOI94 


I.OOI98 


^ i6 


„ 196 


„ 200 


II 205 


II 209 


II 214 


^ ^Z 


,» 212 


V 217 


„ 221 


,|225 


» 230 


: «8 


„ 231 


» 235 


II 239 


II 244 


11248 


«» 19 


„ 250 


„ 254 


,,258 


II 263 


II 267 


e 200 
S 21 


1.00270 


1.00275 


1.00279 


1.00283 


1.00288 


„ 291 


» 295 


,1 300 


1, 304 


II 309 


h 22 


N^, 313 


» 318 


II 322 


1,326 


n 331 


23 


,, 336 


w 340 


II 344 


1, 349 


H 353 


24 


n 3w 


« 364 


,,368 


„ 373 


,1 377 


250 


1.00384 


1.00389 


1.00393 


1.00398 


1.00402 


26 


„ 410 


«4i4 


II 419 


,,4^3 


1,428 


27 


» 437 


» 441 


II 445 


„ 450 


,» 454 


28 


,,464 


„468 


ii473 


n 477 


^ 482 


29 


n 492 


If 497 


n 501 


II 505 


» 5^0 


300 


1.0052 1 


1.00525 


1.00530 


1.00534 


1.00538 



Speciflo Volumes. Tables 28-28 B. 



876 

'8d. Spaee in «ii. en. ocevpied by a qaantity of Water weighing 1 fpnm 

in Yacno. 



25® 1.00287 r>if. 

26 I.00313 » 

27 1.00339 JJ 

28 1.00367 » 

29 1.00395 » 
30® 1.00424 ^ 

31 1.00454 f 

32 1.00485 '* 

33 1.005 17 » 

34 I 00550 M 
350 1.00585 » 

36 1.00620 ** 

37 1.00656 * 

38 1.00693 »7 

39 1.00731 » 

40° 1.00769 38 

41 1.00808 » 

42 1.00848 *o 

43 1.00888 *o 
-4 1.00928 ^ 

eO UQO97O ^ 
I.OTO13 ^^ 
I.OTO56 <* 
I.OIIOI *5 
I.OII47 *• 
I.OII94 *^ 



O© I.000I2Dif. 
I 1.00006 — * 
1.00002 — * 

-.a 



3 

4 

r 



I.OOOOO 

0.99999 

I.OOOOO 
1.00002 
1. 90006 

8 i.oooti 

9 1.00017 
10® 1.00025 
II 1.00034 

1.00044 
1.00056 
1.00069 
1.00083 
1.00099 
1.00115 
I.OOI33 
I.OOI52 
ao» 1.00173 

21 I.OOI94 
^2 1.00216 

23 1.00238 

24 1.00262 
25® 1.00287 



12 

13 
14 
150 

16 

17 

18 

"9 



— 1 

+1 
t 

4 
5 
6 
8 

9 
10 

la 

13 

u 

16 
16 
18 
18 
81 
81 
88 
88 
2i 
85 



44 

46 

47 
48 

49 

500 



50®i.OIi94Dif. 

51 I.OI242 « 

52 I.0129I ^ 

53 1.01340^ 

54 1.01389^ 

550 1.01438 « 

56 1.01487 ^ 

57 i.oiS36« 

58 1.01586 «> 

59 1.01637 »» 
60® 1.01690 ® 

61 1.01743 M 

62 1.01797 ** 

63 1.01851 ** 

64 1.01907 56 
65° I.OI963 56 

66 1.02020 ^ 

67 1.02077 5^ 

68 I.02136 M 

69 1.02195 5* 
70° 1.02255 W 

71 I.O2315 ® 

72 1.02377 « 

73 1.02439 O 

74 1.02502 w 

75° 1.02565 « 



75« 

76 

77 

78 

79 

80® 

81 

82 

83 

84 

850 

86 

87 
88 

89 
900 

91 

92 

93 

94 

95° 

96 

98 

99 

too® 



.02565 Dif 
.02629 ** 
.02693 ^ 

.02757 2 
.02821 «* 

.02886 « 
.02951 65 
.03017 ®* 
.03084 CT 
.03152 ® 

r .03220 ® 
[.03288 « 
■.03357 ® 
.03426 ® 

.03496 TO 

1.03566 TO 
r .03637 Ti 
•.03709 w 
.03781 « 
.03855 « 
.03930 w 
.04005 T* 
..04081 w 
t.04157 3; 

.04234 TI 

.04311 w 



B, A. Space in en. cm.-occnpied by 1 gram of Herenry. 



.0« 


0.073,551 


lOO 


0.073,684 


20° 


0.073,816 




Dif. 




I 


.073.564 


II 


.073*697 


21 


.073,830 




IB 


14 


2 


.073,578 


12 


.073.710 


22 


.073,843 


.1 


I 


I 


3 


.073,591 


13 


.073,723 


23 


.073.856 


.2 


5 


3 


4 


.073,604 


14 


.073,737 


M 


.073,870 


.3 


4 


4 


5* 


0.073,617 


15* 


0.073,750 


25® 


0.073,883 


4 


S 


6 


6 


.073,631 


16 


.073763 


26 


.073,896 


.5 


I 


7 


I 


.073,644 


17 


.073,776 


*z 


.073,910 


.6 


8 


.073,657 


18 


.073,790 


28 


.0731923 


.7 


9 


10 


9 


.073,670 


19 


.073,803 


^%. 


.073,936 


.8 


10 


II 


loo 


0.073,684 


200 


0.073,816 


30« 


0.073,950 


.9 


12 


13 



28« B. Space in en< 
apparently i gram 



cm. occupied by a quantity of Herenry weighing 
when balanced by Brass Weights of Density 8.4 in 
Air of Density .0012. 



oo 0.073,547 


10® 0.073,680 


20® 0.073,812 


Dif. 


I .073,560 


11 .073,693 


21 .073,826 


18 14 


« .073 574 


12 .073,706 


22 .073,839 


.1 I I 


3 .073.587 


13 .073,719 


23 .073,852 


.233 


4 .073,600 


14 -073733 


24 .073,866 


.344 


5* 0.073,613 


15** 0.073,746 


25** 0.073,879 


A 5 6 


6 .073,627 


16 .073,759 


26 .073,892 


.577 


7 .073,640 

8 .073,653 


17 .073,772 


27 .073,906 


.6 8 8 


18 .073,786 


28 .073,919 


.7 9 10 


9 .073,666 


T9^ .073799 


29^ .073,932 


.8 10 11 


to® 0.073,680 


20® 0.073,812 


30® 0.073,946 


^ 13 13 



Tables 24—26. Standard Densities. 



877 



Density of Water, Mercury and Glycerine. 

^, Density of Mercury at different temperatures. 



OO 


. 13.596 


900 


13.377 


1 800 


13.162 


2700 


12.948 


10° 


13.572 


100° 


13.353 


1900 


13.138 


280O 


12.924 


20© 


13.547 


110° 


13.329 


200° 


13.114 


290® 


12900 


30° 


13.523 


120° 


13.305 


210° 


13.091 


3OOO 


12.876 


400 


13.498 


130° 


13.281 


220° 


13.067 


310° 


12.853 


50° 


13.474 


140° 


"3.257 


230° 


13.043 


320O 


12.829 


600 


13.450 


150° 


13.233 


240° 


13.019 


330° 


12.805 


70° 


13.426 


1600 


13.210 


250° 


12.995 


340° 


12.781 


80° 


13.401 


170° 


13.186 


260O 


12.972 


350° 


12.757 



26. Density of Water at different temperatures. 



00 


0.9998? 


25« 


0.99714 


Soo 


0.98819 


75"* 0.97497 


I 


94 


26 


.99687 


51 


772 


76 437 


2 


98 


27 


61 


52 


725 


77 376 


3 


I.OOQOO 


28 


34 


53 


677 


78 315 


4 


01 


29 


06 


54 


629 


79 254 


5^ 


1. 00000 


300 


0.99578 


55^ 


0.98582 


800 0.97193 


6 


0.99998 


31 


548 


56 


534 


81 131 


7 


94 


32 


518 


57 


486 


82 069 


8 


89 


33 


486 


58 


437 


83 006 


9 


83 


34 


453 


59 


388 


84 .96942 


lOO 


0.99975 


35^ 


0.99419 


600 


0.98338 


850 0.96878 


II 


66 


36 


384 


61 


286 


86 814 


12 


56 


37 


348 


62 


234 


87 750 


13 


44 


38 


311 


63 


I8i 


88 686 


14 


31 


39 


274 


64 


127 


89 621 


15° 


0.99916 


400 


0.99236 


650 


0.98073 


900 0.96554 


16 


01 


41 


198 


66 


018 


91 488 


17 


.99885 


42 


158 


67 


.97963 


92 421 


18 


67 


43 


118 


68 


907 


93 354 


19 


48 


44 


078 


69 


850 


94 286 


20O 


0.99828 


4f 


0.99037 


700 


0.97793 


95® 0.96216 


21 


07 


46 


.98996 


71 


735 


96 146 


22 


.99785 


*Z 


954 


72 


676 


97 076 


23 


62 


48 


21^ 


73 


617 


98 005 


^^. 


39 


49 


865 


74 


557 


99 .95934 


25° 


0.99714 


500 


0.98819 


75^ 


0.97497 


looo 0.95863 





26. Density of Oommereial Glycerine (o^3gO. 




00 1.269 


50 1.266 


10° 1.263 


150 1.260 


20® 1.257 


25^ i.2:>3 


lo 1.268 


60 1.265 


iio 1.262 


16° 1.259 


21® 1.256 


26® 1.253 


ao 1.268 


7^ 1.265 


12° 1.262 


170 T.258 


22® I.25:> 


27® 1.252 


3^ 1.267 


8° 1.264 


13<* 1. 261 


1 80 1.258 


23° 1.255 


28® 1.252 


4° 1.267 


9® 1.263 


140 1.260 


19"* 1.257 


24® 1.254 


29® 1.251 


5^ 1.266 


10® 1.263 


I5<* 1.260 


20® 1.257 


25* 1.253 


30® 1.25 1 



878 



Density of Alcohol. 



Table 27. 



yt 


VOL. IS^I 15" 1 


16^ 1 jy'' IS^ 1 19° 1 ao" 1 2I*» 1 22** 


o 


0.00 


.9993 


.9990 


.9988 


.9987 


.9985 


.9983 


•9981 


.9979 


I 


1.26 


.9971 


.9969 


.9967 


.9966 


.9964 


.9962 


.9960 


.9958 


a 


2.51 


.9953 


.9951 


•9949 


.9947 


.9945 


•9943 
.9926 


.9942 


.9940 


3 


3-75 


•9936 


•9934 


•9932 


.9930 


.9928 


•9924 


.9923 


4 
5 


5.00 
6.24 


.9920 
•9903 


.9918 


.9916 


'9897 


.9912 
.9895 


.9909 
.9892 


•989^ 




6 


7-47 


.9887 


.9885 


•9883 


.9880 


.9878 


.9876 


.9874 


.9S72 


I 


8.70 


.9871 


.9869 


.9866 


.9864 


.9861 


.9859 


.9857 


.9S55 


9-93 


.9856 


.9854 


.9851 


.9849 


.9846 


.9844 


.9842 


.9839 


9 


II. 16 


.9842 


.9839 


•9837 


•9834 


.9832 


.9829 


.9827 


.9S24 


lO 


12.38 


.9828 


.9825 


.9823 


.9820 


.9817 


.9815 


•9813 


.9S10 


II 


1359 


.9814 


.9811 


•9809 


.9806 


.9803 


.9800 


.9798 


.9795 


12 


14.81 


.9801 


.9798 


•9795 


•9793 
.9780 


.9790 


.9787 


.9784 


.9781 


13 


16.03 


.9789 


.9786 


•9783 


.9777 


.9774 


•9772 


.9769 


H 


17.24 


.9777 


.9774 


.9771 


.9768 


•9765 


.9762 


•9759 


.9756 


15 


18.45 
19.65 


•9765 


.9762 


•9759 


•9755 


.9752 


.974? 


.9746 


.9743 


i6 


•9753 


.9750 


.9746 


•9743 


.9740 


.9736 


•9733 


.9730 


:^ 


20.85 


•9741 


.9738 


•9734 


•9731 


.9727 


.9724 


.9721 


.9717 


22.05 


.9729 
.9718 


.9725 


.9722 


.9718 


.9715 


.9711 


.9708 


.9704 


19 


23.25 


.9714 


.9711 


•9707 


.9703 


.9699 


.9696 


.9^92 


20 


2445 


.9707 


.9703 


.9699 




.9691 


.9687 


.9683 


.9679 


21 


25.64 


•9695 


.9691 


.9687 


.9683 


.9679 


.9674 


.9670 


.9666 


22 


26.83 


.9683 


.9679 


.9674 


.9670 


.9666 


.9661 


.9657 


.9'553 


23 


28.01 


.9671 


.9666. 


.9662 


•9657 


.9653 


.9648 


.9644 


.9^39 


24 


29.19 


.9659 


.9654 


.9650 


.9645 


.9640 


.9635 


.9631 


.9(^26 


25 


30.37 


.9647 


.9642 


.9637 


.9633 


.9627 


.9621 


.9617 


.9612 


26 


31.54 


.9633 


.9628 


.9623 


.9618 


.9613 


.9607 


.9602 


.9597 


27 


3^-71 


.9619 


.9614 


.9608 


:» 


.9598 
.9583 


.9592 


.9587 


.9582 


28 


33.86 


.9604 


.9599 


.9593 


•9577 


.9571 


.9566 


29 


35.02 


.9589 


^9583 


.9578 


•9572 


.9567 


.9561 


.9555 


•9549 


30 


36.17 


.9573 


.9567 


.9561 


•9556 


.9550 


.9544 


.9538 


.9532 


31 


37.30 


.9556 


.9550 


.9544 


.9538 


.9532 


.9526 


.9520 


.9514 
.9496 


32 


38.44 


.9539 


•9533 


.9527 


.9521 


.9515 


•9508 


.9502 


33 


39.57 


.9522 


.9516 


.9509 


.9503 


.9497 


.9490 


■iSi 


.9478 


34 


40.69 


:SJ 


.9498 


.9491 


.9485 


.9479 


.9472 


.9459 


3| 


41.81 


.9479 


.9^73 


.9466 


.9460 


.9453 


.9447 


.9440 


36 


42.92 


.9467 


.9460 


.9454 


.9447 


.9440 


.9433 


.9427 


.9420 


37 


44.02 


.9448 


.9441 


.9434 


.9428 


.9421 


.9414 


.9407 


.9400 


38 


45.12 


.9429 


.9422 


.9415 


•9408 


.9401 


.9394 


.9388 


.9381 


39 


46.21 


.9410 


.9403 


•9396 


•9389 


.9382 


.9375 


.9368 


.9361 


40 


47.30 


.9390 


.9383 


.9376 


:l^ 


.9362 


.9354 


.9347 


.9340 


4^ 


48.38 


.9370 


.9363 


.9356 


.9341 


.9334 


.9327 


.9320 


42 


49-45 


•9349 


.9342 


.9334 


•9327 


.9320 


.9312 


.9305 


.9298 


43 


50.51 


.9328 


.9321 


.9313 


.9306 


.9298 


.929^ 


•9284 


.9276 


44 


51.57 


.9307 


.9290 
.9278 


.9292 


•9284 


.9277 


.9248 


.9262 


.9254 


jj 


52.62 


.9286 


.9271 


.9263 


.9256 


.9240 


.9233 


53.67 


.9265 


.9257 


.9250 


•9242 


.9234 


.9226 


.9219 
.9198 


.9211 


s 


54.71 


.9244 


.9236 


.9229 


.9221 


.9213 


.9205 


.9190 


55.75 


•9223 


.9215 


.9207 


.9200 


.9192 


.9184 


.9176 


.9168 


49 


56.78 
57.80 


.9201 


.9193 


.9185 


.9178 


.9170 


.9162 


.9154 


.9146 


50 


.9179 


.9171 


.916J 


.9155 


.9147 


.9^39 


.913a 


.9124 



Table 27. 



Density of Alcohol. 



879 



w't 


VOL. 15° 


X5** 


i6*» 


17** 


x8^ 


19° 


ao*» 


21° 


aa* 


5° 


57-^ 


.9179 


.9171 


.9163 


.9155 


.9147 


•9139 


.9132 


.9124 


51 


58.81 


•9157 


.9149 


.9141 


•9133 


.9125 


.9117 


.9110 


•9102 


52 


59.82 


•9135 


.9127 


.9119 


.9111 


.9103 


•9095 


.9087 


.9079 


53 


60.82 


•9"3 


.9105 


.9097 


.9089 


.9081 


-9073 


-9065 


•9057 


54 


61.82 


.9091 


.9083 


•9075 


.9067 


•9059 


.9050 


.9042 


•9034 


55 


62.81 


.9069 


.9061 


•9053 


•9045 


•9037 


.9028 


.9020 
-8997 


.9012 
-8989 


56 


6379 


.9046 


.9038 


.9030 


.9022 
.8998 




-9005 


57 


64.77 


.9023 


.9015 
.8992 


.9007 


.8982 


.8974 


.8966 


58 


6574 


.9000 
.8977 


.8984 


-8975 


.8967 


-8959 


.8951 


•89^3 


59 


66.70 


.8969 


.8961 


.8952 


.8944 


.8936 


.8928 


.8920 


60 


67.65 


•8954 


.8946 


.8938 


.8929 


.8921 




.8905 


.8897 


61 


68.60 


•8931 


.8923 


Xt 




.8898 


.8882 


.8873 


62 


6955 


.8908 


.8900 


.8875 


.8867 


.8859 
.8836 


.8850 


63 


70.49 


.8885 


.8877 


.8868 


.886^ 


.8852 


.8844 


.8827 


64 


71.42 


.8862 


.8854 


.8845 


-8837 


.8829 


.8821 


.8813 


.8804 


65 


72-34 


.8838 


.8830 


.8821 


.8813 


.8805 


.8797 


.8789 


.8780 


66 


73.26 


.8815 


.8807 


.8798 


.8790 


.8782 


•8773 


.8765 


.8756 


•67* 


74.18 


.8792 


.8784 


.8775 


-8767 


•8759 


.8750 


.8742 


•8733 


68 


7508 


.8768 


.8760 


.8751 


-8743 


-8735 


.8726 


.8718 


.8709 


69 


75-98 


.8744 


.8736 


.8727 


.8719 


.8711 


.8702 


.8694 


.8685 


70 


76.88 


.8721 


.8713 


.8704 


.8696 


.8688 


.8679 


.8671 


.8662 


71 


77-77 


.8698 


.8689 


.8681 


.8672 


.8664 


.8655 


.8647 


.8638 


73 


78.65 


.8674 


.8665 


.8657 


.8648 


.8640 


.8631 


.8623 


.8614 


73 


79-51 


.8649 


.8640 


.8632 


.8623 


.8615 


.8606 


.8598 


.8589 


74 


80.37 


.8625 


.8616 


.8608 


-8599 


.8591 


.8582 


.8574 


.8565 


75 


81.23 


.8601 


.8592 


.8584 


.8575 


.8567 


.8558 


.8550 


.8541 


76 


82.08 


.8576 


.8567 


•8559 


.8550 


.8542 


•8533 


.8525 


.8516 


77 


82.92 


.8553 


•8543 


.8535 


.8526 


.8518 


.8509 


.8501 


.8492 


78 


83.76 


.8528 


.8519 


.8511 


.8502 


-8494 


•8485 


.8476 


.8468 


^ 


84.59 


.8503 


.8494 


.8486 


.8477 


.8469 


.8460 


.8451 


.8443 


85.41 


.8478 


.8469 


.8461 


.8452 


.8444 


•8435 


.8426 


.8418 


81 


86.22 


.8453 


.8444 


.8436 


.8427 


.8419 


.8410 


.8401 


-83|3 


82 


87.03 


.8428 


.8419 


.8411 


.8402 


.8394 


.8385 


.8376 




83 


87-84 


.8404 


•8395 


.8387 


.8378 


.8370 


.8361 


.8352 


'8344 


84 


88.63 


.8379 


.8370 


.8362 


Zl 


.8345 


.8336 


•8327 


.8319 


85 


89.42 


.8354 


.8345 


.8337 


.8320 


.8311 


.8302 


.8294 


86 


90.20 


.8329 


.8320 


.8312 


.8303 


.8295 


.8286 


.8277 


.8269 


87 


90.97 


.8303 


.8294 


.8286 


.8277 


.8269 


.8260 


.8251 


.8243 


88 


91.72 


.8277 


.8268 


.8260 


.8251 


.8243 


.8234 


.8225 


.8217 


89 


92.47 


.8251 


.8242 


.8234 


.8225 


.8217 


.8208 


.8199 


.8191 


90 


93.22 


.8225 


.8216 


.8208 


.8199 


.8190 


.8181 


.8173 


.8164 


91 


93.96 


.8199 


.8190 


.8182 


.8173 


.8164 


.8155 


.8147 


.8138 


92 


94.68 


.8172 


.8163 


.8155 
.8128 


.8146 


.8137 


.8128 


.8120 


.8111 


93 


95.39 


.8118 


.8136 


.8119 


.8110 


.8101 


.8093, 


.8084 


94 


96.09 


.8109 


.8101 


.8092 


.8083 


.8074 


.8066 


•8057 


95 


96.78 


.8090 


.8081 


.8073 


.8064 


.8055 


.8046 


.8038 


.8029 


96 


97.45 


.8061 


.8052 


.8044 


-8035 


.8026 


.8017 


.8009 


.8000 


97 


98.11 


.8032 


.8023 


.8015 


.8006 


•7997 


.7988 


.7980 


•7971 


98 


98.75 


.8002 


•7993 


.7985 


-7976 


.7967 


.7958 


.7950 


•7941 


99 


9938 


.7972 


-7963 


.7955 


.7946 


•7937 


.7928 
.7897 


.7920 


:S 


100 


100.00 


•7941 


•7932 


.7924 


-7915 


.7906 


.7889 



880 



Density of Solutions at 16°. Table 2S. 





TJ .^ 


•6 


•s '^ 


**- - 


^ 




*o *G *C 


u -a 


2.. 


og 


S - r? 


i-' 


s h yit- s 


Alcohol! 
in Ethei 

Methyl 

Alcohol. 

CH4O 

Hydrate 
Sodium 
NaOH. 


«.2 

"5.50 




& 


^J II £^Xcg<K hJ 


xS^u 


5J s:eJ 


o 


0.999 0.999 0.999 0-999 0.999 0.719 0.999 0.999 


0.999 


0.999 0.999 


a 


1.002 I.OIO I.OIO ] 


[.010 1.008 


.721 .993 1.02 
.723 ,989 1,04 


^02 


1.004 1.007 


4 


i.oos 1.022 1. 021 1 
1.008 1.03s 1.032 1 


[.024 1.017 


1.03 


1.009 1.015 


6 


[.039 1.026 


,724 .985 1.06 


1.05 


1 .01 4 1.023 


8 


1.011 1.047 1.044 1 


1.053 1.036 


.726 .982 1.09 


1.07 


1.019 1.031 


10 


1.014 1.059 1-056 1 


1,068 1.045 


0.728 0,980 1,1 1 


1.09 


1.024 1.039 


12 


1.017 1.071 1.068 1 


,084 1.055 


,729 .978 I.13 


I. II 


1.030 1.047 


14 


Ij920 1.083 1.080 1 


[.099 1.065 


.731 .9761.16 


I.I2 


1.035 1.056 


i6 


1.023 1.096 F.O93 ] 


[.1141.075 


.733 .9741.18 


I.I4 


1.040 1.065 


i8 


1.026 I.IO8 1. 106 1 


[.129 1.085 


.734 .9721.20 


I.16 


1.045 1.073 


20 


1.028 I.I2I I.I 19 


[.144 1,095 


0.736 0.970 1.22 


I.18 


1.050 1.082 


22 


I.03I 1. 134 I.I32 


[.160 1. 106 


.738 .9681.24 


1.20 


1.055 1091 


24 


1.034 I.M7 i.H6 


[.175 1,116 


.739 .965 1.27 


1.22 


1.061 I.IOO 


26 


1.036 1. 160 I.I 59 


[,191 1.127 


.741 .963 1.29 


1.24 


1.066 I.I 10 


28 


1.039 i.t73 1.174 


[.207 1.138 


.743 .961 1.31 


1.26 


I.07I I.I 19 


30 


1.041 1.186 1.188 1 


t,224 I.T49 


0.745 0.959 1.33 


1.29 


1.076 I.I 29 
I.08I 1.138 


32 


1.044 1.T99 1.204 1 


r.240 1. 160 


.746 .9571.35 


1.31 


34 


1.046 1.213 1.218 1 


1.257 I.I7I 


.748 .955 1.37 


1.33 


1.086 1. 148 


3^ 


1.048 1.226 1.233 1 


[.274 1. 182 


.750 .953 1.39 


1.36 


1.092 1.158 


38 


1.050 1.239 1.248 1 


[.290 1. 193 


.751 .950141 


1.39 


1.097 1.168 


40 


1.052 1.252 1.264 


r.306 1.205 


0.753 0.947 1.44 


MI 


1. 102 I.I78 


42 


1.054 1.265 1*280 1 


1,323 I.217 


.754 .945 M6 


143 


1.107 1.189 


44 


1.056 1.278 1.297 


[.340 1,229 


.756 .943148 


'•^§ 


I.I 12 I.I99 


46 


1.058 1.292 1.313 


t.361 1.240 


.757 .9401.50 


1.48 


I.I 17 I.2I0 


48 


1.060 1.305 1.330 


1.380 1.253 


•759 .9381.52 


1.51 


1. 122 1.221 


SO 


1.062 T.318 1.348 


[.399 1.26 


0.760 0.935 1.54 


1.53 


I.I 27 1.232 


52 


1.063 1.330 I 365 1 


[.418 1,28 


.761 .932 1.56 


1.56 


1.132 1.243 


S4 


1.065 1.342 1.383 


t.438 1.29 


.763 .929 1.58 


1.58 


1.137 1,254 


56 


1.066 T.353 1.401 


r.459 1.30 


.764 ^26 1.60 


I.6I 


1.143 1.266 


58 


J. 067 1.364 1.420 


[.480 1.32 


.765 .923 1.62 


1.64 


I.I48 1.277 


60 


1.069 1.375 1439 1 


[.502 


0.766 0.919 1.64 


1.66 


1.153 i.2«9 


62 


1.070 1.386 1 


[.525 


.767 .9151.66 


1.69 


1. 158 1.301 


64 


I.071 I.3q6 ] 


^'Ho 


,769 .911 1,68 


1.72 


1.163 1.313 


66 


1.072 1.405 1 


t.568 


.770 .905 1,70 


1.75 


I.I68 1.325 


68 


1.073 1.414 1 


1.591 


.771 ,9001.73 


1.77 


1.173 1.337 


70 


1.073 1.423 1 


1.615 
t.638 


0.773 0.896 1.75 


1.79 


I.I78 1.350 


72 


1.074 1.431 1 


.774 .890 




1,183 1.363 


74 


1.074 1438 1 


r.662 


.775 .885 




1.188 1.375 


76 


1.075 1.445 


1.686 


.777 .880 




1.193 


78 


1.075 1.453 


[.710 


.778 .873 




I.I98 


80 


1.075 1.460 1 


f.734 


0.779 0.868 1,8? 


a.0? 


1.203 


82 


1.075 1.467 ^ 


1.758 


.781 .862 




1.209 


b 


1.074 1.474 


^774 


.782 .857 




I.2I4 


86 


1.074 1.481 


t.791 


.784 .851 




1.220 


88 


1.073 1.488 


1.807 


.785 .846 




1.225 


90 


I.071 1.495 1 


r.819 


0.786 0.840 1.9? 


2.1? 


I.23I 


92 


1.070 1.502 


1.829 


.788 .835 




1.237 


94 


1.069 1.509 


r.836 


.789 .829 




1.242 


96 


1.064 1.516 


1,840 


.791 .823 




1.248 


98 


1.060 K523 


r.841 


.793 .817 




1.254 


100 


1.055 1.530 


1.839 


0.794 0.810 2,0? 


a.2? 


1.260 



Table 28. Density of Solutions at 15°. 881 



& X 

O 0.9Q9 0.999 0.999 0*999 0*999 ^•999 0.999 0.999 0.999 0.999 O.999 

2 1.009 .990 I.OI7 I.OI6 I.OI9 1.010 I.OII 1.012 I.OIO I.OO9 I.0I2 

4 I.OI9 .982 1.036 1.033 1.036 1.020 1.024 ^•025 1.020 I.OI8 T.O23 

6 1.029 .974 T.0S4 1.051 1.052 1.031 1.038 1.039 '•031 1.028 r.034 

8 1.039 '9^6 '•073 1.068 1.07 1 1. 04 1 1.052 1.053 '•042 1.038 1.046 

10 1.049 0.958 1.092 1.086 1.090 1.052 1.065 1*067 1.053 1.048 1.058 

12 1.059 '95^ ^'^^^ ^-'OS 1. 109 1.063 1.077 1.082 1.064 ^.058 1.072 

14 1.069 -944 ^'^S' ^''23 1. 127 1.074 1.091 1.O96 1.076 1.068 1.084. 

16 1.079 •937 ^-'S^ ^'H^ 1. 145 1.085 1.105 i.iio 1.087 ^.078 1.096 

18 1.089 .930 1. 171 1.162 1. 164 1.097 I.I 19 1.125 1.099 J.088 1.109 

20 1. 100 0.924 1. 192 1.181 1.185 1.108 1. 134 1.141 I. Ill 1.099 1.123 

22 i.iio .918 1.213 1.202 1.206 1.120 1.151 1. 157 1,124 1.109 1-136 

24 1.120 .912 1.234 1.222 1.227 1.131 1.171 1.173 1. 136 1.120 1.149 

26 1.130 .907 1.256 1.244 1*248 1.143 1. 191 1.189 1.148 1.131 1.163 

28 1.140 .902 1.278 1.265 1.269 1.155 1.211 1.206 1. 160 1.142 1.178 

30 1.150 0.897 1.300 1.287 1.290 1.167 1.231 1.223 1.173 1.153 1.193 

32 1.160 .892 1.323 1.310 1.315 1. 179 1.250 1.240 1.186 1.164 1.208 

34 1.170 .887 1.346 1.332 1.339 1.191 1.270 1.258 1.199 1.175 1.223 

36 I.I 80 .883 1.370 1.355 1.365 1.204 1.290 1.276 1.212 1.186 1.239 

38 1.190 1.394 1.379 1.391 1.216 1.310 1.295 1*225 1.198 1.254 

40 1.200 1.418 1.402 1.419 1.229 1.331 1.315 1.238 1.210 1.270 

42 1.442 1.445 1.242 1.352 T.335 1.222 1.287 

44 1.467 1.472 1.255 1.374 T.355 1.234 1*303 

46 1.492 1.499 1.268 1.397 1.375 1-246 1.319 

48 1.518 1.532 1.281 1.420 1.396 1,259 1*336 

50 1.543 1.565 1*294 1.443 1.417 i«27i 1.352 

52 1.570 l.;99 1.468 1.284 1.369 

54 1.63s 1.493 1.297 1.387 

56 1.668 1.519 1.405 

58 1.703 1.424 

60 1.739 5 1.444 



_ _ to 

•So oPJ o§ ^§ 'S Ofl ^J^ «Md c^ Xjf S« 



fa <£!,«? CJo^; UOC D22 DtoZ CJ»fai4 m^i< Zfai^ CO a'? co "^ ' co<S 

o 0.999 0.999 0.999 0.999 0.999 0.999 0.999 0.999 0.999 0.999 1*999 

2 1.013 1.021 1.005 1.016 1.014 1.012 1.01 I.OII 1.012 1.007 i»004 

4 1.027 1.042 1.012 1.033 1.028 1.025 1.03 1.023 1.024 1.015 1*009 

6 1.042 1.063 1*018 1.050 1,043 1.038 1.04 1.035 1*037 1.023 i»oi5 

8 1.057 1.084 1.024 1.067 1*058 1.052 1.05+1.048 1.051 1.031 1*021 

10 1.072 1.106 1.030 1.085 1*072 1.065 1*07 1.061 1.064 1*039 1*027 

12 1.088 1. 128 1.036 1.103 1.088 1.079 1*08 1.074 1*078 1.047 1.033 

14 1. 105 1.150 1.042 1. 121 1. 103 1.093 1*10 1.088 1.091 1.055 '*040 

16 1.121 1.175 1.047 1.140 I.1 18 1. 107 1.102 1.105 1*063 1.047 

18 1. 132 1.200 1.053 I.I 58 1.134 1.121 1.116 1. 120 1.072 1.054 

20 1.155 1.058 1. 177 1.150 1. 135 1. 131 1.134 1.080 1.062 

22 1.173 1.064 1.197 1.167 1. 150 1. 146 1.149 1.^88 

24 1,192 1.069 1*217 1.183 1.16s i«i6i 1.165 1.096 

26 1.075 1.237 1.200 1. 181 1.105 

28 1.258 1.197 1.113 

30 1.278 I.2I4 I.I22 



882 





¥3 


•0 


.a 


1 


imonia 

rbonatc 
tassium 
CO, 


fad 


ife 


|i= 


i-i 






§x 


cg^af 5 J ^c33?c3£tJc3<22S5'3 


Sc25 


x^S 


o 


100 


100 


too 


100 


100 


100 


100 


100 


100 


100 


100 


a 


102 


• • 


lOI 


98 


92 


too 


100 


100 


100 


100 


100 


4 


103 


• • 


lOI 


96 


86 


100 


100 


lOl 


101 


100 


100 


6 


los 


• • 


102 


94 


79 


lOI 


100 


lOI 


lot 


lOI 


lOI 


8 


107 


• • 


102 


93 


71 


lOI 


101 


lOI 


101 


lOI 


lOI 


10 


109 


• • 


103 


91 


65 


lOI 


lOI 


lOI 


102 


lOI 


lOI 


12 


HI 


• • 


103 


90 


59 


lOI 


101 


102 


102 


lOI 


102 


H 


109 


• • 


104 


89 


53 


lOI 


lOI 


102 


103 


lOI 


102 


i6 


106 


,, 


104 


88 


47 


101 


101 


103 


103 


102 


103 


i8 


102 


• • 


105 


87 


41 


102 


102- 


103 


104 


103 


103 


20 


88 


104 


«05 


86 


36 


102 


102 


104 


105 


103 


104 


22 


73? 


104 


106 


86 


30 


102 


102 


los 


105 


104 


lOS 


M 


59? 


105 


106 


85 


25 


103 


103 


106 


106 


105 


106 


26 


48? 


106 


107 


85 


20 


103 


103 


107 


107 


106 


107 


28 




107 


108 


84 


15 


104 


104 


108 


108 


107 


108 


30 




108 


109 


s* 


10 


104 


104 


109 


• • 


109 


HO 


32 




109 


III 


84 


S 


105 


105 


HI 


• • 


HO 


112 


^ 




HO 


112 


83 





106 


105 


113 


• • 


H2 


114 


36 




110 


114 


83 


-5? 


107 




114 


• • 


114 


116 


38 




III 


116 


83 




108 




116 


• • 


116 


118 


40 




112 


118 


2^ 




109 




118 


• • 


119 


120 


42 




113 


120 


82 




110 




120 


• • 


122 


123 


44 




114 


122 


82 




11! 




122 


• • 


125 


125 


46 




115 


124 


82 




112 




124 


, , 


128 


128 


48 


is? 


116 


126 


82 




113 




126 


• • 


131 


131 


50 




"2 


128 


82 




114 




128 


• • 


134 


134 


52 




118 


130 


81 




IIi> 




I3» 


• • 


137 


137 


54 




119 


133 


81 




ii6 




134 


• • 


140 


140 


56 




119 


137 


81 




• • 




138 


• • 


143 


M3 


S8 




119 


139 


81 




.. 




141 


• • 


146 


146 


60 




120 


142 


81 




• • 




144 


• • 


149 


149 


62 




120 


145 


81 




• • 




148 


, , 


• • 


• • 


64 




120 


150 


81 




• • 




T52 


, , 


• • 


• • 


66 




119 


156 


80 




• • 




!S6 


• • 


• • 


• • 


68 




116 


163 


80 




• • 




160 


• • 


• • 


• • 


70 




• • 


170 


80 




• • 




164 


• • 


,, 


• • 


72 




• • 


176 


80 




• • 




160 


, , 


, , 


• • 


74 




• • 


183 


80 




• • 




»73 


, , 


• • 


• • 


76 




• • 


190 


80 




, , 




17^ 


, , 


, , 


• • 


78 




• • 


198 


80 




• • 






• • 


• • 


• • 


80 




100? 


206 


79 




• • 




• • 


• • 


• • 


• • 


82 




• • 


214 


79 




, , 




, , 


, , 


• • 


316? 


84 




• • 


225 


79 




, , 




• • 


• • 


• . 


• • 


86 




• • 


236 


79 




, , 




, , 


• • 


316? 


• • 


88 




• • 


248 


79 




.. 




• • 


• • 


• • 


• • 


90 




• • 


260 


79 




• • 




• • 


• • 


, , 


• • 


92 




• • 


274 


79 




• • 




• • 


« , 


• • 


• • 


94 




, , 


288 


78 




• • 




• • 


, , 


• • 


• • 


96 




40? 


303 


78 




• • 




• • 


• • 


,, 


• • 


98 




• • 


318 


78 




• • 




• • 


• • 


red 


red 


100 




• • 


333 


78 




• • 




«• 


• • 


heat heat 



Table 30. 



Specific Heat of Solutions. 



883 



^ 



u 
U 



B 

m 



SI . . « 






^^ 






rgc 



1?" 









■ QOOO^CfOO ■ *0Q 



-00 OO *O0 QOQO ' ■ X^ 






^ [V qO go 0^< d O " " ~^ 



ihTti a^. t^ c^»^" "H-'r^ qQ 



^.gO CT' t> C^C? O ojc^ o* ■ bo so 3D O^ Cfli C* Cf^ o^ 



DO C^ 

. Of J S 






r^ *- ^ .._ ^ S 



Qo rn u-i inoo r* '^oo t*>H»i ^ fn "^ f*i ^^ xo ■+ *a r^. ^ 
^" O^ " ' * 



o- q^ , « 






cr CCS O^ 0-- O O 0^53 O^ O*. C ■CT^ O^ ^^ "^ ^^, '^ ^^. '^- ^ 



' ^' -■ q- 






rvX «j t^ l>-W l^«3 t-HlX: C^ QQ oo OQ r^ac oo 

c^3s c^ct CCS <sv o^ qoi a* C" q*. ^- o% o a% 



u 

C 

2 

.A 

o 






c a c c^ c o^- Cr^ c^ c^ ^. cj". Qv 



S?§=5*BiS;S'SgS.S|8SSSl||SS8" 




a:xxxxouxziu^;:!^i^2:x^2:ucjjLjH:s^N 



fJ o ., 

<XZ 
< 









^ ^ :'^^ e 



;s - 



e ^ 3 






og:2 



o 



3; 



^<d;w 



■ o o 
:<UJ:;;S.[^N 






;5 3 3 



884 Electrical Condactivity of Solutions. Table 81 A 



8 



§^ 



ceo 



s 



o 

a 



Q •*»■ iH»l 



s; 



ss^ ^ 



3S S 



g- 



osM vo r>. 






Qooca 



^^, 









^fi 



:itSS 






sss rs5^ SSS'': 



*agas ssa 



MT'.vH O »i^O ^ 






O M fN. ON O^ fO O'CO 



Ct^m 0»V0 00 

>o c> "•' ^ 



« t>.00 «^00 I- »r» ^ %r> 






nO>e9 t^»/^ONM ^t^^ 



O moo t>* ON «^vO O^VO 00 »M f>0 * 00 • 



^fO aOD u^M 00 00 OsvO ^^ O ^ w^vO 0^»r>0 ^>0 00 PO *r>.vo fo 



c< ^fH NO ^ »« r^vO r>, «r> *-• ONOO >0 O* ^ t^ ^ »r> ro «r>0 « eo «r> *n « 



^g>ovo <^ce>9«^At>»r«ko«oo:0^^ed^kOO]Oii^^Oii 



•*-vO * M « to to rO too ^^«*>ONM rOOPOM eorON « « « 
^ ^ ^ O 



^<*^o wp<>c«ooNWNr«*roror«vo«N««M««MwwMi 
Q ^ ^ O 



go 



oooooooeooeoooooeoooooooooe 






=■§1 

-•2 5-2 



.H^.9 






■51 



^ ON «s vO ^'ir» ^ ^ S -, 

m ^ ^ - QO 2 



K-oH 



■fi: 

td9ivo to A 00 o D r« 00 ^ i-i t^fM r>. M >o 00 IT) t^ &eo ^ X r« 09 o^^ 



h-B 



S'^^ 



O N <-«0 rONO ^ ^^ ^OO^O w^^N rO^rO ^C<« ^^€4 N rOrOft >,« 









■5 « 

g'5 




Tabieg8l,B-i>v Refractive Indices, eta of Solutions. 



885 



81. B. Refractiye and DisperslTe Indices of Solatlons at abont 18**. 



Name and Symbol 



1^*3 M u*& Si «* c* 



f9 



•— • o o 

Acid, acetic HCaHsOj 01.333 ^OH 
„ ,» 201.348 .. 

401.362 .. 
601.371 .. 
801.378 .. 
,, ^, 100 1.374 .017 

Hydrochlonc,HCl 35 1.413 .023 
Nitric, HNOs 501 401 .024 

Sulphuric, Ha SO4 o 1.333 .014 
201.338 .. 
„ n 40 1.382 .016 

„ 601410 .. 
801434 .018 
100 1.434? .017? 

o 1.333 .014 
401.358 .015? 
100 1.360 .015? 



Alcohol, SaHsOrf 



Nam« and Symbol 



Chloride Amm. H4NCI 10 1.351 .016 
It *9 w 20 1.370 .0x8 

„ Calcium CaCla 20 1.384 .019 

»• o Ji K?' ^1 *^ ^-^^ •°23 
„ Sodium NaCl 10 1.350 .016 

' „ „ „ 20 1.368 .018 

,» Zinc ZnCla 20 1.370 .018 

„ „ ,, 40 1.410 .021 

Hydrate Potas. KOH 40 1.403 .018 

„ Sodium,NaOH 10 1.359 .016 

n „ „ 20 1.384 .018 

„ p „ 30 1404 .020 

Nitrate Sodium NaNOs 20 ^'355 .017 

,» i9 «> 40 1.380 .021 

Sugar, CiaHfflOu 10 1.348 .015 

}, „ 20 1.364 .016 

„ „ 30 1. 381 .017 



81. C. Table for preparing IQxtnres of any Desired Streng^tlk 






O 
I 

2 

3 
4 

5 

6 



9 
10 
II 

12 
13 
14 

15 
16 

17 

18 

19 

'20 



fic« 


i 


^ 


2o« 


Mm 


i Son ^ 


t 




^ fi2« ^ 


a^-s 


u2 


i5< 


.S^-s 


o< z<-o at 


o< 


'o< £;^ cS2 


ps 


r 


r 




r 


r io8 fe^ 


r 


ro§ r 


r t^^ t 


0.000 


100 


20 


25.000 


80 


40 66.667 60 


60 


150.00 40 


80 40000 20 


1. 010 


99 


21 


26.582 


79 


41 69.492 59 


61 


15641 39 


81 426.32 19 

82 455.56 18 


2.041 


98 


23 


28.205 


78 


42 724M 58 


62 


163.16 38 


3.093 


97 


23 


29.870 


77 


43 75.439 57 


63 


170.27 37 


83 488.24 17 


4.167 


96 


24 


31.579 


76 


44 78.571 56 


64 


177.78 36 


84 525.00 16 


5.263 


95 


25 


33.333 


75 


45 81.818 55 


65 


185.71 35 


85 56667 15 


6.383 


94 


26 


35.135 


74 


46 85.185 54 


66 


194.12 34 


86 614.28 14 


7.527 


93 


27 


36.986 


73 


47 88.679 53 


67 


203.03 33 


87 669.23 13 


8.696 


92 


28 


38.889 


72 


48 92.308 52 


68 


212.50 32 


88 733.33 12 


9.890 


91 


29 


40.845 


71 


49 96.078 51 


69 


222.58 31 


89 809.09 II 


ii.iii 


90 


30 


42.857 


70 


50 100.00 50 


70 


233.33 30 


90 900.00 10 


12.360 


89 


31 


44.928 


69 


51 104.08 49 


71 


244.83 29 


91 1,011.1 9 


13.636 


88 


32 


47.059 


68 


52 108.33 "48 


72 


257.14 28 


92 1,150.0 8 


14.943 


87 


33 


49.254 


67 


53 112.77 47 


73 


270.37 27 


93 1,328.6 7 


16.279 


86 


34 


51.515 


66 


54 117.39 46 


74 


284.62 26 


94 1,566.7 6 


17.647 


85 


35 


53.846 


65 


55 122.22 45 


75 


300.00 25 


95 1,900.0 5 


19.048 


84 


36 


56.250 


64 


56 127.27 44 


76 


316.67 24 


96 2,400.0 4 


20.482 


83 


37 


58.730 


63 


57 132.56 43 


77 


334.78 . 23 


97 3,233.3 3 


21.951 


82 


38 


61.290 


62 


58 138.10 42 


78 


354.55 22 


98 4,900.0 2 


23.457 


81 


39 


63.934 


61 


59 143.90 41 


l^ 


376.19 21 


99 9,900.0 I 


25.000 


80 


40 


66.667 


60 


60 150.00 40 


80 


40000 20 


100 00 



81. D. Coefficients of DiiTnsion of Saline Solutions in Water. 

Hydrochloric Acid 000,0100 Sulphate of Potassium . .000,0037 

Hydrate of Potassium . . .000, 0070 Sulphate of Sodium . . .000, 0030 
Sulphuric Acid . . . • .000, 0052 Sulphate of Magnesium . .000, 0020 

Nitrate of Potassium • • .000,0052 Sugar 000,0019 

Common Salt .000,0046 Gum Arabic • • • • • .000,0010 

Nitrate of Sodium • • • .000, 0040 Albumen • • • 000, 0002 

, •••• Caramel • • • 000,0001 



886 



Rotation and Polarization. Table8 8l,E— F 



81, E. Rotation In de^crees of the Plane of Polarization for the Frannliofer 

Lines A— H, produced by passing through 100 cm, of varions solations* 

eontainlng in each case 1 in^am of a given substance in 100 en. cm.. 



Name and Symbol of Substance 



t 



Acid Malic, H-rC+HiOs+aq . , , 
,, Tartaric HsC^HiOfi + aq, , 
Camphor CioHieOH- alcohol . . 
Chotcstcrine €3^1^440 + ether . * . , * — 
Cinchonidine C^tbiNaO+aicohol. . — 

Coni:hinineCsioH*^Nt022iHyO+alcohol4' 

Glycocol CnaNH,COOH + fllcohoL + 

Mai ate of Ammon{ H4N )3C4H40^^+aq, — 

^ „ Lithium Lij Q H40ii+ aq* , — 

,^ „ Potassium KsC4H405+aq. — 

„ „ Sodium Na2C4 Hi Oa-f-aq, * ^ — 

Morphine chLCnHieNOsHClj HaO+aq - 

Qu i n m e h yd r, C^ H 24 NgO 3* 3 HgO+alcoho! — 
„ sulph. CsoUsiN^Oa H3S047H5jO+aq.— 
Salicine Cli;^! li^U^ + ^q, .*,.*.,, — 
Sa n to n id , para - ; Q^ 1 1 laO j-l-alcohol < 4- 
Santonine CtsfHaOa + alconol* , , , , ^ 
Sugar (Cane-) CisH/jOu+aq. 

11 i^i^ape „ . , . 

44 milk , . - , 

,, maltose . . . , „ ... 

,^ Inctose „ . . • 

„ inverted , , . * „ , , * 



BjGID 

-.13.04.2 
2»oi,63-3 



58 
II 

3.B4.B 



^4 

H 

M 

0.7 
i.o 
10 

IS 
t6 

6-5 

16 



66 

s 

5 
H 

S 
3,9 



E F G 



1.9 2.0 



6.1 

4.01 



33 



8.0 
4.9 



4.9 



lafi 167 
22 26 
8,5 to.i 



6.2 



5.7 



H 



13.2157 



81, F. Rotation In degrees of the Plane of Polarization for Frannhofer 
Lines A— H produced by plates of various substances 1 cm. thick. 



Name and Symbol of Substance 



Benzil, C14H10O9 

Bromate of Sodium NaBrOa 

Chlorate „ NaClOa 

Cinnabar, HgS 

Diacetylphenolphthaleine 

Ethylenediaminesulphate 

Guanidine Carbonate 

Hyposulphate of Calcium CaS206.4H30 

„ Lead PbSa06.4H20 . . . 

„ Potassium K2S2O6.2H3O 

„ Strontium Sr Sa Oe.4H20 

lodate sodium, per- Nal04 

Nicotine (liquid) C10H14N3 

Quartz (ordinary right handed) SiOj + 
Strychnine (sulphate)2C8iHa2N202 .H2SO4 
TartaricEther (liquid) (Q H 5)2 Q H4 0,-4- 
Turpentine right handed CioHja . . . + 

n (liquid) left handed QoHie — 



24 

3000? 

i68l? 



B 



23?. 



24a 
z8 
32 40 



46 



S9 



1 97 246 1 

1461781 
21 

4' S5| 73 
62 841051 

..| ..| 16 
1942332851 

127157 173 217 275 32; 425 51 1 

0.8 
37.0 



342,471 



H 



6g 



• • 

• • 


168 

• • 


? 

• ■ 



Tables 81, 6-L. 



Miscellaneous Data. 



887 



81, G. 



Botatton of the Pltine of Polarization cansed by a Unit Magrnetio Field 
(C. G. SO in Unit Thicknesaeg of Different Substances. 



Bisulphide of Carbon (sodium light) cP.ooTo 
(thallium „ ) cP.oo$6 



Water (white light) .... cP.oooi 

„ , - Coal gas oP.ooo,ooo^ 

Note. In these, and in nearly all cases, the rotation is with the current 
producing the magnetic field. A solution of ferric chloride in methyl alcohol is 
mentioned as one of the exceptions to this ru|e (Deschanel, § 839). 



81, H. Magnetic Moment of 1 en. cm. of varions gnbstances (C. G. S.) 



Name 

of 

Substance 


Magnetization 
induced by 
Unit Field 


Maximum 
Magnetization 


Maximum 

Permanent 

Magnetization 


Name 

of 

Substance 


Magnetization 
induced by 
Unit Field 


Iron 

Steel 

Cobalt. . . . 
Nickel .... 
Iron Oxide . 


300? 
70? 
300? 
140? 
0.2? 


1400 

'ioo? 
500 


<'8oo 

• • 


Nickel Oxide 
Water .... 
Bismuth . . . 
Phosphorus . 


-4-0.1? 

— O.OI? 

— O.OI? 

— 0.004? 



81, L Coefficients of Friction (f) for -water corresponding to Velocities (v) in 
centimetres per second (From Weisbach). 



V 


/ 


V 


/ 


V 


/ 


V 


/ 


V 


/ 





00 


100 


00299 


200 


.00264 


300 


00247 
00246 


400 


.00239 


10 

20 


.00554 

.00445 


no 

120 


OOT© 


3IO 
220 


.00262 
.00260 


l^ 


410 
420 


.00230 
.00238 


30 




130 


00284 


230 


.00258 


330 


430 


.00238 


40 


.00308 


140 


.00280 


240 


.00256 


340 


00245 


440 


.00236 


g 


.00347 


'£ 


00276 


^ 


.00255 


iS 


.00244 


% 


.00236 


.00333 


100 


00274 


.00254 


00243 


.00235 


s 


.00321 
.00312 


gs 


'^9 


ss 


.00253 

.00251 


^ 


00242 
00241 


X 


.00235 
.00234 


90 


.00305 


190 


.00266 


290 


.00250 


390 


00240 


490 


.00234 


100 


.00299 


200 


.00264 


300 


.00249 


400 


00239 


500 


.00233 



81, J. Cpefacients of Friction of Solids on Solids. 





Oak 


Hard 
Wood 


India 
Rubber 


Leather 


Hemp. 


Bronze 


Iron 


Cast 
Iron 


Oak 

„ soaped 

Bronze 

Iron (cast, smooth) . 

„ „ wet . . . 
„ greased . 


".T6-^ 

.48 
.49 


.38 


• •• 
••* 

.20 


.30 

.3 
.36 
.15 


.52 


:1i 

.20 
.2 
.31 
.15 


.Vs 

.2-.4 


^9 
.19 
.21 

.2-.4 
.31 
.15 



81, K. Action of Plates (1 cm thick and bounded by plane snrfiices) upon nor- 

mally indiUiit Ruaiftnt H<?&t. 



Subs lance 






Ab- |Triin!i-j| 
sorbs mits I 



Substince 



Re* 
fltcta ) 



Ab- 
sorbs 



Trfina- 
mits 



81, L. 



Lampblflck. . . 
Ii^ia Ink « » , 

Ice . 

Alum. , . . ■ ■ 
White Lead , . 
Gl;i£&. ..... 

Sbdfac, .... 

Pulished Metal9 
Rock Salt . . . 

I^ampblack.. , * 
While Lead . . 
Ice « 

Alum 

Glass. ..... 

India Ink. , . . 

Estimates of 



5 

t 
^' 

8 
80? 

o 
o 

8 



TOO^ 

95 



30? 



100 
JOO 

96 

90 



0% 

o 

l\ 



30 

45 




^Weier ... 

A q u e ousSoluii ons 
Alcohol . . 
Elher. . . . 
Oils .... 
Chloroform 

^ BisulpliideCarbon 
^Mercury . . 




4% 


86% 


4 


ti6 


5 


S3 


i 


75 


6 


no^ 


6 


^ 


13 


35, 


75' 


25? 


s 


72 


gr 


=5 J 

30? 


8 


0? 





0—10? 





0— .0(1?' 



10 

13 

21 

30 

53 







the nnmber of Units of Heat radiated in 1 sec. by 1 sq. em* 
blackened snrface in space at 0°. 



Temp.Rad. 



Temp. Rad. ITemp. Rad. Temp. Rad.!Temp. Rad.! Temp. Rad. Temp. Rad. 



-273^- .019? 

— 200P— .015? 

— lOoP— .009? 

fP .000 



-locP-f 


-.012?* 


500P .1? 


20rf» 


.028?* 


6orf> .2? 


^ 


■^V' 


^11 



900P .5? 

loooP .7>// 
iiocP ,g?// 

I20gPi.2?+§ 



1300P 2?* 
1400P 2?* 
i5ai?3? 



iTooP 5? 2500P 6o?"f 

1800P 7?fti3oooP 270? 4 

1900P10? g5O0Pl2OO?§ 

I200CP13? l4ax/>5400?f 



* Dark, f Dull red. S* "Red Heat'*. § Cherry Red. ^ Orange. t§ 
-"White Hear, ff Flame, t* Voltoic Arc Light §§ Sunlight 



Yellow 



888 



Heat of Combustion. 

89ft* Hf^ats of Comhoatlftn in Oxj^rph* 



Tables 



Name 
of Subsiflnc* 
Consuroffd. 



Chtmiiral Reaction 

inrolviiog 16.0 era mi of Oifgea 

in each case. 







ActivU-nc. 
AlconoL , 
Arsenic , . 



Barium . . , « , 
Bismuth . . , . 
CuLcium , p , . 
Carbon . . . * . 
Carbonic OxtdE 
Chlorine , , . , 
Copper 



Ethane . m 
Elher. . , 
Eihvk'tfcc - 
Hyafogcn. 
JoUine . . 



Lead * * , . 
Miigneiium . 
Mercury . , 



Methane 
^itroacn 



Phosphoruii. , » . 
Potassium » * , . 

Sek'ninm 

Silver. ,,.,,. 
Sodium *.,,,, 
SpermacelJ . . . . 
Stearine, , , * . . 

Strontium 

Su I p h idcCarli ouM'i 
Sulphur, . . . . . 
Thallium , , . * . 
Tin, ....... 



C,H.O + 
A&j + 

ilU -*- 

Bi, H- 

a Ca -+- 

C -h 

aCQ "»- 

2 CI, + 

4 Cu + 

J C, H, 
C|TTnO + 

^Ph 
^Mg 
4Hs 

■iHg 

aN, 
N, 

1^: 

Sc, 



^ 70,= 
+ 6 0t= 



S0t = 

0,= 
p,= 
p. 3 

aO,- 
0,= 

0,= 
aO,- 
50p = 

y' = 



4CC\+aH,0 
3CO, + 3H,0 
aA^Oi 
aAf,0, 
:iBaX> 
aBi,0, 
aCaO 
CO, 
a CO* 
aCl,D 
a Cu, O 
aCuO 

^COi + GH.O 
4C0, + '^H,0 
4CO, + 4H,0 
a H, U 

aKeO 

Fc,0, 
aPbO 
2 Ug O 

aHflO 
CCV+2H,0 

a NO 
a NO. 

ap.a 

aK.O 

3As,0 
2Na,0 



7*7 13-7 
50.0 <3o.a 
30.0/ 46. P 

r0.S 153.8 

13S.71 

38.0 

laD.a 
63-1 



Titrpentinc , 
M'ax , , , . 
Wood. . , . 

ZlDC . < . . 



aSr 

cs, 

aSn 

■iSn 
Cm H|i 

^Zu 
HSb. 



-h 0,= aSrO 

+ 30.:^ COi + aSOi 

-h aO,= aSOt 

+ 0,= 2T1,0 

-»- Os- aSnO 

+ aO,= aSnO, 

+ T.iOi = ioCUi'i-BHiO 



i54'7< 
55-9 

23.0 

143.3 
79 
4*31 30.3 
6p3 33. a 

30.6 

18.0 
66.6 
71.9 
57*9 
133.4. 
40-0 

]99'S|ai5.B 
2o.a 
44^0 
3D.- 



4*7 
2.9 
so. 6 

55-9 

ao&.4 



14, 

13.41 

78.1 
39-4' 



a8.3 
94 

55.4^ 



ij.4,23''^ 
40. 0| oa.Q 



87.3 

ia.7 

16.0 

4aS,a 

niJ.o) 



32.0 

134-0 
b9M 7S-0 
4.9 30. 



fiajiMo 13,000 
54,oool 7^000 
3t,.Soq i^ 
44,000 i,>po 
l3L>.(Boi 950 
13,300 q6 
i3iVMoi 3.^80 
4H/300J 8,00a 
67An| 2,4ctJ 
tg^— 350- 



I ll,5ClO I 

6,000 9J0OO ; 
5 e>poa 1 2,jDod [ 
SyjMO 34,5o^i44dl,^ 

- - 177 - ■ 
1350 

|,™j 1575 
5o,cmoi 343 
i46jooaj a^too 
4^ooti 105 
boot! T53 

53.000 13,100 

18.0001 ^' 



-33,000 
— 1,000 

71,000 
136.000 
a(>,ooo 

153,000 



131/J00 
41000 
35^000 
.400 



-^ Oa= a ZnO 64.9 So.i 

Heatfl of Cfomlvn^tioii in CMorinei 



7a^ 
53,000 

S4y4oo 



— 6<io— 37 

— 150-6," 

5750 

1745 

730 

37 

3.30Q 

]q;3oo 
0,700 

T.SOO 

3*400 
a;i50 

575 

1^330 
10.700 
101500 

4,000 

1,300 



Name 
of Substjince 
Consumed. 


Chemical Reaction 

invohing 70,7 gramf of Chlorine 

in each case. 


II! 


ill 


n 




Antimony. . 
Arsenic . . . 
Copper . . , 
HvdroEcn. . 
Iron, . . . , 
Potassium . 

Tin 

Zinc .... 






Sb,-f (^CU=-^SbCls 
As4"F6CI,^4AsCU 
Cu -h Cli^ CuCf, 
H, ^ Cl, = aHQ 
ale -i-3Ci,= FejCl. 
K, -H CU^itKtl 
Sn -H3Cli= SnCU 
Zn -f- Cl|= ZnCli 


8D.7 
3.0 


151.4 

riO.6 
133-S 
73.7 
loS.o 
148.8 
139.7 
J3si 


J7/JD0 

SO^OOQ 

oijood 
47,000 
63,000 
3i:f?,ooo 

6^hOOD 

99.000 


707 go 

ii 

I.53oi S 


1.33 
1.01 

k 



88. Heats of Combination. 






Name 
of Substance 
Acted upon. 



Chemical Reaction 

inrolving 16.0 grams of Oxygen 

or its equivalent 



J|i 



m 




Copper. . . . 
Nitnc Oxide . 
Nitrous Acid . 
Zinc 



2 Cu + O, + 2 SOi -»-Aq. = 2 Cu SO4 • Aq. 
4NO+0, + 2H,0-»-Aq.=4HNO,'Aq^ 
iHNO,Aq. + Ot == 2HNO,Aq. 
iZn-l-0,+3SO,+Aq.= 2ZnSO«-Aq? 



w.o 
47.0 
04.8 




ig.300 
108,500 



Tabl« 84. 



Electromotive Force. 



(pajBjjnaonoo) 
PPV oui\H 




« 

f 


00 




(ajnd) jajB^ 




Q «- M r^oo M 

1 °- f 1 f 1 1 1 1 





1 

M 


XjnoJdj^ 




N 00>O N 






uoqjB3 




oosooomu^^6ro d 

7,1 1 1 1 1 1 


M 


CO 

0. n^^ 


ranai;Bij 




W OO t^ S\VO 00 fO i-i u- 

71 1 1 1 1 1 


t 

•7 


*0 


j9ddo3 


>0 inO 

If ■ 


rO 'th N NO >0 t^ 00600 

• r n r f 1 


eo 

M 
M 


ssejg 


00 u-> 


1 r Iff ■ ' ■ 


t 


NO 

q 


' HO-*! 


f.f 


^ 6 •-« VO "*vo 00 ^ 




i ! °Ii 




POM o\ ror^vo 10 

NOoo 0\ M t^»r» C^Ox 


15 

• 


f 


i P«91 


n 




M 


8 


oojz 


t>.xn 

1 r 


T^ M Of o\o M so 


*f 


r 


OUIZ 

pajBOlBSlBUIV 




© M fO Tf t>.00 00 •^ rt 


8 


00 9> 00 


•bVOW'^OS-nO 

jaddo3-iidins%6z 




t 


% 




•bV*0*H*^ 




VO OMr> fO T^ t^vo 
ro ro N u^M M ^ 
u^M NVO M « 

r 




VO 

t 

M 


*4 U 


Iff 


u u 


8 

• 


1 '" 






•bvO*H^'OSuZ 


g' 


»l 


00 


1 

Ml 




2 ^S 


^•. •••••••• 


• • 




^ 






^«. Q • 




* * 3 


<i>3 




* * * * <;> 


°i::::::::: 


IS 

1" 


s 


ifsi.sgillll 


v'»os«h|. 

ppy f 



889 

I 



W5 

I 



890 



Electromotive Force. Tables 85—80. 



-otuojioai:}! 



ajn)iU9d 

-U19X 



> 



l-i 



II 
^1 



z 



cS 



O .3 

^ Q o 



a.i 



a ^ 



O" 00 00 "^ '<■ o c^ "^ •-. •^, •^, o ^ On fo q 



o o ^ 
a ^ -f So 85 



« 



s&sssssse 



so 



o 
O 



S S £ 



S § S u 



v7 P4 S S S K 



r !3 



o 2 






■ I 



O S £ K S & 



00 
C/3 



u 3 



ScQcn 



^ « K e e 



s 

s 

R g " -g T3 a 

{^CO 2 & C/} 




§-3 

C/5 



I 






_ - 



I 

-a 



ft IS ft s It s.r: 



N 



:^... 



►4 ftK '^ * «^3 






4> 



ft It ft ft 






1 

U 

I 



9 



86. Electromotive Force In Volts and Striking Distance In lllllimetres. 



mm. 


.0 


"T^ 


.2 


.3 .4 


rTI 




1 -7 


~J^ 


.9 




I 
3 

3 




4340 
7600 
10320 


500 
4700 
7890 
10580 


1000 

S050 

8170 

10830 


1470 1920 
5400 5740 
8450 8730 

iioSo 1 1320 


2360 
6070 
9000 
11560 


2780 
6390 
9270 
1 1800 


3190 

6700 

9540 

12040 


3580 
7000 
9810 

12270 


3960 

7300 

10070 

12500 



The Talues in this table are tubiect to a probable error of about no volts. 



Tables 87-88. Specific Resistance. 



87a. Speeifle Electrieal Besistftnees of Condneton at 0^. 

Resistance Resistance Resistance* 

of a 
centimetre- 
cube in 
microhms.* 
1.50* 
i.6o 



in ohms of 
a wire i m. 
long weigh- 
ing Iff. 
>.I6 



Name 

of 

Substance 

Silver, annealed . • 

„ hard drawn . 

Copper, annealed. . • • 1.58 

„ hard drawn . • 1.61 

Gold, annealed . . . • • 2.0 

„ hard drawn ... 2.1 

Aluminum annealed • • 2.8 

Brass • • S.S 

Zinc, pressed • S*^ 

Platinum, annealed. • • 9.0 

Iron* annealed 9.5 

Gold (2) Silver (1) alloy • 10.9. 

Nickel, annealed .... 12.4 

Tin, pressed 13.0 

Lead, ^ 19.0 

German Silver 20.8 

PlatinumCa) Silver( ,) alloy 24.0 

Antimony 35.2 

Mercury (liquid) .... 94.2 

Bismuth 130.0 1.656 

Electric Light Carbons • 

(about) 6000. • . • • 

* These results must be multiplied by 1000 to reduce them to the C G. 
87b. Speeifle Electrical Beristanceg of Insnlatorg. 



in ohms of 
a wire i m. 
long I mm. 

diai9» 

0.019 

o.oao 

0.d20 
0.Q20 

0.026 
0.026 
0.036 
0.070 

0.073 

O.I IS 

0.I2I 

0.138 
0.157 

0.166 
0.242 
0.265 
0.306 



0.1 

0.17 
0.14 
ai4 

0.39 
0.41 
0.07 
0.46 
0.40 
1.90 
0.74 

I.IO 

0.9s 

2.16 

1.77 
3.3? 
2.36 

1217 



891 

Per Cent of 

Increase 

per degree 

centigrade 

0.377 
0.388 
0.365 



0.365 



0.065 

0.365 
0.387 
0.044 
0.031 
0.389 
0.072 
0.354 



S. system 



Name of Substance 



Selenium 

Gutta Percha . . . . 

Shellac ••••••. 

Ebonite 

ParafHne 

Glass 

Air and other Gases. 

t These results must be 

88. 



Resist in Ohms of a centimetre-cube f 



*/o increase per^. 



(about) 



+ 1. 
— 10? 



60,000. 
7, 000, 000, 000, 000, 000. 
9, 000, 000, 000, 000, 000. 
„ 30, 000, 000, 000, 000, 000. 
„ 30, 000, 000, 000, 000, 000. 
Greater than any above. 
Practically Infimte. 
multiplied by 1,000,000,000 to reduce them to' the C G. S. System 



great, negatiyei 



Speeifle Electrical BeslBtajice in Ohms, of a Centtmetre-enbe of different 
Electrolyteg (aee Table 81). 



Per 


Hydro- 


Nitric 


Sulphuric 
Acid 


Sulphate 

^cu^r 


Sulphate 


Chloride 


Chloride 


Cent 


chloric 


Acid. 


of Zinc 


Ammonium 


Sodium 


% 


Acid,Ha 


HNO, 


H,SO. 


ZnS04 


H4NCI 


NaCl 


5 


2.6 


3.» 


5-2 


56.0 


55.0 


1 1.6 


16.0 


10 


1.6 


2.2 


2.6 


330 


26.0 


6.0 


9.0 


IS 


14 


16 


1.9 


25.0 


4.0 


6.5 


20 


1-3 


1.4 


«S 


20.0 


23.0 


H 


S-S 


25 


1.4 


1-3 


1.4 




22.5 


2.8 


5.0 


30 


1-5 


1-3 


1.4 


. . 


25.0 






35 


1.7 


1-3 


1.4 




30.0 






40 


2.0 


1.4 


\i 










45 




\i 










50 




19 










60 




2X> 


:^ 










^ 




2-5 












3-7 


9.0 










90 






10.0 










100 






12.5 











Ifote, The results in 
the C. G. S. System. They 
probable error of about lo^. 



this table mnst be mqltiplied by 1,000,000,000 to reduce them to 
are intended to be accurate at about i9P, but are subject to a 
See Table 31. » r- 



892 



Arbitrary Scales. Tableg 89—41. 







8 








c. 


F. 


C. 


F. 


c. 


F. 

77.0 


c. 


F. a 

122.0 75 


F. 


C 


F. 


C 


F. 


C. 


—125 


-193U, 





32.0 


25 


50 


167.0 


100 


212.0 


225 


437.0 


350 


120 


184.0 


I 


33.8 


26 


78.8 


51 


123.8 


76 


168.8 


105 


221.0 


230 


446.0 


400 


1^5 


175.0 


2 


35-6 


27 


80.6 


52 


125.6 


77 


170.6 


no 


230.0 


235 


4550 


450 


110 


166.0 


3 


37.4 


28 


82.4 


53 


127.4 


78 


172.4 


"5 


2390 


240 


464.0 


500 


105 


157.0 


4 


39.' 


29 


84.a 


54 


129.3 


79 


I74.« 


120 


248.0 


245 


473.0 


550 


; 100 


148.0 


5 


41.0 


30 


86.0 


55 


131.0 


80 


176.0 


125 


257.0 


250 


482.0 


600 


\ 95 


139^ 


6 


42.8 


31 


87.8 


56 


132.8 


81 


177.8I 


130 


266.0 


255 


491.0 


650 


u. 


130^ 


7 


44.6 


32 


89.6 


57 


134.6 


82 


<79.6, 


135 


275.0 


260 


500.0 


700 


1 21.0 


8 


46.4 


33 


91.4 


58 


136.4 


P 


181.4' 


140 


284.0 


265 


509.0 


750 


80 


112.0 


9 
10 


48.3 


34 


93.» 


59 


I38.a 


84 


l83.a' 


145 


293.0 


270 


518.0 


800 


75 


103.0 


50.0 


35 


95.0 


60 


140.0 


85 


185.0' 


150 


302.0 


275 


527.0 


850 


70 


85!o 


" 


51.8 


36 


96.8 


61 


I4I.8 


86 


186.8 


155 


311.0 


280 


5360 


900 


65 


12 


53.6 


37 


98.6 


62 


143.6 


^l 


188.6 


160 


320.0 


285 


545.0 


9S0 


60 


76.0 


U 


55.4 


38 


100.4 


t^ 


145.4 


88 


190.4' 


165 


329.0 
338.0 


290 


554.0 


1000 


SS 


67.0 


H 


57.» 


|39 


I02.a 


64 


I47.a 


89 


192.8 


170 


295 


563.0 


1050 


SO 


58.0 


15 


59.0 


40 


104.0 


^5 


149.0 


90 


194.0' 


»75 


347.0 


300 


■572.0 


IIOO 


4S 


49.0 


16 


60.8 


4^ 


105.8 


66 


150.8 


91 


195.8 


180 


3560 


'305 


581.0 


1200 


40 


40.0 


17 


62.6 


,42 


107.6 


67 


152.6 


92 


197.6 


185 


365.0 


3»o 


590.0 


1300 


3S 


31.0 


18 


64.4 


'43 


109.4 


68 


154.4 


93 


199.4 


190 


374.0 


"315 


599.0 


1400 


30 


22.0119 


66.a 


i44 


Ill.a 


69 


156.8 


94 


201.8 


»95 


383.0 


320 


608.0 


1500 


*5 


13.020 


68.0 


'45 


1 1 3.0 


70 


158.0 


95 


203.0 


200 


392.0 


325 


617.0 


1600 


20 


—4.021 


69.8 


46 


1 14.8 


71 


159.8 


96 


204.8 


205 


401.0 


330 


626.0 


1700 


fS 


+5.022 


71.6 


47 


1 16.6 


72 


161.6 


97 


206.6 


210 


410.0 


335 


635.0 


1800 


10 


+M.023 


73-» 


48 


II8.4 


73 


163.4 


98 


208.4 


215 


4190 
428.0 


340 


644.0 


1900 


s 


+23.0' 24 


75.«;:49 


120.3 


74 


i65.» 


99 


21 0.8 


220 


|345 


653.0 


2000 


-— 


+3^.0 


I25 


77.0 


150 


122.0 


75 


167.0I 


100 


212.0 


225 


437.0 


350 


662.0 


2100 



662 

753 

842 
933 

1022 
I III2 
) 1202 
I 1292 

• 1382 
H472 

• 1562 
I 1652 

1742 
•1832 
I 1922 
>20I2 
>2I92 
•2372 
>2552 
>2732 

^2912 

^3092 

>327a 
>3452 
•3632 
► 3812 





4C 


), Hyilrometer Scales. 






41 


. Wave-lengrths in Air. 




\ 


CT 


. 












4* 


jj 






|l .: 


>^ 1 


i 








P 






"5 


nt 


Whtc- 


? 






1 


■1 


tj 


s 



-E 


i 


^ 




n 





Lcnijili 


"3 


ii 


Ji 


■2 






1 






1 

if 


a 


J§_ 




Cm- 





t.ooo 




T, 000 'r, 000 


. 


1.000 





Ka 


K 


,_ 


17 


383 


.00007685 


5 


'.OJS 




1.030 


■97 J 




1,02c; 


A 


*- 


^ 


— 


18 


404 


.OOQO7605 


10 


1.073 


1.0 kO 


T.063 


.944 




1. 0^0 


B 


— 


— 


Red 


28 


S93 


,00006870 


IS 


1,114 


.9^7 I.O97 


.9x9 


970 


1.07^ 


— 


Liflf 


Li 


— 


32 


645 


.00006708 


20 


rjc^a 


.95V.133 


.8951936 


r.ioo 


! C 


Hn 


H 


— 


, ^^ 


^94 


.00006^63 


25 


1,105 


.907 


U172 


J72".Q05 


1,125 




Na 


N-^ 




50 


/1003! .oocx>^896 


J^ 


1.3^7 


Mq 


r2[4 


.850;'.876 


1.X50 


l^a 


lia 


Yellow 


(1007 


.00005890 


35 


>.3U 


.854 


1.259 


.82g'J49X.»75 


— ■ 





TI 


Urccn 


68 


— 


,001^05350 


40 


"■375 


.830 


I.30S 


.810.824ll.200 


E 





— 


— 


71 


1523 


,00005270 


45 


1442 


.607 


1.360 


.791 




1.225 


F 


m 


H 


— 


go 


2080 


,00004862 


SO 


».S^7 


.7fiS 


1.417 


■773 




1.250 




Srrf 


Sr 


Blue 


105 


2j86 


.00004607 


55 


K599 


.764 


1.478 


.756 




1.275 


f 


Hj- 


H 





137 


^-- 


.00004341 


60 


r.691 


■745 


T54S 


-73^J 




1.300 


G 




, — 




128 


2854 


.00004309 


65 


^795 




1,6x9 


■723 




1.^5 


B 





Ca 




135 


2870 


.00004227 


70 


t.gii 




1,700 


.708 




1.350 




Hif 


H 


Vioiti 


>Sf 


— 


.00004102 


75 


2.045 




1.790 






I.37S 


— 


K^ 


K 


-.- 


153 


— 


,00004060 


So 












1,400 


iia 


H 


Ga 


— 


ib3 




.0000 ^96(> 


TOO 


\ 










1.500 


— 


— 


^ 


160 


I 


.00003934 



Tables 42—48. Arbitrary Scales. 



893 



6/0 

5/0 
4/0 

3/0 

2/0 
O 



42. 

£§ 

.i| 
1.270 

I.179 
1.097 
1. 01 6 

.945 
.884 
.823 



a. English 

No. Diam. 

I 0.762 

.701 



Board of Trade (Imperial) 

No. Diam. No. Diam. 



WIrfr Gange. 
No. Diam. No. 



2 

3 
4 
5 

6 

7 



11 0.295 

12 .261 



.6|0 

.589 
.538 
.488 

.447 



8 .406 

9 .366 
10 0.325 



13 
14 
15 
16 

18 
19 



.234 
.203 
«i83 
.163 
.142 
.122 
.102 



20 0.091 



21 .0813 

22 .0711 

23 .0610 

24 .0559 

25 .0508 

26 .0457 

27 ,0417 

28 .0376 

29 .0345 

30 .0315 



31 .0295 

32 .0274 

33 ^254 43 

34 -0234 44 

35 -0213 

36 .0193 

37 ^»73 

38 .0152 

39 •0132 

40 .0122 



Diam 
.0112 
.0102 
•009 1 
•0081 
.0071 
.0061 
.0051 
.0041 
.0031 
.0025 



42 b. Birmingham Wire Gange (B* W.Q.) 



> Qfi 

oO 

o a 
Z^ 

0000 
000 

00 
o 



!§ 



B «> 

1.2 
l.I 

1.0 
0.9 



No. Diam. No. 
I 0.80 10 

.74 
.68 
.62 
.57 
.53 
.48 

.43 

0.39 



2 
3 
4 
5 
6 

7 
8 



II 

12 
13 

15 
16 

17 
18 



Diam. 

0.35 
.32 
.23 

.25 
.22 
.19 
.17 
.15 
0.13 



No. 

19 
20 
21 
22 

23 

24 
25 
26 

27 



Diam. 

O.IIO 
.OqI 
.083 

.073 
.065 
.057 

.051 

.046 

0.041 



No. 
28 

29 
30 
31 
32 
33 
34 
35 
36 



Diam. 

0.037 
.034 
.031 
.026 
.023 
.021 
.018 

.013 
O.OIO 



48. Musical Pitch (Tempered Scale— complete YibrationB per second). 

Physical 32 foot 16 foot Great Little afoot i foot 6 inch 3 inch ^ *• 



Pitch Octave Octave Octave Octave Octave Octave Octave Octave 
G 16.0 
16.5 
G# 17.0 

17.4 
D 18.0 

D# 190 

19.6 
E 20.2 

20.7 
F 21.4 

22.0 
F# 22.6 

23.3 
G '24.0 

24.7 
G# 25.4 

26.1 
A 26.9 

27.7 
A# 28.5 

29.3 
B 30.2 

31.1 

32.0 



32.0 


64.0 


128.0 


256.0** 512.0 


1024 


2048 


32.9 


65.9 
67.8 


13^8 


263.5** 527.0 


1054 


2108 


33.9 


1356 


271.2** 542.4 


1085 


2170 


34.9 


69.8 


1396 


279.2 558.3 


1117 


2233 


35.9 


71.8* 


143.7 


287.4t 574.7 


1 149 


2298 


37.0 ' 


73.9* 


147.9 


295.8- • 591.5 


1 183 


2366 


38.1 


76.1* 


152.2 


304.41 608.9 


1218 


2436 


39.2 


78.3 


156.7 


313.4 626.7 


1253 


2507 


403 


80.6 


161.3 


322.5 645.1 


1290 


2580 


41.5 


83X> 


166.0 


332.0 664.0 


1328 


2656 


42.7 


w-^ 


170.9 


341.7 683.4 


1367 


2734 


44.0 


88.0 


175.9 


351.7 703.5 


1407 


2814 


45.3 


90.5 


181.0 


362.0 724.1 


1448 


2896 


46.6 


93.2 


186.3 


372.6 745.3 


1491 


2981 


47.9 


95.9 


191.8 


383.6 767.i§ IS34 


3068 


494 


98.7 


197.4 


394 8 789.6§ 


1579 


3158 


50.8 


101.6 


203.2 


406.4 8i2.8§ 


1626 


.3251 


52.3 


104.6 


209.1 


418.3 836.6 


^^11 


3346 


53.8 


107.6 


215.3 


430.5- 


• 861. 1 


1722 


3444 


55-4 


110.8 


22 r. 6 


443.2- 


• 886.3 


1773 


3545 


57.0 


1 14.0 


228.1 


456.1- 


• 9»2.3 


1825 


3649 


58.7 


120.8 


234.8 


469.5 939.0 


1878 


3756 


60.4 


24 T. 6 


483.3 966.5 


1933 


3866 


62.2 


124.4 
128.0 


248.7 


497.4 994-8 


1990 


3979 


64.0 


256.0 


512.0 


1024.0 


2048 


4096 



G 

G# 

D 

D# 

E 

F 

F# 

G 

G# 

A 

A# 

B 



The Paris Conscn'atoire standard of pitch, recently adopted by 
-I /' . ir: jj ^25 vibrations per second for tfie note 



iVOf0. 

ihe International Congress at Vienna, , 

A of the treble staff. This gives Cs-=26i on the natural scale. American instru- 
ments tuned to "Concert Pitch" give C =- 270 -♦-. 

* Lowest D of Bass Voice. ** Middle C of Piano, f Lowest D of 
Flute, ft Violin A $ Higiiest G of Treble Voice. 



894 



Astronomical Tables. Tables u A.— E. 



Thousandths of degrees. 

oooooDoodooo 9tov9t0^cKo!cSo?cKcSo 

IT • O O O O* ^ O' O O^^ 



?8 

en** 



• Thoosandths of degrees. 



I 



o o o e e o^ o^ 

? 



^ Thousandths of degrees. 

^ O O ©00 O O' O* 0^ O 

^ iHoiMeeee^aOM 



I Thousandths of degrees. 



■5 + + 













I 



^ ^ 



^^S^2?!^ 






I— i§ 



o 



as 



1:^ 

5"^ + +^ 000000 g « r^f^oo ovoi 
OOPOOO 






-22?! 
+ + 



?5 



St 



«^ on m CO 



IS ^ § 



o Thousandths of degrees. 



I 






5;, 



t^ 00 

o ©• 



3 * = «2 



^ A t Js « « « er>tne«^ •-< ehio«- 









Thousandths of degrees. 

8 S S^3"o*8 BIS 8*2 = 2 S?r :i7S ^ 

o" 6 ^ ? ^ 



e ^ M 00 



sss^sliss 




_ a= JESS'S 



as???< 



Oeoesoe 



2 < 



II 

II 

If 
I 






I 



Table84i,F-a. 



Astronomical Tables. 



895 



44, F. 

Day Jan. 

23.091 

1 23011 
S 22.925 
8 22.831 

4 22.729 

5 22.619 

6 22.502 

7 22.378 

8 22.246 

9 22.108 

10 21.961 

11 21^ 

12 21.648 
18 21^ 

14 21.306 

15 21.125 

16 20938 

17 20743 

18 20.542 

19 20.335 

20 20121 

21 19.901 

22 19^75 
28 19443 

24 19^20^ 

25 1&960 

26 18.710 

27 18455 

28 1&194 

29 17927 

80 17^55 

81 17.378 



Declination of the 
Feb. March April 
— di -4- 

17.3^ -7.958 4.157 
«7|96 7.579 4.54r 

1&515 &8i5 5.312 
16.219 6432 

15x^18 —6046 6074 

15.611 5^59 6453 

" ^i 6^30 

7.J 

I4.tti6 4491 7-5: 

14.343 -4.099 7.949 

14.016 3.707 8.318 

13.684 3.314 

i3'35o 2.920 

13.01 1 2.525 9409 

12.669 -3.131 91768 

12.324 1.736 10124 

".975 1.341 10 

11.023 0945 10 



Son in Degrees at 

May June July 



Greenwich Mean Noon for 1891. 

August Sept Oct Nov. Dec. Day 



4- — — — 



15.301 



11.269 



1091 1 -<xi55 
10551 -fo.240 
10188 0635 
9^22 1.029 
9454 1423 
QX)83 -♦-1.816 
8.710 2.206 
8.335 2.600 

[7.579] 3%> 
3.769 
4.157 




ia293 +8.664 

1&044 8.r ~ 
17.790 7. 



a.701 14.117 
3.180 14440 
3.5ffi 14.759 



17.531 7.57a 3.956 i5x»74 
17.268 7.203 4.343 15.: 



16.999 +6^33 

16.726 6401 

£287 




7.773 17.978 
8.147 18^42 



0.550 11.175 



ii.5i< 
II. 
12.198 
12.532 

13.190 
13.514 
13.834 
14.151 
14463 
14.772 



18.869 
19.102 
19.331 
19.553 
19.770 

19.982 
20188 
20388 
20582 
20771 

20953 
21.130 
21.300 
21464 
21.623 

21.775 
21.920 



23405 
23.376 
23.340 
23.297 
23.247 
23.191 



16.167 5.711 
15.881 5-334 

15.590 +4.955 6A|3 17.157 

4.575 7.021 17436 

4.194 7.398 17.710 

3.811 

3427 

+3J043 as 18 18.500 

2^57 8.888 18.752 

13445 2^1 9.255 18.999 

13.124 1.883 9.^ 19.241 

" 1495 9^983 19476 

12473 +1.107 10343 19.706 

12.141 0718 10701 19.930 

11.806 +0328 11.056 20148 

1 1468 — OU061 II40Q 20359 

11.128 0451 11.758 20505 

10784 —0841 12.105 20764 

10437 IJ32 12449 20957 

10.088 1.^2 12.789 21.143 

91736 2^12 13.127 21.323 

9.381 2401 13^160 21495 

2.791 13.791 21.6B1 

14.117 



21.661 

21.820 1 

21.972 2 

22.118 8 

22.255 4 

22.386 5 

22.S0Q 6 

22.026 7 

^ § 

33.929 10 

23.015 11 

23:014 12 

23.165 18 

33:2^ 14 

33.283 15 

23.331 1« 

23.371 17 

23403 18 

23!^ 19 

33414 W 

23453 81 

33454 ii 

23447 |8 

23432 24 

33409 25 

33.379 26 

33.341 27 

23.294 28 

33.241 29 

23.179 80 

23.109 81 



44, G. Equation of Time in Hinntes and Seconds at Greenwich Mean Noon for 1891. 
Day Jan. Feb. March April May June July August Sept Oct. Nov. Dec Day 





1 
2 
8 
4 

5 
6 

7 
8 
9 

10 
11 
12 
18 
14 

15 
16 
17 
18 
19 

20 
21 
22 
28 
24 

25 
26 
27 
28 
29 
30 



3 16 
3 45 
413 

m 

628 
654 
7 19 

832 
854 

9 17 
938 
959 

10 19 

1038 

1057 

II 15 

II 32 

II 48 

" i 
12 19 

1233 

12 46 
1258 
13 10 

13 21 
13 31 
1340 



1340 
1348 

13 50 

14 2 

14 8 

14 13 
14 18 
14 21 
1424 
14 26 

\tU 
1427 
14 20 
1424 
14 22 
14 18 
14 14 
14 9 
14 3 

1357 
1350 
1343 
*3 34 
13 26 

13 16 
13 6 
12 56 
12 45 
[13 33] 



12 45 
12 33 
12 21 
12 8 
11 55 
II 43 
II 38 
II 14 

1059 
1044 

10 29 
10 13 
9 57 
940 
924 

in 

757 

739 
7 21 

6 8 
549 
531 
5 12 
454 
4 35 
417 



m > 
+4 17 
3 59 
341 
323 
3 5 
+348 
3 30 

V^ 

1 39 
+1 33 
I 7 
051 
035 

30 

+0 5 

— O 10 

o 24 

038 

53 

-i.s 

1 30 
142 
153 

-2 4 

315 

3 35 

334 

243 

3 51 



251 

3 13 
3 19 
325 
330 

341 

'I 

348 
3 49 
3 49 

It 

3 47 
3 45 
3 43 
340 
3 37 
3 33 
329 
324 

318 
3 13 

306 

359 

3 53 
344 
335 



m s 

-235 

1% 

3 8 

158 

—I 48 

\U 

I 15 
I 3 
-051 
039 
o 27 

015 
-03 



+0 
o 
o 
o 
I 

4-1 

I 

I 
1 

3 

4-3 

3 
3 
3 
3 

3 



10 



% 20 
332 

3 44 

4 

417 
427 

4 37 

5 5 

513 

^H 
528 

5 35 

542 
548 

I? 

6 6 

6 12 
6 14 
615 
616 
6 17 
616 
6 16 
6 14 
6 12 
6 10 



6 10 
6 7 
6 3 
5 59 
554 
548 
542 

520 
5 12 

Ul 

4 43 
432 

4 31 

350 
3 43 
330 

3 16 
3 46 

3 31 
3 15 

159 

IS 

I 9 
051 
033 
015 



+0 15 
-o 4 

33 
043 

1 I 

—I 31 
141 
3 I 

3 31 
342 

-3 3 
323 

344 

4 5 
427 

-448 

5 9 
530 
552 
13 

-^34 
655 
7 iB 

?g 
-IJ 

859 
9 19 
9 39 
959 



1037 

10 56 

11 14 

II 32 

11 50 
13 7 
13 24 

12 40 

12 56 

13 12 
1327 
1341 
1355 



1618 
16 20 
16 21 
16 21 
16 20 



—II 14 

1052 1 

1029 2 

10 5 B 

941 4 



16 19 — 9 
16 16 8 



16 13 
16 9 
16 5 

1559 
1553 
1545 
1537 
1528 



-l 



17 

826 
7 59 
7 33 
6 



14 9 


rt 


1422 


111 


1457 


14 44 


1458 


14 31 


'5 § 


14 17 


1518 


14 3 


11 U 


1347 


1331 


1544 


13 13 


1551 


1255 


1557 


1237 


10 3 
16 8 


1217 
11 57 


16 12 


1136 


16 15 


II 14 


1618 





5 
6 
7 
8 
9 

10 
11 
12 
18 
14 

15 
16 
17 
18 
19 

.3 18 20 

148 21 

X 18 22 

048 28 

- o 18 24 

4- o 13 25 
042 26 
I 13 27 
141 28 
3 II 29 
340 8f 
3 9 » 



6 10 
542 
5 13 

-4 45 
4 16 
346 
3 17 
347 



896 



Astronomical Data. 

44, H. Solar System. 



Tables 44H, 45. 





_ irt 


u 












N«mi» 


m 


III 


& 


5s 


III 

q2 


ill 
nt 

If. 


1^ 

Is 

J? 


Sun . . . 


... 


• . . 


320.000 


• . • 


1.392 


2,000,000 


^4 


Mercury . 


87.97 


.387 


0.07? 


58. 


4.8 


0.4? 


6.? 


Venus . . 


224.70 


.723 


0.8? 


108. 


12.2 


5.? 


6.? 


Earth . . 


365.26 


1.000 


I.OO 


149. 
•0.39 


12.74 


6.1 


<;.6 


Moon . . 


27.32 


.0026* 


0.012 


3.48 


0.07 


34 


Mars . . . 


686.98 


I.S24 


0.1 1 


227. 


8. 


0.7 


4. 


Jupiter. . 


4332.53 


5.203 


310. 


777. 


H2. 


1900. 


«.3 


Saturn . . 


10759.22 


9-539 


93. 


1424. 


119. 


570. 


0.7 


Uranus . 


30686.82 


19.18 


14. 


2864. 


50. 


85. 


1.3 


Neptune. 


60126.71 


30.05 


17. 


4487. 


60. 


100. 


0.9 



* Distance from the Earth. 



45. Mean Position of Fixed Stars, , 


Fan. 1891. 




Names 


Designation 


•0 

2 


a 


s 
8 


j 


6 






rt 


•S «' 


►f 




if 






S 


< 


1 





1 








h m s 


s 


e 





Sirrah 


a Andromedae 


2 


2 45.2 


+3.09 


--28.489 


- -.0055 


Polaris 


a Ursae Minoris 


2 


I 18 53.4 


2.36 


--88.727 


- -.0053 

- -.0048 


— 


a Arietis 




2 I 1.7 


3.37 


--22.947 


Aldebaran 


a Tauri 




1 1 3^:^ 


3.44 


- -16.289' 


- -.0021 


Capella 


a Aurigae 




4.43 


--45.886 


--.001 1 


Rigel 


fi Ononis 




5 9 17.9 


2.88 


- 8.328 


- -.0012 


Beteigeuze 


a Orionis 




5 49 16.2 


3.25 


4- 7.386; 


--.0003 


Canopus 


a Argus , 




6 21 31.9 


1.33 


-52.636 


—.0005 


Sirius 


a Cams Majoris 




6 40 20.6 


2.64 


-16.567 


—.0013 


Castor 


a'Geminoriim 


2-1 


7 27 38.7 


3.84 


- -32.127 


—.0021 


Procyon 


a Canis Minoris 


I 


7 33 35.7 


3.H 


— ?;.'^c>4 


—.0025 


Pollux 


fi Geminorum 


1-2 


7 38 38.7 


3.68 


--:• 8.289 


—.0023 


Regulus 


a Leonis 


1-2 


10 2 34.0 


3.20 


--12,500 


-.0049 


Denebola 


fi Leonis 


2 


II 43 30.0 


3.06 


--tS.iSi 


—.0056 


— 


a Crucis 




12 20 32.6 


3.30 


—62.495 


-.0056 


Spica 


a Virginis 




13 19 27.0 


3.15 


-10.592 


-.0053 


— 


/J Centaun 




13 56 8.0 


4.18 


-1^9.847 


-.0049 


Arcturus 


a Bootis 




14 10 41.3 


2.73 


+ J 9.750 


-.0053 


— 


«»Centauri 




14 32 12.5 


4.04 


--60.383' 


—.0042 


Antares 


a Scorpii 


1-2 16 22 43.4 


3.67 


- 26.1 89I— .0023 


Vega 


a Lyrae 


I 18 33 14.8 


2.03 


+JS.683! 


--OOO9 


Altair 


a Aquilae 


1-2 19 45 27.9 


2.93 


--44.801! 


- -.0026 


Deneb 


a Cygni 


2-Ij20 37 42.9 


2.04 


--.0035 


Formalhaut 


a Piscis Aust, 


1-2 


22 51 37.6 


332 


— 3CI.2CO 


--.0053 


Markab 


a Pegasi 


2 


22 59 19.8 


+2.98 


+14.619 


--.0054 



Note. Tho yearly precession of the equinoxes is about 5o".35, or GPX10345-K 
The mean (not apparent) obliquity of the ecliptic for 1891 is about 23**, 2/, 13 , 
or 27^.45^ The mean obliquity decreases annually by o'^S, or cPjxxxi. 



Tables 4g-48. Latitude, Longitude and Gravity. 



897 



46. Latitudes and Longitudes Measured ft*oni Greenwicb. 



Latitude Longitude Elevation 
o h m s Metres 



Aberdeen . . . O 57.14^ No 8 23 W 
Amsterdam . . T $2:37 1 N O 19 39 E 
Antwerp . . . T51.221 N O 17 37 E 
Athens .... O 37.972 N I 34 55 E 
Baltimore. . . T 39.298 N 5 628W 55 
Belfast .... 54.66 N 23 . . W 
Berlin . . . . O 52.505 N 53 35 E 40 

Bonn .... 50.729 N o 28 23 E 50 

Boston. . . . T42.358N 44415 W 63 
Brussels . . . 50.853 N O 17 29 E 90 

Calcutta . . . T 22.557 N 5 53 19 E 39 
Cambridge U. S.O 42.380 N 44431 W 
Cambridge Eng. . O 52.215 No O 23 E 
CapeofGoodHope O 33-934 S I 13 55 E 
Christiania . . O 59.9 r 2 N O 42 54 E 42 
Copenhagen . . O 55.687 N O 50 19 E 53 

Cork . . . .T 51.90 N 03351 W 
Dublin .... O53.387 N o 25 21 W T 24 
Edinboro . . . 55-956N o 1243 W T139 
Geneva, . , . 46.200 N o 24 37 E 
Genoa. . . . T 44.419 N 035 41 E 
Glasgow . . . O 55.879 N O 17 1 1 W 
GOttlngeh. . . 5«.53oN 3946E 130 
Greenwich. . .O 51.477 No T 64 
Heldelbuiig . . 49.40 N o 34 32 E lOO 
Leipzlc. . . . 5L335N04934E 100 
Usbon . . . . O 38.705 N o 36 34 W 
Liverpool . . . O53.4OI N O 12 I7 W 
Magnetic Polo . 77.83 N 4 14 W 



50 
663 



44 

525 

43 



Latitude Longitude Elevation 

' b h m 8 Aletres 

London . . 5«.5I4N o O 23 W 

Madrid . . 040:408 N 01445W 

Manchester . 5348 N O 9 . . W 

Melbourne . 37-831 S 9 39 54 E 

Montreal, . T45.52 N 4 54 13 W 

Munich . . 48.146 N 04626E 

Naples . .0 40.863 N 057 iW 

NewOrieans. T29.963N 6 014W 

New York. . O 40.730 N 4 55 57 W T .86 

Paris. . . 048:836 N o 921E 60 

Philadelphia. T39.953N 5 039W 50 

Quebec . .0 46.805 N 44449WT108 

Queenstown. T 51.85 N 033 6W 

Rio de Janeiro O 22.907 S 2 52 41 W T 69 

R:me. . . 41.898 N 04954E 29 

Rotterdart . T51.908 N O I7 55 E 28 

San Francisco O 37.790 N 8 9 43 W Ti 1 1 

Savannah. . T32.081 N 5 24 21 W 42 

St.John(N.S.)T45.262N 4 24 15 W 38 

SfPetersBurg O 59.942 N 2 I 14 E ' ll 

Stockholm . O 59.343 N I 12 14 E 20 

Strassburg . O 48.582 N 031 2E 150 
Sydney . . O 33-861 S 10 450E T 65 

Triest. . . 045-643 N 055 2E T 17 

Venice . .O 45.430 N 04925E 

Vienna . . 48.210 N I 5 32 E 182 

Washington . O 38.894 N 5 8 12 W T 63 
Wellington . T41.288 S II 39 II E T I* 

[Kote. T= Time Signal. = Observatory 



47. Aceeleration of Gravity In DiiTerent Latitudes (cm. per see. per sec).* 

Lat. +0° +10 +2® +E** +4° +5** +§** +7** +8** +9^ Dif 

0** 978.10 978.10 978.11 978.12 978.13 978.14 978.16 978.18 978.20 978.23 I 

10° 978.25 978.29 978.32 978.36 978.40 978.44 978.48 978.53 973.58 978-63 4 

20° 978.69 978.75 978.81 978.87 978.93 979.00 979.06 979.13 979.21 979.28 7 

30° 979.35 979.43 979.51 979-59 979.67 979.75 979.83 979.92 980.00 980.09 s 

40° 980.17 980.26 980.34 980.43 980.52 980.61 980.69 980.78 980.86 980.95 <^ 

50° 981.04 981.13 981.21 981.30 981.38 981.46 981.54 981.62 981.70 981.78 9 

60° 981.86 981.93 982.01 982.08 982.15 982.21 982.28 982.34 982.4! 982.47 7 

70° 982.52 982.58 982.63 982.68 982.73 982.77 982.82 982.86 982.89 982.93 4 

80° 982.96 982.99 983.01 983.03 983.05 983.07 983.08 983.0^ 983.10 983.11 1 

48. Length of Seconds-Pendnlnia In Different Latitudes (cm.).* 
Lat. +0° +1° +2° +3° +4° +5° +6^ +7° +8<> +9^ Dif 
0° 99.103 99.103 99.103 99.104 99.105 99.106 99.108 99.110 99.112 99.115 
10° 99.118 99.121 99 125 99.128 99.132 99.137 99.141 99.146 99.151 99.156 
20° 99.162 99.168 99.174 99.180 99.187 99.T93 99.200 99.207 99.214 99.222 
30^ 99-229 99.237 99-245 99-253 99.261 99.269 99-278 99-286 99.295 99.303 

iTr. 99.312 99.321 99.330 99.338 99.347 99,356 99.365 99.374 99.383 99.391 

50° 99-400 99.409 99.418 99.426 99.435 99.443 99.451 99.459 99.467 99.47s 

gO° 99.483 99.491 99.498 99-505 99.512 99.519 99.526 99.532 99.539 99.545 

70° 99.550 99.556 99.561 99.566 99.571 99.576 99.580 99.584 99.588 99.591 

80^ 99.594 99.597 99*600 99.602 99.604 99.606 99.607 99.608 99.609 99.010 

* These values are calculated for the sea level. A deduction of 0bO3 % should be 
made for each kilometre of elevation above the ground and a deduction of oioa % should 
be made for each kilometre of elevation of the ground above the sea* 



898 Bednction of Measures. Table 49a 

49ft. Beduetioii of Meaanres to and from tbe C. 6. S. Syatem. 



Lenf^ths in centimetres 

f inch 

I link = 7-9* in. ... . 

I foot =: 12 in. . • • • • 

I yard = 3 ft. 

I fathom =: 6 jft .... 

I rod = i67j ft 

I chain = loo links = 66 ft. 

I statute mile = 5280 ft. . 

I nautical mile = i«5,200(?) 

Area« in square centimetres 

I square inch , . . 

I square foot = 144 sq. in. . 

I square yard = 9 sg. ft. . • 

I acre = 43»S6o sq. ft. . . . 

I square mue = 640 acres . 
YolLmes in cubic centimetres 



Equivaient 

2.53997 
20.1 16s 
30.4796 

91.4389 
182.878 

502.914 

2011.6s 

160,932 



6.45H 

929.01 

8361.1 

4.046Pxio^ 

2.5899Xio«> 



Logarithm 
040483 

1.96113 



Reciprocal 

.393705 
0497103 
.0328088 
.0109363 



2.9760 
3.0418 
3.5781 
3.6572 



I cubic inch = 16.386 

I cubic foot = 1728 cu. in. . . = 28316 
I cubic yard = 27 cu. ft. . . . = 764526 
I U. S. pint = 1.04} lbs. water = 473 
I U. S. quart = 2 pmts . . . . = 946 

I dry quart = 1101 

I U. S.gallon = 231 cu. in. = 4 qts. = 3785 
I imperial gallon = 10 lbs. water = 4541 

Massed in grams 

I grain . . = .0647987 

I ounce (Avoirdupois) = 716 lo. = 28 3494 
I ounce (Troy) = 480 grains . = 3».«034 
I pound (Troy) = 12 oz. Troy. = 373240 
I pound (Avoir) = 7000 grains . = 453590 
I English ton = 2240 lbs. . . . = i.oi6o4Xio« 

Times in mean solar seconds 
I year (tropical) = 365.24222 days = 31,556.928 
I sidereal year = 365.25637 days z= 3^558,I50 

I (mean solar) day = 86,400 

I hour . = 3i6oo 

I minute = 60 

I so-called sidereal second . . = 0.9972695666 
I true sidereal second . . . . = 0.9972696721 

Velocities in cf>iitimetres per second 
I kilometre per hour. . . . . = 277778 

I foot per second = 30 4796 

I mile per hour = 44-7033 

I nautical mile per hour . . .= 51.44 
I kilometre per minute . . . . = 1666.67 
I mile per minute = 2682.20 

Accelerations in cn>. per sec. per. see. 
I foot per sec. per sec = 304796 

Densities in ^^rams per cu. cm. 

I grain per cubic inch = .0039544 

I ID j)er cubic foot = .016019 

Heat Units in ergs. 
I unit of heat = i gram-degree C. = 4.17x10' 7.620 
I Ib.-degree Fahrenheit . . . . = 1.051X10W 10.022 
I Ib.-degree Centigrade . . . . = 1.89X10W 10.277 

' I Calorie = 1000 g* = 4-i7Xio>« 10.620 



2.26216 00546813 
2.70149 .00198841 
3.30355 .000497103 
5.20664 6.2i378xio-< 
5.2676 540x10-6 

0.80966 .15500 
2.96802 .cx)i0764 
3.92226 .00011960 
7.607 1 1 2.4711X10-8 
1041329 3.861 ixio—" 

1.2 1449 .061026 
445203 3-53i^x»o-5 
5.88339 » 3o8oxio-« 
2.6750 .002114 

.001057 

.000908 

.0002642 

.0002202 



2.81157 15.4324 
1.45254 .0352741 
1.49281 .0321509 
2.57199 .00267924 

2.65666 2.20463X:iO— 5 

6.00691 9.842iox:io— 7 

749809 3.i6888x:io— * 
7.49811 3.i6875xio-» 
4.93651 .000011574074 
3.55630 .00027777778 
1. 778 1 5 .016666667 
T.99881 10027379091 
1.99881 1.0027378030 

1.44370 .0360000 
148401 .0328088 
1.65034 .0223696 
1.7113 .01944 
3.22185 .0006000000 
342849 .000372827 

148401 .0328088 

3:59708 252.88 
X20463 62426 



240x10—8 
9.52x10—11 
5.29x10—11 
2.40x10— «« 



Table 49b 



Seduction of Measures. 



899 



49 b. Continnatioii. Rednctton of Keasnres to and from tiie 0. G. S. Sjstem. 

Values marked with an asterisk (*) are independent of the acceleration of gravity (g)^ 



■A «D «e Ok Ok 

I I L L L 



^m T^O O 6 O O 

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Constants. 



Table50. 



50. Knmben Freqnentlj Required In Calculation. 

HaUiemafleal Constants. Number Logarithm Reciprocal. 

Ratio ot circumference to diameter .... w = 3-14*5927 f^91^i -SiSsoga 

Square of Ditto • • • . 7f* = 9.8690044 a9943o .1013212 

Square Root of Ditto V^r = 1.772453Q 0^24857 '^^^^ 

Square Root of 3 « 72 = 14143136 0113051 .7071063 

Square Root of 3 V3 = 1.7320508 033856 •5773503 

Square Root of 10 Vio= 3*io33777 050000 .31TO37S 

Logarithmic Base « = 3.7182818 043439 •3678794 

Logarithmic Modulus M =1 04342945 i-0377o 3.30258s 

I degree in circular measure 1° = '01745329 _2 24188 57'2957» 

I minute in circular measure i' = J00039089 440373 3437*747 

I second in circular measure i* = joooco^ B^oSsg 20020^ 

Probable error (mean error = 1) 067449 1.828^ 14820 

AstrcniomiGal Constants. _ 

Sidereal time in Mean time 0^973696 X9d88i 1.0037379 

I (tropical) year - Sec Table 49 — in days .... 305-24222 2r50258 00027379 

Annual precession of equinoxes (5o".25) in days . . 00141 S .2*1509 706 

Aberration constant (20" 45) in degrees 0-00568 .^754+ 176- 

Sun's mean angular semidiameter (16' 2") in degrees 0267 1427 3-74 

Solar parallax (8*.83?) «»45 339o 4o8. _ 

Earth* equatorial radius in kilom 6378. 3^8047 iO0oi56B 

Earths polar „ „ „ 6356.+ 3*9* ^0001573 

Gravity. Attraction between two unit 

masses (1 g) at unit distance (i cm.) in dynes . . . 6.5X10— ■ "g:8i3 i.54Xk/ 

Seconds>pendulum (lat 45") in cm 99-356 i-997i9 /)ioo65 

I gram (lat. 45*) in dynes ^ooi 2.99149 ^0010198 

Atmosplierie mean molecular weight 28.86 14603 

„ Pressure (76 cm. Paris) in megadynes per sq. cm. i. 01360 000587 

Density of Air, (o», 76 cm. Paris) 00012932 511167 773.3 

„ t. « (0^, 1 megadyne per sq. cm.) 00012759 ^.10581 783-8 

^ „ Hydrogen (0^, 76 cm. Paris) 000008057 j^.95216 11 164. 

m n ^ (o't I megadyne per sq. cm.) . 000008837 5-94630 11310. 

« » Water at 4° 1.00001 01000004 099999 

« « ^ V apparent at 2i« 0997 T9987 i-oo3 

« M Crown glass, about ••••.. 2.5 0400 040 

„ » Pfass „ 8.1 0.924 •i»9„ 

„ Mercury at igP 13.550 1.1319* 073800 

Sound Velocity in dry air at (f in cm. per sec. 33,330 or 33,300 4.531 .000^1 

1 mean semitone involves ratio ''V2 1^9463 003509 '^387 

Light Velocity in cm. per sec 3.00X10" 10477 3.33Xior-** 

sodium, wave length in air cm.) .00005893 3'.7?^3 16^ 

Refractive Index of water for ditto 1.333 01248 •75o 

Dispersive - ^^ ^« «, « • v.*.- • • • oi)i4+ 2.« 70 ^ 

Rotation of ditto by Quartz plate 1 cm. thick . . . 217^ 2.3365 xx>40l 

Candle power of i sq. cm. melted platinum (2 Carcels) 20 1.30 «o5 

Heat. Conductivity of Copper C G. S 09-f- T96 i*i 

Coefficient of Expansion of glass (cubical) .... ^00035 340 «agooob 

„ n n n stccl (linear) jooooi3 ^x& 03i00O 

H 1* «. » brass „ ..... xxxniq ^.28 $3i00O 

„ „ » « mercury (cubical 20'') . «»i8o+ 4:2S5 55,ooo 

. *- * u ." * w- " 8**" " • • • -"^^ 'S554+ 273 

Latent Heat of Water 79 1.900 ^*27 

- ^ Steam (looP, 76 cm.) 536 2.729 «)i87 

Specific Heat of Brass ,. 0)094 SL973 loo 

H « C'ass 019 1.279 5-3 

„ „ „ Water (c^—too*) 1.005 0002 .995 

„ n „ Air (0»— looP, 76 cm) 02383 or. . . 0238 "i~377 4-20 

„ n «« Gases, (ratio of 2 Sp. Hts ) . . . . 1408 o 1480 .710 

Mechanical Equivalent of 

I unit ot heal (i g*) in ergs, 4.166X10' or. 4.17X10? 7^620 2.4oXior-a 

I kilogramraetre (laL 45») 9^1X10' 7^149 iJOi98xio-t 

I foot-pound („ „)^ 1.3557X10' 7.13217 7.376X10- -8 

Electro-Chemical equivalent of Hydrogen 

in crams per ampfire per sec. /xxx>ip38 3Joi6a 9634 

I Electrostatic Unit of E. M. F. in volts 300 2477 ^00333 

Electromotive Force of Daniell cell in volts ... ijo to t.2 000 to 006 i^ to 0^8 

Internal Resistance of Quart „ „ „ ohms . . . ito2 00 to 03 Xjoto«5 

I H. A. unit in legal ohms 09889 "1-9952 i^u 

I Siemens „ „ ,, „ 09434 ~i.9747 >-o6t> 

Specific Electrical Resistance of Mercury (C.G S.) 0942X10^ 4.974 iJo6iXia*^ 

Uagnetio susceptibility of iron .•••••••••• 300? 2.5? J003? 

Total Intensity of Earths Field •••••••••• .3 to .7 1.5 toX8 3 to 1.5 



."H\I 11IR04 



i 



' f 




TPhys i^c^x^o 




H^atbartr College ILi&rar2 

FROM 



%J jo^y^. lyp. 



SCIENCE CENTER LIBRARY 




IVitb the Compliments of the Author. 



f 

I 

PHYSICAL MEASUREMENT. 



A SHORT COURSE 

OF 

EXPERIMENTS 

PHYSICAL MEASUREMENT. 



By HAROLD WHITING, 

INSTRUCTOR IN PHYSICS AT HARVARD UNIVERSITY. 

5n JFour Parte. 



Part II. 

SOUND, DYNAMICS, MAGNETISM, AND 
ELECTRICITY. 



CAMBRIDGE: 
JOHN WILSON AND SON. 

SSntbnsitg ^Press. 
1891. 






JAN 27 1891 



Copyright, 1891, 
Bt Harold Whiting. 



TABLE OF CONTENTS, 



MEASUREMENTS RELATING TO 

SOUND, DYNAMICS, MAGNETISM, AND 
ELECTRICITY. 



SOUND (continned.) 

MEASUREMENT OF TIME. 

Page 

LI. Yelocity of Sound % . • 279 

LII. Graphical Method 288 

Lin. Beats 291 

LTV. LissAjous' Curves 295 

LV. The Toothed Wheel 301 



VI TABLE OP CONTENTS. 

DYNAMICS. 

Different Methods of Measuring Velocity in 

Dynamics (IT 147) 308 

THE PENDULUM. 

LVI. Falling Bodies 313 

LVII. Law of Pendulum 316 

LVIII. Method of Coincidences 320 

FOROE. ^ 

LTX, Inertia, 1 330 

LX. Inertia, II 334 

LXI. Composition of Forces 337 

LXn. Centre of Gravity 348 

ELASTICITY. 

LXIIT. Bending Beams . - 350 

LXIV. Twisting Rods 354 

LXV. Stretching Wires 360 

COHESION. 

LXVI. Breaking Strength 307 

LXVII. Surface Tension 369 

WORK. 

LXVIII. Coefficient of Friction 373 

LXTX. Efficiency 379 

LXX. Mechanical Equivalents 387 



TABLE OF CONTENTS. 



vu 



MAGNETISM. 

Paob 

LXXI. Magnetic Poles ^ . 394 

LXXII. Magnetic Forces * 398 

LXXIII. Magnetic Moments 402 

The Magnetometer. 

LXXIV. Magnetic Deflections 405 

LXXV. Distribution of Magnetism, 1 411 

MAGNETO-ELECTRIC INDUCTION. 
The Ballistic Galvanometer. 
LXXVI. Distribution of Magnetism, II 414 

The Earth Inductor. 
LXXVII. Magnetic Dip 422 



ELECTRICITY. 

ELECTRICAL CURRENT MEASURE. 

General Precautions in the Measurement of 

Electric Currents (IT 193) 431 

The Tangent Galvanometer. 

LXXVIII. Constants of Galvanometers .... 437 

LXXIX. Comparison of Galvanometers . . . 448 

♦ 

LXXX. The Dynamometer 451 

LXXXI. Electro-Chemical Method 456 

LXXXII. Method of Vibrations 460 

LXXXIII. The Ammeter, 1 466 

LXXXIV. The Ammeter, II 468 



Vlll 



TABLE OF CONTENTS. 



MEASUREMENT OF ELECTRICAL RESISTANCE. 

Page 

LXXXV. Method of Heating 471 

LXXXVI. Comparison of Resistances .... 474 

LXXXVII. Wheatstone's Bridge 480 

LXXXVIII. Specific Resistance 484 

LXXXIX. Thomson's Method 487 

XC. Mance's Method 490 

XCI. U3E OF A Shunt 493 

XCII. Ohm's Method 498 

XCIII. Beetz' Method 501 

measurement of electromotive force. 

Classification op Methods (IT 230) 511 

XCIV. Wiedemann's Method 518 

XCV. The Thermo-Electric Junction . . 520 

XCVI. The Volt-Meter, 1 524 

XCVII. The Volt-Meter, II 528 

XCVIII. Clark's Potentiometer 529 

XCIX. Poggendorff's Method 531 

C. Electrical Efficiency 533 

-I 

EXPERIMENTS FOR ADVANCED 

STUDENTS 537 

INSTRUMENTS OP PRECISION .... 568 



PHYSICAL MEASUREMENT. 



^act ^etottH. 



MEASUREMENTS IN SOUND, J)YNAMIOS, 
MAONETISM, AND ELEOTRIOSTY. 

SOUND — Continued. 



EXPERIMENT LL 

VELOCITY OP SOUND. 

^ 135. Determinatioii of the Velocity of Sound. — 
(1) Two data are required for the detennination of 
the velocity with which sound passes frpm one point 
to another: 1st, the distance between two stations 
(see % 136) ; and 2d, the time occupied in traversing 
this distance (see % 137). To make use of the results, 
the temperature of the air must be found at various 
points between the two stations (see Part I. ^ 15) ; 
and if precision is required, the humidity of the air 
should also be determined.^ The velocity of sound 
is not affected by barometric pressure. 

1 At ordinary eummer temperatures (20° to 30°) the effect of hu- 
midity upon the velocity of sound may amount to one half of 1 %. 
See Table 16, B. 



280 VELOCITY OF SOUND. [Exp. 51. 

(2) If the path traversed by the sound is at right- 
angles with the direction of the wind, the velocity of 
sound will not be perceptibly affected by any ordi- 
nary atmospheric disturbance. It is, however, in- 
creased by the velocity of the wind when the two 
move in the same direction, or diminished by the 
same amount when they move in opposite directions.^ 
When the directions are oblique, the velocity of sound 
is always more or less affected. It is therefore best 
to arrange an experiment so as to find the time oc- 
cupied by sound in traversing a given distance first 
in one, then in the opposite direction. In this case, 
if the velocity of the wind is small and tolerably 
constant, the average result will not be perceptibly 
affected by it. 

(3) Two or more determinations of the velocity 
of sound should be made between stations at differ- 
ent distances. Any constant error in the estimation 
either of distance or of time will be shown by a disa- 
greement of the Several results. The true velocity 
of sound is to be calculated in such a case from the 
difference in ^me required to traverse two given dis- 
tances (see formula II. below). 

(4) Let d be the distance traversed by sound in 
the time t ; then the velocity of sound, t?, is to be cal- 
culated by the equation 

v=^. I. 

t 

1 A velocity of the wind amounting to 10 metres per second, or 
about 22 miles per hour, would affect the velocity of sound by about 

3%. . 



I 



1 136.] LONG DISTANCES. 281 

Distinguishing by subscript numerals 1 and 2 the 
results in the two cases, we should have 

\ hence, ^=1 

Subtracting 1 from both sides of the equation we 
have 

'^i _ 1 — ^1 _ 1 . 

or, reducing to a common denominator, 

whence d, — d,_d^ 

Finally, substituting equals for equals, we find 

v=^^-^\ II. 

t, — t^ 

By the use of this formula, constant errors (§ 24) 
are eliminated. 

^ 136. Measarement of Long DiBtances. — The 
measurement of long terrestrial distances is in general 
a problem for which the student must be referred to 
works on surveying. No particular difficulty will, 
however, be found in measuring approximately a dis- 
tance along a moderately straight path ; for even 
variations as great as 8° (nearly 1 foot in 7), either in 
the direction or in the slope of the path, will in- 
troduce an error of less than one per cent in the 
result. 



282 VELOCITY OF SOUND, [Exp. 51 

Distances may also be determined indirectly by 
means of a sextant. To measure a distance, for ex- 
ample, across a valley, from an observing station, -4, 
(Fig. 123) to an object B^ we place (or select) an ob- 
ject (7, so that the lines joining B with A and with C 



±iyi. 123. 

may be approximately at right-angles. The distance 

BO is then measured directly, and the angle CAB 

is determined from the observing station. Since (by 

definition) 

BO -^AB = tangent CAB, 

we have AB = - — -—r^- 

tan CAB 

To obtain with an ordinary sextant (see ^ 124) re- 
sults accurate within 1 per cent, the distance BC 
actually measured should be at least a hundredth 
part as great as the distance AB to be determined. 
In regard to the direction of C from J?, great accu- 
racy is not required. If the corner of a squnre be 



Fig. 124. 

placed at B (Fig. 124) with one side directed to- 
wards -4, any object, C, nearly in range with the other 
side of the square, will answer for our purpose. An 
error of 8° in the angle ABC will introduce an 
error of only 1 % in the result. The object C may 



t 136.] LONG DISTANCES. 283 

be on a level with B or above it, as may be more 
convenient. The distance BO and the angle CAB 
must be accurately measured. 

In one part of the experiment the distance AB 
should be as great as possible considering the space 
at the disposition of the observer, and the distance 
through which the signals at his command can be 
seen or heard. If the method of difference is to be 
employed (^ 135, 3), it is necessary, in a second part 
of the experiment, to make use of a much shorter 
distance. The second distance should be in no case 
greater than half of the first, and always as small as 
is consistent with the accurate determination of the 
time occupied by sound in traversing it. When the 
time is to be found by an ordinary watch (^ 137, 1.), 
the smaller distance should be several hundred, the 
greater several thousand metres. In the pendulum 
method (^ 137, IV.), distances of 300, 600, and 900 
metres may conveniently be employed. When sound 
signals are, to be sent back and forth between two 
stations (^ 137, III.) 5 the minimum distance may be 
reduced to about 150 metres. The velocity of sound 
has been determined by the use of echoes (^ 137, II.) 
between the Jefferson -Physical Laboratory and the 
Lawrence Scientific School, the walls of which are 
about 80 metres apart. Long corridors, tunnels, and 
conduits of various sorts frequently give rise to 
echoes suitable for the determination of the velocity 
of sound. 

It must be remembered that in the time required 
for a signal to go from one station to another, then 



284 VELOCITY OF SOUND. [Exp. 51. 

back to the first, the distance traversed is twice that be- 
tween the stations. When the sound is reflected back 
to the observer the distance traversed is twice that 
of the observer from the object causing the reflection. 
Care must be taken to identify the object in question. 
In the interval between two successive echoes, sound 
must obviously traverse twice the distance between 
two objects which reflect it, as for instance two par- 
allel walls or the two ends of a conduit. 

^ 137. Measurement of Short Intervals of Time. — 
I. One of the oldest methods of estimating the 
time required for sound to traverse a given distance 
is to count the ticks of a watch which occur between 
the flash and the report of a cannon discharged at 
that distance from the observer (see ^ 138). When, 
owing to obstructions in the field of view, it is im- 
possible to see the flash, an electric telegraph may 
serve in the place of light to inform the observer 
of the exact moment of the discharge.^ Instead of 
counting ticks, a " stop-watch " may be used, or a 
chronograph may be employed (IT 266). Amongst 
various ingenious devices for the measurement of 
small intervals of time may be mentioned the use of 
a stream of mercury from a Mariotte's bottle (see Fig. 
275, ^ 250), which may be directed into a receptacle 
at the beginning of the interval, and diverted at the 

1 The velocity of light is about 30,000,000,000 cm. per sec. ; hence 
the time lost in traversing terrestrial distances may generally be dis- 
regarded. An electric current is practically instantaneous in its ac- 
tion ; but an allowance must be made for the slowness of telegraphic 
instruments to respond to the current, unless a method of difference 
be employed. See 1 135, S. 



IT 137.] SHORT INTERVALS OF TIME. 285 

end of the interval. The quantity of mercury col- 
lected serves to estimate very precisely the interval 
of time in question. 

II. In certain localities the velocity of sound may 
be similarly determined by timing the interval be- 
tween a sound and its echo. When a series of echoes 
may be heard, the interval between them may be de* 
termined by adjusting a pendulum or a metronome 
so as to keep time with the echoes while they last, 
then afterward finding the rate of the pendulum or 
metronome, by timing 100 or more oscillations. 
Again, a method of multiplication may be used 
(§ 39). When the last audible echo reaches the 
observer, a new sound may be made ; so that the in- 
terval of time to be measured may be indefinitely 
increased. One of the earliest determinations of the 
velocity of sound is said to have been made by a 
monk, who made use of the echo in a cloister caused 
by clapping his hands. The sounds thus produced 
were, it is said, so timed as to alternate regularly 
with the echoes. 

III. The effects of an echo may be imitated by a 
series of sound signals interchanged between two 
stations. Let us suppose that two observers, each 
provided with a hammer and a plank, place them- 
selves at suitable distances (see ^ 136). The first 
gives a blow with his hammer, then the second re- 
turns the signal as soon as the sound reaches him. 
When the first hears the response, he gives another 
blow, etc. As in the last method (IL), the interval 
of time to be measured may be indefinitely multiplied. 



286 TIME. [Exp. 51. 

With practice, each observer will learn to anticipate 
the return signal, 'so that very little time will be lost 
in the act of repetition. The time thus lost is to be 
eliminated by making two experiments, as has been 
suggested above (^ 136, 3). 

IV. Another method ^ is to station two observers 
let us say 300 or 350 metres apart, and to provide 
each with a telescope, if necessary, so that he may 
watch a pendulum, or any other object having a peri- 
odic motion, in sight of both observers. Either the 
length of the pendulum, or the distance between the 
observers is then varied until a sharp sound made 
by A, when -the pendulum is at the middle point 
of its swing, is heard by B at the moment when the 
pendulum, after completing one or more oscillations, 
again passes the middle point. The distance is then 
measured, and the time of the pendulum determined. 
Measurements must also be taken in which sounds 
made by B are heard by A as the pendulum passes 
its middle point. The experiment is then repeated 
with a distance between the observers (^ 135, 3) 
two or three times as great as before. 

Other methods of measuring short intervals of time 
will be considered in experiments which follow. 

^ 138. Proper Methods of Coiintlng. — In counting 
the ticks of a watch (which usually occur at inter- 
vals of one-fifth of a second), it will be found diffi- 
cult, if not impossible, to repeat, even mentally, the 
names of numbers which contain more than one 

1 See Ejc. 30, Elementary Physical Experiments published by 
Harvard University. 



1138.] METHOD OF COUNTING. 287 

syllable.^ In the following method of counting, this 
difficulty is avoided: — 



1 


2 


3 


4 


5 


6 


7 


8 


9 


10 


1 


2 


3 


4 


5 


6 


7 


8 


9 


8 


1 


2 


3 


4 


5 


6 


7 


8 


9 


8 



By counting the ticks which actually occur within 
a given interval of time, the length of that interval 
will on the whole be fairly estimated. There is, 
however, a tendency in most persons to count one 
too many ticks. When a given interval contains a 
whole number of ticks, one occurring at the begin- 
ning of the interval should be counted " nought,*' or 
not counted at all. Obviously the first and last tick 
should not both be counted. 

With intervals of time (as with intervals of space), 
care must be taken to distinguish the number of in- 
tervals from the number of divisions between which 
they lie. In the same way that the zero of a scale 
should not be counted "one," the beginning of an 
interval of time should not be called one second 
or one-fifth of a second. A miscount may generally 
be avoided by pronouncing the word " now " at the 
beginning of the interval, then beginning the count 
immediately afterward. 

An accurate method of counting is important in a 
great variety of measurements, especially those which 
involve rates of vibration or revolution. The student 
should consider carefully what habits he has formed 

1 The difficulty is greatly lessened by counting every other tick; 
but on account of the greater inaccuracy, this method of counting is 
Dot generally rec6mmended. 



288 



TIME OF VIBRATION. 



[Exp. 52. 



in this respect, and if they are not good, whether 
it is preferable to change them, or to make an al- 
lowance for *' personal error " in each separate 
determination. 



EXPERIMENT LH. 
GRAPHICAL METHOD. 



^ 139. Determination of Rates of Vibration by the 
Oraphical Method. ^ — A tuning-fork (ae, Fig. 125) 




Fig. 125. 

making from 100 to 300 vibrations per second, and a 
pendulum (Jf), made of an ounce bullet (/) and a 
piece of clock-spring (6), are mounted as in the fig- 
ure, so that when the tuning-fork and pendulum are 
in vibration, two short and fine brass wires attached 
one to each may make marks (h and i, Fig. 126) as 
close together as possible on a piece of smoked glass. 

^ The experiment here described is essentially the same as that 
given in Exercise 81, Elementary Physical Experiments, Harvard 
University. This application of the graphical method is due to 
Prof. Hall. 



^ 139.] GRAPHICAL METHOD. 289 

The tuning-fork and the spring are then firmly 
clamped by the screws c and d. 

The smoked glass is now drawn slowly out from 
under the pendulum and the tuning-fork. The points 
of the wires e and / should draw a single line (hix^ 
Fig. 126) upon the surface of the glass. If they do 
not, the wires should be bent, or their relative posi- 
tion otherwise adjusted. The smoked glass is now 
to be replaced, and both the pendulum and the 
tuning-fork are to be set in vibration, — the latter by 
drawing a violin-bow across one of the prongs. The 
bow must be drawn slowly at first, and always in a 




Fig. 126. Fio. 127. 

direction nearly parallel to the vibration which it is 
desired to create. That is, the bow should be held at 
right-angles to the prongs, but nearly parallel to the 
plane containing them. The smoked glass is again 
di-awn out from under the pendulum and the fork, 
with a slow but uniform velocity. 

The wire attached to the tuning-fork, partaking of 
its vibration, will trace upon the glass a series of 
waves. The wire attached to the pendulum would 
similarly trace a series of much longer waves, were 
it not that owing to the amplitude of its oscillation, 
the wire usually leaves the glass at the extreme 
points of a swing. The result is a series of marks 
(y,A, i,etc., Fig.127). 

19 



290 TIME OF VIBRATION. [Exp. 52. 

The time required for one complete oscillation of 
the pendulum is represented by the distance between 
alternate marks {j and Z, or k and w, Fig. 127). The 
number of complete vibrations made by the tuning- 
fork in the same length of time is to be found by 
counting the waves executed in the same distance. 
Thus between j and I there are (in the figure) about 
6 J complete, or 12 J half-waves; and between / and n 
there are similarly about 7 waves. In practice, a 
much greater number would be counted. 

If the waves are perceptibly closer together at k 
or at I than at m or at n (or the reverse), the glass 
has not been drawn with sufficiently uniform veloc- 
ity. In this case, instead of depending upon the 
marks (;, A;, Z, etc.) actually made by the pendulum, 
it is necessary to draw a line at a distance from each 
mark equal to that between h and i (Fig. 126), and 
at the left or at the right of it, according to whether 
h is at the left or at the right of f. The new lines 
show where the wire attached to the pendulum would 
have eroaaed the glass, provided that it could have been 
made absolutely coincident with the wire attached to 
the pendulum. By the use of lines drawn as above, 
we may in counting the waves avoid errors due to 
irregularity in the speed of the glass. The number 
of whole waves included between two alternate lines 
should be recorded in each case, together with an 
estimate of the fractions of a wave left over at each 
end of the series. This fraction should be expressed 
in tenths § (26). 

To find the rate of vibration of the tuning-fork, 



1 140.] THEORY OF BEATS. 291 

the time occupied by one complete oscillation of the 
pendulum must now be determined. This is done 
by timing, let us say, one hundred complete oscilla- 
tions. Having given a signal, one observer begins 
to count the oscillations of the pendulum, while a 
second observer, as soon as the signal is perceived, 
begins to count the ticks of a watch (see ^ 138). 
When the pendulum has completed a given number 
of oscillations, the first observer signals to the second 
to stop counting. 

The number of complete oscillations of the pendu- 
lum per second is found from the time required for 
100 or 200 oscillations (as the case may be), by simple 
division, and the result is multiplied by the average 
number of waves made by the fork during one of these 
complete oscillations to find the " vibration number," 
or " pitch " of the fork, — that is, the number of com- 
plete vibrations made in one second. 



EXPERIMENT LIII. 

BEATS. 

^ 140. Theory of Beats. — When two musical notes, 
nearly but not quite in unison, are sounded together 
with about the same degree of loudness, the effect 
upon the ear is by no means uniform. At regular 
intervals the sound swells out, and these intervals 
are separated by moments of comparative silence. 
Each rise and fall of the sound constitutes a " beaC 



292 TIME OF VIBRATION. [Exp. 53. 

The increase is due to the mutual re-enforcement of 
the two sets of vibrations communicated to the air ; 
the decrease is caused by the interference of these 
vibrations. 

Let us suppose that two tuning-forks, one making 
256, the otiier 255 vibrations per second, are started 
at a given instant by forcing their prongs together 
and suddenly releasing them. The prongs of both 
forks will spring apart simultaneously, and each fork 
will cause a slight condensation of the air on each 
side of it. This condensation will be followed by a 
rarefaction when the prongs rebound, then by sev- 
eral alternate condensations and rarefactions, nearly 
though not quite synchronously performed. The re- 
sult is that the vibrations reaching the ear at the 
same distance from both forks are very much greater 
than if one fork were sounding alone. At the end of 
half a second, however, the first fork will have made 
256 -f- 2, or 128, complete vibrations ; so that, as at 
the start, its prongs will be springing apart; but the 
second fork will have made only 255 -f- 2 or 127J 
vibrations, so that its prongs will be approaching 
each other. The condensation produced by one fork 
will tend to oflfset the rarefaction produced by the 
other. The effect on the ear will accordingly be less 
than if one of the forks were sounding alone. This 
interference of the vibrations will evidently continue 
as long as the forks are vibratinc^ in opposite ways. 
At the end of a second, the first fork will have 
made just 256, the second fork just 255 complete 
vibrations, and the direction in which the prongs 



% 141.] METHOD OF BEATa 298 

are moving will be in each case the same as at the 
start, and hence the same for both forks. The 
sounds will therefore re-enforce each other as at first. 
It is evident that, with the forks in question, periods 
of re-enforcement must occur every second, separated 
by intervals of interference. In other words, two 
forks making 256 and 255 vibrations per second 
must give rise to 1 ^^ beat'' per second when sounded 
togetlier. 

In the same way it may be shown that two forks 
differing by n vibrations per second give rise to n 
beats per second. In other words, when two musical 
notes are nearly in unison, the number of beats per 
second is equal to the difference between the vibration 
numbers corresponding to the two notes in question. 

^141. Determinations of Fitch by the Method of 
Beats. — The special apparatus required for this ex- 
periment consists of a series of tuning-forks with 
differences of from three to five vibrations per second, 
covering an interval of one octave (^ 134). The 
first and the last of the series are to be sounded 
together, to make sure that the musical interval is 
exact. If the forks are nearly but not quite an 
octave apart, faint beats may be heard. In this case 
one of the forks must be loaded with small bits of 
wax near the end of its prongs until the beats dis- 
appear. If the wrong fork is loaded the beats will 
become more frequent than before. The same effect 
may be produced if too much weight is added to 
either fork; hence care must be taken at first to add 
very little weight at one time. 



294 TIME OF VIBRATION. [Exp 58. 

The simplest way in general to tell whether a fork 
is higher or lower than may be required for the pur- 
poses of harmony is by the method of loading sug- 
gested above. The effect of the additional weight is 
to lower the rate of vibration of the fork to which it 
is attached. Whenever by loading a fork it may be 
brought into harmony with a given musical note, we 
know that fork to have a higher rate of vibration 
than the purposes of harmony require. 

If, for instance, the first fork in the series gives 61, 
and the last 120 vibrations per second, the first will 
have to be loaded until it gives 60 vibrations per 
second, in order to be in harmony with the other 
fork. Again, if the second fork gives 64 vibrations 
per second, it will have to be loaded to bring it in 
unison with the first fork. We may generally as- 
sume that the forks are arranged by the instrument- 
maker in an ascending series. 

The experiment consists in a determination of the 
number of beats produced in a given length of time 
by sounding together each pair of consecutive forks 
in the series , that is, the first and second, the second 
and third, the third and fourth, etc. The student 
will do well to begin counting with one of the beats 
which happens to occur when the second-hand of his 
watch indicates a round number. The beginning of 
this beat should not be counted (see ^ 138), One 
hundred beats should be timed if possible. The time 
of the last beat should be observed to a fraction of a 
second. The number of beats per second should be 
calculated in each case. 



t 142 ] LISSAJOUS' CURVES. 295 

The results represent dififerences between each pair 
of consecutive forks in the series ; hence when added 
together we have the difference between the first and 
the last in the series , for the whole difference in 
question must be equal to the sum of all its parts. 

Now two notes an octave apart are to each other, 
in respect to their vibration numbers, as 2 is to 1 
(^ 184) ; hence the last number in the series is 
twice the first. It follows that the difference be- 
tween the first and last numbers is equal to the fii'st 
number in the series. The result of adding together 
the numbers of beats per second is therefore to find 
the number of vibrations executed by the first fork 
in one second. 

By adding to this number the number of beats per 
second between the first fork and the second fork we 
find the pitch of the second fork , and in the same 
way, successivel}^ the pitch of each fork in the series 
can be calculated. 



EXPERIMENT LIV. 

LISSAJOUS' CURVES. 

^ 142. Theory of Lissajous' Curves. — We have 
seen, in Experiment 52, that when a piece of smoked 
glass is drawn beneath a pointed wire attached to a 
vibrating tuning-fork, a wave-line is traced upon it. 
If instead of drawing the glass completely away 
from the tracer, the motion be suddenly reversed, we 
shall evidently obtain a double wave which will re- 



296 TIME OF VIBRATION. [Exp. 54. 

semble one of the figures below (Fig. 128, 1, 2, and 
3) according to the point (a) in the curve at which 
the reversal takes place. In the first curve the two 
waves happen very nearly to coincide. We may im- 
agine the reversal to take place so that there should 
be a perfect coincidence. 

Now let us suppose that when the tracer reaches a 
certain point, J, a second reversal takes place, and a 
third reversal occurs when the tracer returns to the 
former point, a. Evidently, if the reversals are prop- 

XAAAAAA/ 

^.^ /X\ /TOv /X\ /X\ /9\^ 





V vy ^^' v/ ••..' v/ \y vy \.*' 

Fig. 128. 



erly timed, the tracer will follow the same path over 
and over. 

In practice we obtain a similar result by attaching 
a small piece of smoked glass to the larger of two 
tuning-forks. When the larger fork makes one vibra- 
tion in the same time that the smaller fork makes for 
instance 8, we obtain tracings as in Fig. 129, 1, 2, 
or 3, according to the relation which happens to 
exist between the forks at the start. 

These are examples of Lissajous' curves. The re- 
versal of the smoked glass is not sudden, as in the 
case previously supposed, and its velocity is greatest 



^ 142.J LISSAJOUS' CURVES. 297 

when the middle of the figure is being drawn. This 
accounts for the difference in appearance between 
these curves and those represented in Fig. 128. 

It may be shown that whenever two vibrations at 
right-angles are compounded graphically, as in Fig. 
129, unless the times of the vibrations are incom- 
mensurate, a Lissajous' curve results. Each musical 
interval (^ 134) has, accordingly, its characteristic 
curves. These curves are in general too complicated 
to be discussed in an elementary work. We shall 
confine ourselves to such cases as are represented in 

■\Ay^\A/ 

Fig. 129. 

Fig. 129, where one fork makes a certain whole num- 
ber of vibrations while the other makes one. 

To find in such cases the musical interval between 
the forks, we have to experiment until a figure like 
the third is obtained (Fig. 129, 3). If this figure 
contains n lobes, then the higher fork niakes n times 
as many vibrations as the lower fork. 

It has been so far assumed that the two forks are 
separated by an exact musical interval, so that at 
the end of a certain period they find themselves in 



298 TIME OF VIBRATION. [Exp 54. 

exactly the same mutual relation as at the start. If 
this is not the case, it is evident that the tracer will 
not follow the same path in all cases, but that this 
path will be continually changing. 

Let us suppose that the tracer reaches its highest 
point, as seen in the figure, when the glass reaches 
its extreme right-hand or left-hand turning-point. 
Then the curve traced will be represented as in Fig. 
129, 1. If the small fork is a little behind-hand we 
shall have a tracing as in Fig. 129, 2 ; and if the 
small fork has only reached the middle of its course 
when the glass turns, we shall have a tracing like 




Fig 130. 

Fig. 129, 3. Evidently, if the small fork starts as in 
(1) and falls slowly behind the other, we shall have 
a series of tracings represented by (1), (2), and (3). 
It is not until the higher fork has fallen one complete 
vibration behindhand that the same figure will be 
repeated. 

If the smaller fork is gaining instead of losing, 
a similar series of changes will be produced. There 
is in fact no way to tell which fork is too high for the 
musical interval in question, except as in the last ex- 
periment, by loading it and observing the result. A 
complete cycle of changes in the case of two forks 
one octave and one fifth apart (^ 134) is shown in 
Fig. 130. 



1[ 143.] LISSAJOUS' CURVES 299 

The symmetrical lobed figures (3 and 7) appear 
twice in a cycle ; the serpentines appear also twice ; 
but one of them is left-handed (1), the other right- 
handed (5). The interval between two left-handed 
(or that between two right-handed) serpentines 
always represents one complete cycle, and is ac- 
cordingly equal to the time in which the higher 
fork makes one whole vibration more or less than 
would be required to give a perfect musical interval. 

Let p be the pitch of the lower fork, that is, the 
number of vibrations it makes in one second, and let 
n denote the approximate musical interval between 
the forks; then the pitch of the higher fork, which 
we will call P, must be equal to wp, nearly. If, how- 
ever, we observe c cycles per second, the true pitch of 
the higher fork is np ± c. Here c is positive if by 
loading the higher fork the musical interval may be 
made perfect ; if on the other hand the lower fork 
must be loaded, c will be negative. With this under- 
standing we have 

P=znp + c. I. 

and p = . II. 

n 

These formulae apply only to cases in which, as we 
have supposed, w is a whole number. 

^ 148. Determination of Pitch by Lissajous' Curves. 
— A tuning-fork of known pitch (Exps. 52 and 53) 
and one approximately an octave above or below it 
are to be mounted, as in Fig. 131, with their prongs 
at right-angles. The prongs of one fork (^A) are to 
be coated with lampblack, except at a small point 



300 



TIME OF VIBRATION. 



[£xp. 54. 



where, by the touch of a pin, the bright metallic sur- 
face is made visible. Opposite this point on the other 
fork (jS) a lens, 6\ of about 1 inch focus, is to be at- 
tached with sealing-wax, at such a distance that a 
highly magnified image of the point may be seen 





■— f 



Fig. 131. 



through the lens. When a violin-bow^ is drawn 
across the fork -4, the bright spot partaking of the 
vibration will be apparently extended into a hori- 
zontal line, Fig. 132. 




Fig. 132. Fig. 133. Fig. 134. 

When the fork B is set in vibration, the motion of 
the lens will cause the spot to be apparently elon- 
gated into a vertical line, as in Fig. 133. When both 

1 In practice, it will be found convenient that one or both of the 
forks should be maintained in vibration by electrical means. 



IT 144.] THE TOOTHED WHEEL. 801 

forks vibrate Bimultaneously the vertical and horizon- 
tal motions will be combined, and if the forks are sep- 
rated by an exact octave, one of Lissajous' curves 
will be formed, as for instance in Fig 134. 

If this curve is permanent in form, the experiment 
is now finished; but if, as is generally the case, it 
passes through a series of cycles, as in Fig. 130, ^ 142, 
it becomes necessary to count the number of com- 
plete cycles which take place in a given length of 
time. It is also necessary to load one of the forks, 
as in ^ 141, until the changes in the cycles become 
less frequent.^ 

We thus find whether c is positive or negative 
in the formulae of ^ 142. The pitch of one of 
the forks is finally to be calculated by one of the 
formulae in question from the pitch of the other 
fork, previously determined. 



EXPERIMENT LV. 

THE TOOTHED WHEEL. 

^ 144. ConBtruction of a Toothed-'Wheel Apparatus. 
— A toothed- wheel apparatus capable of giving fairly 
accurate results is represented in Fig. 135, as seen 
from above. A vertical cross-section is shown also 
in Fig. 136. The works (c) of an ordinary eight-day 

1 It is po»8ib1e to load a fork so that a flprure of a certain class 
(see Fig. 130, 1-9) may preserve its characteristics until the vibra- 
tion dies awaj. 



302 



TIME OF VIBRATION. 



[Exp. 55. 




Fio. 135. 



spring clock, from which the escapement has been re- 
moved, are mounted on a piece of wood, and a disc 
of cardboard (a) is attached to the axle usually carry- 
ing the second hand. Two pieces of watch-spring are 

also attached to this 
axle at 6, and bent into 
loops so that two small 
loads (<? and rf) whicli 
they bear may hang 
quite close together 
when the wheel is 
at rest. The friction 
which the springs ex" 
ert against the air acts 
as a governor upon the 
speed of the machine. 
The velocity of rotation will be found to vary very little 
as the force of the main-spring grows less and less. To 
make the w^heel turn faster, the loads 
(c and d) may be decreased ; or a 
slight change may be produced by 
winding up the main-spring. To 
make the wheel go slow, the load may 
be increased ; or a slight decrease in 
speed may be had either by waiting for the main- 
spring to unwind itself, or by applying friction to one 
of the more slowly moving wheels. The upper sur- 
face of the disc, a, should be painted black. The 
number of revolutions which it makes in a given 
time may be counted by watching a white spot 
upon it, or still better by listening to the sound 






iB3--iai. 



Fig. 136. 



1 145.] THE TOOTHED WHEEL. 303 

made by an object striking lightly against a projec- 
tion from the wheel or from the axle upon which it is 
mounted. At equal distances around the circumfer- 
ence of the wheel, narrow radial slits should be cut 
out. The number of slits must be made with refer- 
ence to the usual speed of the machine and the num- 
ber of vibrations per second which the toothed wheel 
is intended to measure. The wheel represented in 
Fig. 135 makes about 8 revolutions per second with- 
out any load, — the speed being reduced to 4 revo- 
lutions per second by a load of a few grams at c 
and d. With twelve notches in the disc, this ap- 
paratus affords from 48 to 96 nearly instantaneous 
views of objects seen through the rim of the 
wheel. The instrument is accordingly suited to the 
determination of the pitch of tuning-forks making 
from 48 to 96 vibrations per second. It may also 
be used for much higher forks, as will be presently 
explained. 

^ 145. Theory of the Toothed Wheel. — By the ap- 
paratus just described we are able to obtain at reg- 
ular intervals a series of instantaneous views of a 
vibrating object. If the intervals between the views 
correspond to the period of vibration in question, the 
same view will evidently repeat itself over and over. 
If the intervals are sufficiently short, the effect will 
be a continuous impression upon the eye. Thus 
when the eye is held close behind the rim of the ro- 
tating disc (Fig. 135), the speed of which is prop- 
erly adjusted, we may obtain a series of views of a 
tuning-fork, in all of which the prongs are, for in- 



304 



TIME OF VIBRATION. 



[Exp. 55. 



stance, at their greatest elongation. The result is 
tliat the fork appears to be at rest. To obtain this re- 
sult the number of slits which pass in front of the eye 
in one second must be equal to the number of vibra- 
tions executed by the fork in the same time. If the 
wheel is moving a little too fast or too slow, the suc- 
cessive views of the fork will not be exactly the same. 




Fig. 137 

The position of the prongs will seem to change as if 
the fork were executing a very slow vibration. When 
the fork is held close behind the rim of the disc, as in 
Fig. 137, a different effect is produced. 

Let us first consider the effect of a single slit mov- 
ing along the fork. Let 1, 2, 3, 4, 5, 6, 7, 8, Fig. 138, 




Fig. 138. 

be views of the fork seen through such a slit when 
occupyinioj the successive positions a, 6, c?, c?, e,/, g^ A, 
and L These views are evidently situated along the 
dotted line aL Let us now supply the intermediate 
views. We shall evidently have the curve shown in 



t 145.] THE TOOTHED WHEEL. 806 

Fig. 137, or in a6, Fig. 139. Now let another slit pass 
along the fork. We shall have similarly a carve, cd 
or ef (Fig. 139), which may or may not coincide with 
ab. If it does not coincide with ai, we shall prol>- 
ably not see either of the curves, since the light re- 
flected through the slits will hardly have time to 
affect the eye. If, however, several such curves coin- 
cide, the joint effect will be similar to that shown in 
Fig. 137. 

In order that successive curves may coincide, it is 
necessary that successive slits should reach a given 
point in the curve (as a, Fig. 138) at the same instant 
that the prong of the tuning-fork reaches that point. 




FiQ. 139. 



In other words, the interval of time between the 
arrivals of successive slits must correspond with the 
period of the tuning-fork. 

It will be found, if a toothed wheel is adjusted so 
as to show waves, as in Fig. 137, that when the speed 
is increased the waves will seem to follow the direc- 
tion in which the wheel is moving , while if the 
speed is lessened, the waves will move in the oppo- 
site direction. This is the result of a series of wave 
images (see Fig. 139), each of which is situated in a 
slightlt/ different place from the one preceding it. 
The direction in which the waves seem to move is a 
valuable guide in adjusting the speed of the wheel. 

20 



806 TIME OF VIBBATION. [Exp. 55. 

It is easy to trace out in a similar manner the ap- 
pearance of a vibrating fork for any speed of the wheel. 
Usually it will appear blurred, as if looked at in the 
ordinary manner. If, however, the wheel is moving 
twice as fast as it ought, a double wave will be visible, 
as in Fig. 140. If, again, the fork makes in one sec- 
ond a number of vibrations twice as great as the num- 
ber of slits which pass a given point, the appearance 
of the fork will be as in Fig. 141. Care must be taken 
not to mistake this curve for the double curve of Fig. 
140, nor for the regular curve of Fig. 137. We no- 
tice that in Fig. 141 there are two complete waves in 
the distance between two successive slits (a and 6). 




Ay^%(A 



Fig. 140. Fig. 141. 

In the same way this distance will be divided into n 
waves if the fork executes n vibrations between suc- 
cessive views from a given point. 

By this principle we may find the rate of a fork 
too high to be measured by the ordinary method. 

^ 146. Determination of Pitch by means of a Toothed 
Wheel. — The experiment consists simply in adjusting 
the speed of a toothed wheel (Fig. 185, ^ 144) so 
that a fork held behind the rim of a wheel (as in Fig. 
137,^145), and making about 64 vibrations per sec- 
ond, will be apparently thrown into simple stationary 
waves, the lengths of which will be equal to the dis- 
tance between the teeth of the wheel, then finding 



t 146.] THE TOOTHED WHEEL. 807 

how many teeth pass by a given point in one second. 
We have ah-eady considered (^ 144) the manner in 
which the speed of the wheel may be adjusted and 
how the number of revolutions may be counted.^' 
The number of revolutions made in one second mul- 
tiplied by the number of teeth gives the number of 
teeth per second. This is (see ^ 139) the " pitch " of 
the tuning-fork. 

^ If it is fouDd impossible to adjust the speed exactly, or to keep 
it adjusted, accurate results may still be obuined by counting the 
number of waves which in one second traverse the field of view. 
This number is to be added to the number of slits passing a given 
point in one second if the motion of the waves is opposite to that of 
the wheel ; if both move in tlie same direction the first number is 
to be subtracted from the second. 



808 DTNAMICS. 



DYNAMICS. 

^ 147. Different Methods of Measaring Velocity in 
Dynamics. — When a body is moving so slowly that 
it is possible to make a series of observations of its 
position at different points of time, no particular diffi- 
culty is met in the measurement of its velocity. 
Thus in Exp. 60, to find the average velocity of a 
ring rotating about its axis, we observe the distance 
traversed between two ticks of a clock, and divide it 
by the interval of time in question. Such slow mo- 
tions are, however, the exception in dynamics. In 
certain cases instantaneous photography has been 
employed for the study of rapid motions. The esti- 
mation of velocity generally requires, however, spe- 
cial devices, such as have been employed for the 
velocity of sound (Exp. 51). 

(1) In rough measurements, we frequently make 
use of the sounds produced by a moving body when 
it strikes different obstacles in its course. A familiar 
example of this method consists in the determination 
of the speed of a railway train by counting the num- 
ber of rails crossed in a given length of time. To 
find the velocity of a marble rolling in a groove, 
small tacks may be driven into the groove at such dis- 
tances that the successive sounds made by the marble 
in crossing them correspond, with the ticks of a clock. 
The regular increase of velocity caused by a steady 



IT 147.] 



3iEASUBEMENT OF VELOCITY. 



309 



incline is then easily demonstrated by measuring the 
distances between the tacks. 

(2) By substituting for a series of tacks a series 
of electrical connections which are made or broken by 
a moving body, we may make use of any of the devices 
by which time is measured by electrical agency.^ 

The velocity of a rifle bullet has been measured by 
the interval of time be- 
tween the rupture of 
two wires a known dis- 
tance apart. The time 
of rupture is usually re- 
corded " graphically ' 
by means of a chrono- 
graph (see ^ 266). 
Curves traced simul- 
taneously by the arma- 
ture of an electrical 
sounder and by a tuning- 
fork (see Exp. 52) en- 
able us to estimate pre- 
cisely exceedingly small 
intervals of time. 

(3) There are various 
devices in which the 
motion of a body may 
be directly recorded by the graphical method. Thus, 
in Morin's Apparatus (Fig. 142), a pencil (c) at- 
tached to a falling body marks directly upon a 
revolving cylinder covered with paper. If the rate 

1 See Trowbridge's New Physics, Exp. 71, 72, 73. 




Fig. 142. 



310 DYNAMICS. 

of revolution is known, we may obviously infer the 
position of the body at different points of time from 
the tracing (a6) made by the pencil. 

Another device in which the vibrations of a tuning- 
fork attached to a falling body may be made to indi- 
cate its position, will be found in Trowbridge's New 
Physics, Exp. 74. 

A simple instrument illustrating the graphical 
method of measuring velocity will be described in 
the next section. 

(4) In studying the motion of fluid streams, the ve- 
locity is frequently calculated from the size of a tube 
or orifice, and from the volume which flows through 
this tube or orifice in a given time. Thus if a stream 





Fio. 144. 



Fig. 143. 

of water issues from an orifice J sq. cm. in cross-sec- 
tion at the rate of 25 cu. cm. per sec,^ its velocity at 
the orifice must be 100 cm. per sec. This principle has 
been applied to illustrate the law of falling bodies. 
A stream of water projected horizontally with a 
known velocity must traverse a known horizontal 
distance (J5(7, Fig. 148) in a known time ; hence the 
time required for gravity to deflect the stream through 
a known vertical distance (^AB) is determined. 



IT 147.] MEASUREMENT OF VELOCITY. 811 

(5) The pressure of a stream of gas has been ap- 
plied to the determinatiou of the mass of the gas 
when its velocity is known, and conversely for a de- 
termination of its velocity when the mass is known. 
If, for instance, a mass of gas m, impinging with the 
velocity v, on a scale-pan (a, Fig. 144) causes a force, 
/, to be exerted for a time t, we have from the general 
formula (§ 106) 

V m 

(6) The laws of falling bodies are frequently made 
use of for indirect measurements of velocity. Thus 
since a body is known to fall 4.9 metres in 1 second, 
the velocity of a stream of water projected horizon- 
tally at a distance of 4.9 metres above a certain level 
will be equal numerically to the horizontal distance 
traversed before reaching that level, the time in ques- 
tion being 1 second. Again, the velocity of a pendu- 
lum when it passes its central point may be estimated 
by the distance it has fallen in reaching that point, or 
by the distance it r^ses after reaching that point (see 
§ 109). 

(7) The law of action and reaction enables us to 
make comparisons of velocity. Thus 
if a bullet of mass fn, striking a 
log of mass M^ suspended as in 
Fig. 145, gives it a velocity V (see 
§ 106), the velocity of the bullet (t?) 
may be found by the equation. 






Fig. 145. 



m 



312 



DYNAMICS. 



Changes in velocity may be measured by the same 
principle. If two billiard balls, A and B (Fig. 146), are 
suspended by cords of equal length so as to just 
touch each other without pressure, and if the greater, 
-4, is drawn aside to a position A (Fig. 147) and 
allowed to strike B while resting at JS', the latter will 
reach a position ^', while the former reaches A\ 
The velocity acquired by A in falling from A' to A 
will be proportional to the straight line A A (§ 109) ; 
the velocity after impact will be proportional to AA' 
and in the same direction as before ; hence the loss 




/ 






[A A* J X' 
Fig. 146. Fig. 147. 



Fig. 148. 



will be proportional to A' A — AA\ At the same 
time B gains a velocity represented by RB\ 

If on the other hand B strikes A from a position 
B (Fig. 148), it will rebound to J?'' in the opposite 
direction ; hence its change of velocity will be B'B 
+ B^'B. The corresponding gain of velocity by A 
will be represented by AA\ 

It is easy to show by experiment that the products 
of the masses, and their respective changes of velocity 
are equal, whether the balls are elastic or inelastic.^ 

^ See Ex. 20 of the Descriptiye List of Elementary Physical Ex- 
periments published by Harvard University. 



tl48.] 



FALLING BODIES. 



818 



A comparison of the changes of velocity in question 
gives a simple means of estimating the relative 
masses of the balls. 



EXPERIMENT LVI. 



FALUNO BODIES. 

^ 148. Deteniiination of Diatanoas travened by Fall- 
ing Bodies in Different Lengths of Time. — A wooden 
rod, jp (seen edgewise in Fig. 149), ^ 
about 25 cm. in length, 8 cm, in 
breadth, and 1 cm, in thickness, is 
suspended from the edge, /, of a 
bracket, ef^ by a strap of paper 
forked at A, so that the rod, when 
free, may hang in a vertical posi 
tion. An ounce bullet is next sus- 
pended by a thread from the peg, c, 
and lowered to a position, 9, near 
the bottom of the rod. The bracket 
is then moved (by loosening the 
screws d and g) so that the rod may 
barely touch the bullet. Then the ^'^- 1*»- 

bullet is removed, and either the rod is smoked at j 
and at jt>, or pieces of smoked paper are attached to 
it at these points. 

The bullet is now suspended at a point, i, near the 
top of the rod, by a thread passing over the smooth 
round pegs c, a, and /, to a screw-eye, w, near the 




314 DYNAMICS. [Exp. 56. 

middle of the rod. The rod is drawn one side by 
the pull on the thread, due to the weight of the 
bullet. Care must be taken to ease the thread round 
the pegs, so that the true position of equilibrium may 
be found. A pin m may then be placed so as to 
mark this position of equilibrium. 

To find the height of the bullet a finger is laid 
upon the thread at a, and the thread is slipped off 
the peg /, so that the rod may strike the bullet. A 
mark will thus be made on the smoked surface at j. 
The thread is now carefully replaced on the peg ?, so 
that the tension may be the same as before. When 
the finger is finally removed from a, there should be 
no slipping of the thread. If there is, the experi- 
ment must be repeated, until the bullet, having made 
a mark on the rod, remains unchanged in position. 

Any oscillation of the bullet must now be arrested 
by lightly pushing the thread, just below {?, in a direc- 
tion always opposite to that in which the bullet is 
swinging, or simply by allowing time enough for it to 
come to rest. The thread is then burned at h by 
holding a lighted match under it. The rod and the 
bullet will thus be released at very nearly the same 
instant. When the rod reaches its vertical position, 
jp^ it will strike the bullet at some point, 5, where the 
bullet will make a mark on the smoked surface. 

The distance between the two marks, one near y, 
the other near p^ is now to be measured. This dis- 
tance is equal to that through which the bullet falls 
while the rod is reaching its vertical position ; that is» 
in half the time it takes the rod to swing from one side 



T U8.] FALLING BODIES. 815 

to the other. To determine the time in question, we 
set the rod once more in oscillation and find how 
long it takes it to complete 100 or more swings.^ 

To obtain the best results, the oscillations should 
be timed as will be explained in the next experiment. 
The time of a single oscillation (either from left to 
right or from right to left) is then calculated and di- 
vided by 2, to find the time occupied by the rod in 
reaching its vertical position in the middle of one 
swing. This gives the time occupied by the bullet in 
falling through the observed distance. 

The experiment should be repeated with the same 
apparatus until results are obtained agreeing within 
2 or 8 per cent. The experiment should be then 
varied by using rods of different lengths. The re- 
sults should be entered as follows : in the first column, 
the distance through which the bullet falls; in a 
second column, the corresponding times of falling; in 
a third column, the squares of these times ; in a 
fourth column, the ratios of the distances to the 
squares of the times. Thus : — 

1. Distanoe Fallen. 2. Tim« Ooeupied. & Sqoaie of Time. 1 Ratio of 1 to 8. 
19.2 cm. 0.20 sec. 0040 480 

80.0 0.40 0.160 600 

etc. etc. etc. etc. 

It will be seen by the formula rf = J ^ (§ 108) that 
the ratio of the distance to the square of the time 
must be equal to J g, which is the distance a body 

* The student should notice thftt thouRh the swings jrrow shorter 
and shorter in lenprth. there is little or no perceptiMe chiin>re in the 
rate of oRdllation (see § 111). A more exact method of testing this 
point will be met incidenUllj in Exp. 58. 



816 



DYNAMICS. 



[Exp. 57. 



falls in one second. The numbers in the fourth 
column may be considered, therefore, as different esti- 
mates of this distance, founded on observations lasting 
through different intervals of time. These estimates 
should evidently show an approximate agreement ; 
but the results are modified somewhat by the fact 
that we are not experimenting with a body which is 
perfectly free to fall. A device, similar in many re- 
spects to that shown in Fig. 149, will be found de- 
scribed in Exp. 20 of the Descriptive List of Experi- 
ments in Physics, published July, 1888, by Harvard 
University. A device in which two electromagnets 
are used to set free a pendulum and a falling body 
^^ will be found in Trow- 

Kfi ^2 bridge's New Physics, Exp. 

EXPERIMENT LVII. 

LAW OF PENDULUM. 

^ 149. Determination of 
Times of OsciUation. — An 
ounce bullet (<?, Fig. 150) is 
to be suspended by a waxed 

^^ ^y^ silk thread, passing through 

a notch (J) in the edge of a 
bracket to and round a pin, 
a, by which the thread can be lengthened or short- 
ened. The lower surface of the bracket must be 
horizontal (see J, Fig. 151), and the groove must be 



6 

Fig. 150. 



Fig. 151. 



H 149.] 



THE PENDULUM. 



817 



deep enough to reach this surface. It is now required 
to find the length of the pendulum thus constructed ; 
that is, the distance 
from its point of sus- 
pension, in the surface, 
6, to the middle of the 
bullet, c. This is done 
by means of a wooden 
rod, bdy graduated in 
millimetres. The rod 
is held parallel to the 
thread (and hence ver- 
tical) with its zero at b. 
The height of the cen- 
tre of the bullet is 
found from that of the 
top and bottom by 
taking the mean. To 
avoid parallax (§ 25) 
these heights are 
sighted through a tele- 
scope (e), on the same 
level with them. We 
thus find the length 
of the pendulum in 
question. The time 
occupied by a hundred 
or more consecutive^ 
oscillations of the pen- ^'^- ^^^' 

1 The iniTwrtanoe of observiiiK lonj? series of consecutive observa- 
tions must not be overlooked. A student is apt to imagine that 10 




820 DYNAMICS. [Exp. 58. 

with that obtained for falling bodies in Exp. 56, we 
discover a curious relation. The length of a pendu- 
lum which makes one swing in one second is about 
99 cm. The distance a body falls in one second is 
about 490 cm. The latter is nearly 5 times as great 
as the former. Again, the length of a half-second 
pendulum is not quite 25 cm. . the distance a body 
falls in half a second is about 122 cm., that is, nearly 
5 times as great as the corresponding length of the 
pendulum. This proportion will be found to exist in 
every case. 

It is obvious that if this proportion is known,^ we 
may calculate the distance through which a body falls 
in a given time from the length of a pendulum making 
one swing in the same time. We shall make use of 
this principle in the next experiment. 



EXPERIMENT LVni. 

METHOD OP COINCIDENCES. 

^ 150. Adjustment of a Pendulum of Peculiar Con- 
■truction. — A serviceable device, which conforms ap- 
proximately to the conditions required of a simple 
( pendulum, is represented in Fig. 158 as seen from in 
front, and in Fig. 154, in profile. It consists of a 
cylinder {gj) suspended by two vertical loops of silk 

1 The law of falling bodies gives (§ 108) <f = J i7«*; the theory 
of the pendulum gives Csee Appendix) / = $; hence we have 
d.l:.ifii2 4.936 : 1, nearly. This ratio is not affected by the 
value of g, but is slightly affected by the resistance of the air. 



T 150.] 



METHOD OF COINCIDENCES 



821 



thread passing around the horizontal pins ab and hi. 
The diameter of these pins should be exactly the same, 
and not over 1 cm. Their length should be about 10 cm. 
The upper pin (aft) is driven through a fixed support ; 
the lower pin should pass as nearly as possible through 
the centre of gravity of the cylinder. The ends of the 
thread, after passing over the pin aJ, are carried each 
to one of the pins c, rf, e, and /, by turning which the 
threads may be lengthened or shortened. A disc is 

/ 




Fig. 153. 



Fig. 154. 



also attached to the cylinder, and in this disc are made 
two V shaped holes (g and /). Opposite the lower 
hole (y) may be placed an opening (A), in a shield, 
through which instantaneous views of objects be- 
'hind the pendulum may be obtained at regular in- 
tervals. A small wire loop may be attached to the 
pendulum so as to complete an electrical connection 
between two drops of mercury at I when the pendu- 

21 



322 DYNAMICS. lExp. 5a 

lum is at rest or in the middle of a swing. The length 
of the pendulum thus constructed is found by meas- 
uring the distance between these pins from centre 
to centre. In the absence of a cathetometer (^ 262) 
or other device by which the distance in question may 
be accurately measured, it is well to adjust it by turn- 
ing the pins c?, d, e, and / until a metre rod fits with- 
out looseness or pressure between the pins ab and hi, 
so as to subtend the vertical distances either between 
a and h or between b and i. The diameters of the 
pins at a, 6, h and i are now measured by a vernier 
gauge (Part I. ^ 50). The average diameter added 
to the length of the metre rod gives the distance 
between the pins from centre to centre. 

In regard to the working of this pendulum, it 
may be pointed out that the cords (aA and bi) keep 
the pins (oJ and hi) parallel, hence horizontal, and 
always the same distance apart. The centre of the 
pin hi swings, therefore, in a vertical plane about 
the middle point of ab as a centre. Now equal par- 
allel forces applied by the cords (aA and bi) on each 
side of the pins (a6 and hi) act in all cases like single 
forces applied at the centres of these pins (see Expe- 
riment 61, ^ 159, 1). If the centre of gi*avity of the 
cylinder and disc is in the axis of Ai, we have, as 
in the simple pendulum, a weight acting as if it were 
applied at a single point (in hi), and made by forces 
also applied at the same point (in hi) to oscillate 
about another point (in ab) as the centre. There is 
no rotation either of the cylinder or of the disc to 
complicate the result, as in the case of an ordinary 



1 151 1 METHOD OF COINCIDENCES. 823 

compound pendulum. Evidently no such rotation 
can exist, unless the cords (ah and b%) slip on the 
pins (ai and hi). There is, moreover, no tendency 
to produce such rotation; because forces acting at 
the centre of gravity of a body (in hi) can cause 
only a linear motion of that centre of gravity. A 
line in the disc or cylinder which is vertical in one 
position of the pendulum, remains accordingly ver- 
tical in all positions. Here lies an essential distinc- 
tion between this and other compound pendula.^ 

^ 151. Determinatioii of Times of Oscillation by the 
Method of Coincidences. — A pendulum between 100 
and 101 cm. in length, adjusted and measured as in 
^ 160, is placed, let us say, in front of the pendulum 
of a regulator (Fig. 152, ^ 149) and set in vibration 
in an arc not exceeding 10 cm, in length (that is, 
5 cm. on each side of the vertical — see Table 8, g). 
Each swing will occupy a little over a second ; hence 
the first pendulum will fall slowly behind the second. 
The two pendula will be moving now the same way, 
now opposite ways. The ticks of the regulator will 
occur when the first pendulum is now at its furthest 
right-hand or left-hand point, and now when it is at 
the middle point of its swing. Every such corres- 



1 The student may notice that the time of oscillation of the stick 
used in Exp. 66 is considerably greater than that of a simple pendulum 
(see Table, f 14ft) equal in length to the distance between the centre 
of gravity of the stick and its point of suspension. This is owing to 
the fact that gravity has not only to move the centre of the stick 
through a certain angle about its point of suspension, but also to turn 
the stick through the same angle. For a similar reason all ordinary 
compound pendula are somewhat retarded. 



324 DYNAMIOS.' [Exp. 58. 

pondence involves a ^^ coincidence " of some sort. The 
object of this experiment is to find the average inter- 
val of time between two coincidences of a given 
kind. The student will be surprised to find in the 
reduction of different results (^ 152) how large an 
error may be committed in the method of coincidences 
without introducing any considerable error into the 
result. 

I. Ocular Method. — When the pendula are ap- 
parently swinging the same way, the time is to be 
read by the clock in hours, minutes, and seconds ; and 
again the time is to be noted when the pendula seem 
to be moving in opposite ways. This should be con- 
tinued for half an hour or more, according to the 
length of time that the pendulum may continue to 
swing perceptibly. The two pendula will probably 
seem to coincide for a long time in each case. Every 
effort must be made to determine the middle of such 
periods of coincidence. 

II. Eye and Ear Method (§ 28). — The times 
may be noted when the ticks of the regulator are 
heard just as the pendulum under observation reaches 
its furthest point to the right or to the left ; or better, 
when it reaches the middle point of its swing. In the 
latter method, the time of coincidence may be gener- 
ally found within 10 seconds. It may be convenient 
in some cases to connect an electrical telegraph instru- 
ment with a break-circuit in the clock (Fig. 152, a) 
so that the ticks may be re-enforced or reproduced at 
a distance. 

III. Optical Method. — Instantaneous views of 



T 151.] METHOD OF COINCIDENCES. 325 

the pendulum of the regulator may be obtained 
through the opening, A;, in a fixed shield (Fig. 153), 
and an opening, j^ in the disk of the pendulum* The 
regulator should be illuminated so that these views 
may produce a sufficient impression upon the eye. 
The times are to be noted when the pendulum of the 
regulator is seen at the middle point of its swing. 
Times of coincidence may thus be determined within 
a few seconds. 

IV. Electrical Method. — An electrical current 
is sent first through the break-circuit of ttie clock 
(Fig. 152, ^ 149), then through the break-circuit 
Imno (Fig. 156) attached to the pendulum (see Pick- 




FiG. 155. h\G. 156. Fig. 157. 

ering, Physical Manipulation I. § 41). The ends of 
these wires should be amalgamated by dipping them 
first in nitric acid, then in mercury in order to make 
good electrical connections. The two hollows, n and 
(Fig. 157), must be filled with mercury and raised 
by thin wedges so that the mercury may touch the 
wires (InC) in the middle point of the swing (»i, Fig. 
155). 

When the swings of the two pendula come into a 
certain mutual relation, an electrical connection will 
be made by both break-circuits at the same time, and 
the sounder will respond. After a certain time this 
relation will cease, and the sounder will become 



326 DYNAMICS. [Exp. 58. 

silent. The beginning and end of each period of 
response should be noted, and the middle of the 
period found by calculation. This method, though 
more complicated in detail, requires much less effort 
than the optical method, and is in general equally 
accurate. 

The experiment is to be repeated with a hollow 
cylinder of sheet zinc, instead of the solid zinc cylin- 
der represented in ffj\ Fig. 153 ; then again repeated 
with this hollow cylinder filled with sand or lead 
shot. The weights of the empty cylinder and its 
contents should be noted. 

^ 152. Redaction of Results obtained by the Method 
of Coincidences. — The reduction of results obtained by 
the method of coincidences will be best explained by 
an example. The times of coincidence should be ar- 
ranged (see § 61) in three columns of about equal 
length. These columns should contain an odd num- 
ber of observations, and should be averaged, thus: — 





min. 0eo. 




min. 


BOO. 




min. MO. 


l8t 


13 41 


6th 


24 





11th 


34 32 


2d 


16 44 


7th 


26 


3 


12th 


36 34 


8d 


17 61 


8th 


28 


9 


13th 


38 39 


4tli 


19 56 


9th 


30 


15 


14th 


40 46 


6th 


21 68 


10th 


82 


23 


15th 


42 49 



Average 8d 17 50 8th 28 10 I3th 38 40 

The first average corresponds in the example to 
the time of the 3d observation ; the second average 
corresponds similarly to the 8th observation , and the 
last average corresponds to the 13th observation. 
For reasons stated in § 51, these averages are prob- 
ably more accurate th.in the single observations to 



t 152.] METHOD OF COINCIDENCES. 827 

which they correspond. The difference between the 
first and second averages is 620 seconds ; and since 
between the 3d and 8th observations, to which they 
correspond, there are 5 intervals, the average for each 
interval must be 124 seconds. It appears, therefore, 
that in 124 seconds the first pendulum loses just one 
swing with respect to the regulator ; that is, it makes 
123 swings while the regulator makes 124. Assum- 
ing that 124 swings of the regulator occupy as many 
seconds, one swing of the first pendulum must oc- 
cupy yJ^ of 124 seconds, or 1.0081 sec. In the same 
way, between the 8th and 13th observations, we 
find coincidences on the average 126 seconds apart ; 
hence the average time of one swing is -jj-^ of 126 
seconds, or 1.0080 sec. The student should note that 
the time occupied by one swing (1.0081 sec.) in the 
first part of the experiment differs very slightly from 
that (1.0080 sec.) in the last part of the experiment. 
The difference, due to a decrease in the arc of the 
pendulum, is in fact only about itj^tht ^^ * second 
(see Table 3, g). He should also notice that this 
small difference in the result corresponds to a com- 
paratively large difference (2 seconds) in the average 
interval between coincidences. Even with rough 
methods (^ 151, 1, and II.) such a difference could 
hardly fail to be observed when suflSciently multi- 
plied by a long series of observations. If, conversely, 
the average interval between coincidences can be 
found within 2 seconds, the time of oscillation must 
be accurate within -ji^inr ^' ^ second. 

A comparison of results obtained with a solid and 



328 DYNAMICS. [Exp. 5a 

with a hollow cylinder of a given size and shape 
should show that the resistance of the air (which 
must exert a relatively greater influence in one case 
than in the other) is slight. A comparison of re- 
sults obtained with a hollow pendulum filled with 
different materiala should show that the time of os- 
cillation of a pendulum of given length is indepen- 
dent of the nature of the substance of which it is 
composed. 

^153. Relation between the Iiength and Time of 
OsciUation of a Pendnlum and the Acceleration of Ghrav- 
ity. — We have already seen (^ 149) that a relation 
must exist between the length of a pendulum and 
the distance traversed by a falling body while the 
pendulum is making one swing. To find the distance 
which a body falls in 1.C081 sec. we have only to 
multiply the length of the pendulum, let us say 100.8 
cm. by a certain number (4.935) already determined. 
From the distance which a body falls, and from the 
time occupied, we may calculate the velocity imparted 
to the body (see § 108) ; and from the velocity im- 
parted in a given length of time, we can find that 
imparted in 1 second (§ 108). This is called the 
acceleration of gravity, and is denoted by g in the 
formulsB of § 108. To shorten this calculation, 
which depends solely on the length and time of oscil- 
lation of a pendulum, the following table has been 
computed for simple pendula between 99 and 101 
cm. in length: — 



T153.] 



ACCELERATION OF GRAVITY. 



829 



TIMB OF OSCILLATION. 



J3 
B 



99.0 


1.0000 


0.9996 


0.9990 


0.9986 


99.1 


1.0006 


1.0000 


0.9996 


9990 


99.2 


10011 


1.0005 


1.0000 


0.9996 


99.;^ 


lOOlb 


1.0010 


1.0006 


1.0000 


99.4 


10021 


1.0016 


1.0010 


1.0005 


99.6 


1.0026 


1.0021 


1.0016 


1.0010 


99.6 


1.0U31 


1.0026 


10020 


1.0016 


99.7 


1.00:^6 


1.0031 


1.0026 


1.0020 


99.8 


1.0041 


1.0036 


1.0031 


1.0025 


99.9 


1.0046 


1.0041 


1.0036 


1.0030 


100.0 


1.0051 


1.0046 


1.0041 


1.0035 


100.1 


1.0056 


1.0061 


1.0046 


1.0040 


100 2 


1.0061 


1.0066 


1.0061 


1.0045 


100.3 


10066 


1.0061 


1.0066 


1.0050 


100.4 


1.0071 


1.0066 


1.0061 


1.0056 


100.6 


10076 


1.0071 


1.0066 


1.0061 


100.6 


10081 


1.0076 


1.0071 


l.OOJiO 


100.7 


1.0086 


1.0081 


1.0076 


1.0071 


100.8 


1.0091 


1.0086 


1.0081 


1.0076 


100.9 


1.0096 


1.0091 


1.0086 


1.0081 


101.0 


1.0101 


1.0096 


1.0091 


1.0086 



0.9980 0. 

09985 0. 

0.9990 0, 

9995 0. 

1.0000 0. 

1.0005 1 
1.0010 1 1 



.9975 0.9970 
.9980 0.9976 
.9985 0.9980 
.9990 0.{«85 
.9996 0.^^990 



1.0015 'l 

1.0020 ;i 

1.0026 !l 

i.oasoii 

1.0086 1 1 



10040 
1.0046 
1.0050 
1.0055 
1.0060 
1.0065 
1.0070 
1.0075 



1.0080 1 



.0000 
.0005 
0010 
.0015 
.0020 
.0026 
0030 
0035 
0040 
.0045 
0050 
.0(155 
0060 
.0065 
.0070 
0076 



0.9995 
1.0000 
10006 
1.0010 
1.0016 
1.0020 
1.0026 
1.0030 
1.0036 
1.0040 
1.0045 
10060 
1.0056 
1.0060 
1.0065 
1.0070 



0.9965 
0.9970 
0.9975 
0.9980 
09985 
0.9990 
0.9995 
1.0000 
1.0005 
1.0010 
1.0015 
10020 
10025 
10030 
1.0036 
1.0040 
1.0045 
1.0050 
10065 
1.0060 
1.0065 



g = 077 978 979 980 



981 



982 988 984 



The length of the pendulum is to be found in the 
left-hand column ; then in line with it the number 
nearest the time of oscillation is to be selected. Be- 
neath this number, at the bottom of the column will 
be found the value of g. 

Example I. Given the length, 100.8 cm, and the 
time, 100.81 sec, required g. We find the time of 
oscillation, 1.0081, in the 4th column in line with 
100.8 in the left-hand column and at the bottom of 
the 4th column we find the number 979, which rep- 
resents the acceleration of gravity in question. 

Example II. Given the length, 100.84, and the 
time, 100.81, required g. We notice that the times in- 
crease by the amount .0005 when the length increases 
by 0.1 cm. ; hence 0.04 cm. corresponds to .0002 sec. 



330 DYNAMICS. [Exp. 69. 

If, therefore, the length had been 100.8 instead of 
1.0084 the time would have been 1.0079 instead of 
1.0081. Now 1.0079 comes between two numbers 
opposite 1.008, namely 1.0081 and 1.0076. Under 
the first we find 979, under the second we find 980. 
Since 1.0079 differs from 1.0081 by .0002 sec, and 
a di£ference of .0005 sec. makes a difference of 1 unit 
in g, we must add .0002 -s- .0005 or f of a unit to 979 
to find the value of g. We have, therefore, ^ = 
979.4. 

The object of this calculation is not so much to de- 
termine the value of g^ which is already known with 
suflBcient accuracy for all latitudes (see Table 47), 
and is believed to be the same for all materials, but 
rather to obtain a check upon the standards and 
methods hitherto employed for the measurement of 
length and time. 



EXPERIMENT LIX. 

• IKERTIA, I. 

^ 154. DeterminationB of Mass by the Method of 
Oscmations. — A small glass beaker (d, Fig. 158) is 
to be suspended from a support, a, by a coiled spring 
of steel wire, Jc, as long and as flexible as may be 
convenient. A substance whose mass is to be de- 
termined is placed in the beaker. The beaker is 
then pulled downward to a position cf, vertically 
beneath cZ, then released. It will spring up to a 



T i54.] MEASUHEMENT OF MASa 881 

position d'\ nearly as far above i as (f is below it. 
Then it will return nearly to d\ and thus make a 
considerable number of oscillations before it comes 
to rest. The oscillations should not displace 
the load in the beaker ; if they do, the load 
must be rearranged, or the oscillations must 
be diminished in amplitude. The time of 
oscillation is now to be found as in \ 149. 

The load is next removed from the beaker, 
and in its stead weights from a set are 
placed there, sufficient in quantity to stretch 
the balance to the same point as before. The 
time of oscillation is again determined. If 
it is less than before, more weights are added, 
if greater, weights are removed ; and thus by 
trial (§ 35) the weight is adjusted untU the ^'^- ^*®- 
time of oscillation is the same with the weights as 
with the substance, the mass of which is to be 
determined. 

The student should notice that the time of oscilla- 
tion is nearly independent of the amplitude of oscilla- 
tion as in an ordinary gravity pendulum. It should 
be pointed out, however, that in the vertical oscilla- 
tion shown in Fig. 168, gravity has nothing to do 
with the time of oscillation in question, except in so 
far as it may affect the elasticity of the spring by 
stretching it to a greater or less extent. When a 
spring is already loaded the force required to stretch 
it 1 cm. further may be taken as a measure of the 
stiffness of the spring under the load in question. 

The time of oscillation of a load suspended by a 



332 DYNAMICS. pixp. 69. 

spring depends (1st) on the stiffness of the spring and 
(2d) on the mass to be set in oscillation. When two 
loads give the same time of oscillation under the same 
circumstances, their masses are necessarily equal. 

Having adopted as our standard of mass a certain 
piece of platinum in the French Archives (§ 6), we 
should theoretically use. platinum weights in this 
experiment. It has been found, however, that two 
quantities which have equal masses, estimated as 
above by the dynamical method, have also equal 
weights (in vacuo) ; that is, gravity exerts the same 
acceleration upon them, without regard to the sub- 
stances of which they are composed (see Exp. 58.) 
The use of brass weights will not, therefore, in prac- 
tice^ introduce any error. 

The results of Exp. 59 are to be expressed in 
grams like results obtained by an ordinary balance. 
Strictly, however, the word maBS should be written 
before or after these results instead of the word 
weight (§§ 152, 153). 

^ 155. Relation between "Weight and Mass. — The 
student must not assume that weight and mass are 
necessarilt/ the same. We do not know whr/ a body 
is attracted by the earth, neither do we know why, 
being attracted, it does not move instantly, under 
that attraction, from one place to another. The 
former phenomenon we attribute to gravity (§ 150), 
the latter to inertia (§ 151). 

By the weight in grams of a body we mean the 
number of grams of platinum to which the body is 
equal in respect to weight proper (§ 153), or the 



IT 155] MEASUREMENT OF MASS. 833 

force exerted upon it by gravity. By the mass in 
grams of a body we mean the number of grams of 
platinum to which it is equal in respect to inertia, or 
the necessity of force to set it in motion (§ 152).^ In 
the absence of any explanation of gravity and inertia, 
no reason can be assigned why any proportion should 
exist between them. There is no proportion be- 
tween electrical or magnetic forces and the masses 
upon which they act. The existence of such a pro- 
portion between mass and weight is simply an infer- 
ence from the results of experiment (see Exp. 58), 
It is possible, so far as we know, that a new sub- 
stance may be discovered, the mass of which may 
be disproportional to its weight. It is also possible 
that if masses could be measured with the same accu- 
racy as weights, slight variations might be discovered 
which have hitherto escaped observation. We have 
several instances of physical laws which are approx- 
imately but not exactly fulfilled ^ as for instance the 
law connecting the molecular weights and specific 
heats of elementary substances (§ 86, note). At 
the same time that such variations are possible, as 
far as we know, in the case of gravity and inertia, 
it is by no means probable that any such will ever 
be discovered. It is much more probable that grav- 
ity and inertia are both manifestations of a single 
principle, according to which, for reasons unknown 
to us, one must be proportional to the other. 

1 See Hall's Elementary Ideas, published by C. W. Sever, Cam- 
bridge, Mass. 



884 



DYNAMICS. 



[Exp. 60- 



EXPERIMENT LX. 




INERTIA, II. 

^ 156. Determination of Force by Observations of 
Mass, Length, and Time. — A metallic ring about 20 
cm. in diameter, and weighing about 500 gi-ams 
(C D F E, Fig. 159) is suspended horizontally by a 
spring brass wire AB^ about 0.26 
mm. in diameter (No. 81, B.W.G.), 
and at least one metre long. The 
wire is fastened at the top and held 
at the bottom by a small vice, B. 
This vice, 5, is connected by fine 
iron wires (about No. 31) with four 
points C, D, E, and F of the ring. 
A paper millimetre scale is attached 
to the ring, and the distance through 
which it revolves is indicated by a 
fixed marker ( 0^). 

The reading of the marker is to 
be first observed when the ring is 
at rest. Then the ring is turned 
through nearly 860°, and released. 
All pendular vibration must be stopped by touching 
(if necessary) the wire AB. The ring will then 
have only a rotary movement, due to the " torsion" 
of the wire. As the ring approaches a turning-point, 
several readings of the marker are taken at inter- 
vals of two seconds. The intervals may be deter- 




FiG. 159. 



1 157.] MEASUREMENT OF FORCE. 835 

mined by the ticks of a regulator, or by an electrical 
sounder connected with the regulator.^ 

When the experiment has been repeated a suffi- 
cient number of times, the ring is taken down and 
its weight in grams determined. The vice, £, should 
not be weighed with the ring. It is better not to 
weigh the connecting wiies with the ring ; but their 
weight (which should not exceed 1 gram) will not 
in any case introduce a serious error into the result. 
The material, length, and diameter of the wire AB 
should be noted. The observations are then to be 
reduced as in ^ 157. 

^ 157. Calcnlation of Force from ObservationB of 
MasBj Length, and Time. — The rotation of a ring 
about its axis presents one of the simplest cases 
in dynamics. The whole mass of the ring is at 
(nearly) the same distance from the axis in ques- 
tion, and hence acquires (nearly) the same velocity. 
To find the force exerted upon the ring in the direc- 
tion of this velocity, we have to find (1) the velocity 
acquired, (2) the time required to attain this velocity, 
and (3) the mass acted upon. The force may then 
be calculated by the general formula (§ 106) : — 



jf mv 

^~ t 



1 If greater precision is required than can be obtained by the eye, 
a small bristle attached to the armature of tlie sounder can be made 
to mark the seconds on the edge of the ring, which must be previously 
smoked for this purpose. By employing two such markers on oppo- 
site sides of the ring, slight errors due to swinging of the ring can be 
eliminated. 



\ 



386 DYNAMICS. [Exp. 60. 

In practice we make this calculation as in the ex- 
ample below. The observations are numbered and 
arranged as follows: — 

mm. Biflerence in 2 sec. Mean Velocity. Difference in 2 sec. Acceleiation 



1 


552 


+33 


+16.5 


2 
3 
4 


685 
600 
695 


+15 
-5 

-20 


+ 7.6 
- 2.6 
-10.0 


6 

6 


675 
635 


-40 


-20.0 



8.0 


4.0 


10.0 


5.0 


7.6 


8.8 


10.0 


5.0 



The differences in the 8d column show the dis- 
tance passed over in 2 seconds ; hence these are 
divided by 2 to find the distance passed over in 1 
second, or the mean velocity for a period of 2 sec- 
onds. The velocity is called positive if the ring is 
turning away from its position of equilibrium, other- 
wise negative. The 5th column shows the algebraic 
differences in these velocities ; that is, the change of 
velocity in 2 seconds. To find the acceleration, or 
change of velocity in one second, the numbers in the 
5th column must be divided by 2. This gives the 
numbers in the 6th column, the average of which is 
4.5, nearly. Since we have used mm. throughout, the 
change of velocity in one second amounts to 4.5 mm. 
per ««(?., or 0.45 cm. per sec. 

This is the acceleration strictly of the outer surface 
of the ring. Let us suppose that the outside diameter 
is 20.5 cm, and the inside 19.5 cm.^ so that the mean 
diameter is 20.0 cm, ; then the average acceleration 
will be less than 0.45 in the ratio of 20.0 to 20.5. 
The average acceleration will be, therefore, about 0.44 
cm. per sec. If now a mass of 500 ff. receives this 



T158.] SPRING BALANCES. 837 

acceleration, the force exerted upon it must be 500 x 
.44, or 220 dynes (§ 12). The angle through which 
the steel wire is twisted is given in circular measure 
by the ratio of the arc to the radius. Since the latter 
is 10 cm. (nearly), the minimum deflection (58.5 C9it.) 
corresponds to 5.85 units of angle. The maximum 
deflection (60.0 cm.^ corresponds similarly to 6.00 
units of angle. The mean deflection is accordingly 
not far from 5.7 units of angle. Since one unit of 
angle in circular measure is equal to 57°.8, nearly, 
the mean deflection of the ring is about 57^.8 X 5.7, 
or 827°. 

We note, therefore, that a piece of steel wire of 
given length and diameter, when twisted 827° , ex- 
erts at a distance of 10 cm. from its axis a force of 
about 220 dynes. 

The use which is to be made of this result will be 
explained in ^ 165 in connection with a method by 
which a force similar to the one in question may 
be directly balanced by gravitation. A more ac- 
curate method of reducing results obtained by the 
** torsion pendulum " will be given in the Appendix 
(Part IV). 



EXPERIMENT LXI. 

COMPOSITION OP FORCES. 

^ 158. Correctloii of Spring Balances. — A spring 
balance consists of a spiral spring, cd (Fig. 160), con- 
tained in a hollow metallic case, hhy to which it is 

ss 



338 



DYNAMICS. 



Sxp. 61. 



fastened at c. The spring is connected by a rod, rfi, 
with a hook, ij^ from which weights are hung. A slit, 
eg^ is made in the case so that a pointer,/, attached to 
the rod, rfi, may indicate the elongation of the spring 
on a scale outside of the case. In measuring vertical 
forces with a spring balance, the instrument is gener- 
ally suspended by the ring, a. When 
forces in other directions are to be 
determined, the case (JbK) should also 
be supported, so as not to bear against 
the index, /. If this precaution is not 
observed, large errors from friction 
may be introduced into the results. 
Spring balances are usually graduated 
so as to indicate the weight of a body 
either in kilograms or in pounds. It 
must be remembered that such indi- 
cations are affected by the force of 
gravity. Thus a spring balance, grad- 
uated correctly in England, would give, 
in Brazil, readings too low by about \ . 
of 1 %. Obviously spring balances, however sensitive, 
cannot serve everywhere as standards of maBB (§ 6). 
The readings depend, not directly upon the masses 
suspended, but upon the forces which they exert on 
the instrument. A spring balance once graduated 
correctly in megadynei * should, however, give forces 
correctly (in megadynes) irrespective of locality. A 

1 The student may be interested to cut a scale of megadynes by the 
side of the ordinary scale. In latitude 40^-460, 1 megadyne = 1.02 
kilos. = 2\ lbs. nearly. 




Fio. 160. 



T 158.] SPRING BALAKCEa 889 

Rpring balance is essentially an instrument for meas- 
uring force, and it is only in a given latittide that it 
may be employed for estimating weights either in 
kilograms or in pounds. A pair of 10-kilo. (or 24-lb.) 
spring balances will be suitable for the experiments 
which follow. 

The reading of a spring balance may be corrected 
by hanging known weights upon it, as in Fig. 160. 
Weights provided with a ring, a hook, or an eye will 
be found convenient for this purpose. The reading 
of the balance should be tested with weights of 1, 2, 
8, etc., up to 10 kilos, (or 2, 4, 6, up to 24 lbs.). The 
zero-reading of the spring balance should also be 
found, both in a vertical and in a horizontal position. 
The weights used may be compared by an ordinary 
balance with standards if it is thought necessary. 
From these results we are to calculate the correc- 
tions to be added to the reading of the spring balance 
under different loads, in order to find the true load. 
Thus if the indication with a 4 lb. weight is 8 lbs. 
14 oz., the correction is +2 oz. The results should 
be arranged in tabular form, either in kilos, or in 
pounds, as follows : — 

FIRST TABLE OF CORRECTIONS. 



(1) Load in kilM. 


Correction in kilo«. 


(2) Load in lbs. 







-0.10 





-3 


1 


-0.06 


2 


-1 


2 


■fO.08 


4 


+2 


8 


+0.25 


6 


+6 



10 +0.06 



24 +1 



One of the weights is now to be attached to the 
spring balance by a light but strong cord (ac^ Fig. 



840 



DYNAMICS 



[Exp. 61. 



161) passing over a pulley (6) made to run as freely 
as possible. The readings of the balance are to be 
carefully compared in different positions (a', a", etc.). 
To eliminate the effects of the friction of the pulley, 
the readings are to be made in each 
case (1) when the weight is being 
slowly raised, and (2) when it is 
being slowly lowered. If the two 
readings differ perceptibly, the mean 
is to be taken. 

The object of testing a spring bal- 
ance in different positions is to elim- 
inate the effects due to the weight of 
the hook and spring.^ From the results we are to 
calculate the corrections to be added to the readings 
under different inclinations in order to find the read- 
ing in the vertical position. Thus if a 2 lb. weight 
weighs apparently 2 lbs. 1 oz. in the vertical posi- 
tion, and 1 lb. 11 oz. in the horizontal position, the 
correction for an inclination of 90° is +6 oz. These 
corrections should be the same for all weights, and 
should be entered in a second table, as follows : — 




Fig. 161. 



SECOND TABLE OF CORRECTIONS. 



(1) Inclination of 
Balance. 

90O 
1200 
160O 
ISQo 



Corrpction 


(2) Inclination of 


in kilos. 


Balance. 


+002 


30O 


08 


60O 


0.16 


90° 


024 


120O 


0.80 


160O 


0.32 


180O 



Correction 
in OS. 

1 

8 

6 

9 
11 
12 



^ Tills method was suggested to the author by a similar one em- 
ployed by Mr. Forbes of the Roxbury Latin School. See also Ele- 
mentary Physical Experiments, published by Harvard University, 
page 11, footnote. 



f 



T 159.] COMPOSITION OF FOBCEa 841 

^ 159. Determinatloiui of Weight by the Composi* 
tion of Foroee. — It is frequently inconvenient to 
measure the weight of a body directly, either by 
ordinary scales, or by a single spring balance, as 
when the weight of the body exceeds the capacity 
of such instruments, or when the body forms an 
inseparable part of a combination. In such cases, 
we may sometimes make use 
of principles involved in the 
composition and resolution of 
forces. 

(1) To find the force of ^ 
gravity on a " 28-lb." weight 
with two spring balances, each 
of 10 kilograms' capacity, we Fio^Tel 
hang the weight (e, Fig. 162) 

at the middle of a stick Qcd^ so that it may bear about 
equally upon the spring balances (a and 6) while 
hanging in a vertical position. The reading of each 
balance is to be noted ; then the weight is to be re- 
moved, and the readings again taken with the stick 
alone. The diflference between the two readings of 
a given balance, with and without the weight, cor- 
rected if necessary by Table I., ^ 158, gives the part 
of the load borne by that balance. The sum of the 
two parts is of course equal to the whole load. 

(2) To find the force of gravity on a "56-lb." 
weight with a single spring balance of 10 kilograms' 
capacity, we suspend a lever (cd^ Fig. 163) as before, 
except that a cord, ftd, takes the place of the spring 
balance (6, Fig. 162). The weight is then hung at a 



m 



ri 



342 . DYNAMICS. [Exp. 61. 

point, e^ let us say one-fourth the distance from d to 
<?, and the reading of the spring balance is observed. 
Care must be taken that the cords fy and Ai, by which 

the weight is suspended, 
swing free of the side of 
the lever as in the cross- 
section (Fig. 163). A 
similar precaution should 
be observed in respect to 
the cords by which the 
Fig. 163. spring balance, a, is at- 

tached to the lever at c. 
The cords should both be vertical. The horizontal 
distances cd and ed are to be accurately measured. 
The weight is now to be removed, and the reading 
of the spring balance again noted. If F and / are 
the forces indicated by the spring balance with and 
without the weight, both being corrected by the 
first table of ^ 158, the force (w^ exerted by the 
weight at c is evidently equal to F — /. If we call 
the whole weight JF, then since the couple (§ 113) 
produced by W (equal to Wxde) is balanced by 
the couple produced by the spring balance (equal 
to w X cd), allowing for the weight of the lever, it 
follows that — 

TF=(i^— /) X cd^ed. 

(3) Another method of suspension is represented 
in Fig. 164. It is assumed that the weight will be 
able to lift the lever, so that the balance must be 
applied from under the lever. The reading of the 



i 



I 



T 159 J COMPOSITION OF FORCES. 848 

balance in this position must be coiTeeted both by 
the first and by the second table of % 158. Thus 
since the inclination of the balance is 180^ (compare 
Figs. 164 and 161), we must add 0.32 kilos accord- 
ing to the second table (^ 158), besides the ordi- 
nary coiTCction for the ob- 
served reading from the first 
table (^ 158). In addition 
to the force exerted by the 
spring balance, we have that 
part of the weight of the 
Fig. 164. lever which is felt at a, 

helping to balance the 56-lb. 
weight. To allow for the weight of the lever, we 
remove the 56-lb. weight, and apply the spring bal- 
ance as in Fig. 163, so as to 
sustain the lever at a. The 
reading of the balance in this 
position needs to be corrected 
simply by the first table «_ 
(^ 185), and gives the force | 
(/) exerted by the lever at a. 6 
This is to be cMlded accord- ■ 
ingly to the force (JP) exerted ^'^- *^5- 

by the spring balance with the weight (e) to find the 
total force which balances this weight. Calling this 
force w^ and the load TF, we have w x ab = Wxhc^ 
or — 

W= (F+f^ X ab^bc. 

(4) To test a 4-lb. weight with a 10-kilogram spring 
balance, we fasten one end of a lever ((?, Fig. 165) to 



344 DYNAMICS. [Exp. 61. 

the ground by means of a vertical cord, tre, and suspend 
the lever fiom a spring balance by a cord J, not far 
from c. The force,/, indicated by the balance is to be 
observed. The weight, rf, is then hung from the free 
end of the lever, and the force (F) indicated is again 
observed. Allowing as before for the weight of the 
lever we find the force (-F — f=tv^ exerted by the 
spring which balances the load W at d. Then since 
WXac=wXbc,we have W= (F—f^ Xhc-i- ao. 

If the distance be is one fourth of ac^ every ounce at 
a will produce an effect at h equal to 4 oz. We might 
therefore weigh a small object to ounces with a bal- 
ance graduated only to 4 oz. (or \ lb.). 

(5) Another method of weighing small objects is 
to hang two spring balances, A and B (Fig. 166), from 




Fia. 166. 

nails in the wall, 2 or 3 metres apart, then to connect 
them by a cord acb. At the middle of the cord ({?) 
a ring (C) is hung so that the weight, TF, may be read- 
ily attached. Two pins are driven into the wall op- 
posite points a and 5, on the cords at equal distances 
(let us say just 1 metre) from c. A cord, aJ, is 
stretched between them by means of two small 
weights, / and g. The perpendicular distance, cd, 
between c and ab is then measured. 



IT 159.J COMPOSITION OF FORCEa 845 

The vertical component of the force A registered 
by the spring balance near a, is by the triangle of 
forces (§ 105) equal to Ax cd-^ac. The vertical 
component of the force, B^ due to the spring balance 
near J, is similarly B X ed-r-hd. The total sum of 
these components must balance the combined weight 
of the ring ((7) and of the load ( W). That is, 

W+ 0=Ax cd-i-ac + Bxcd-^bc. 
To eliminate the weight of the ring, the load (TT) 




is removed, and the experiment is repeated with the 
ring alone, as in Fig. 167. We have, similarly, 

C= Ax cd-i-ac+BXcd-^bc. 

Hence subtracting the last value ((7) from the first 
(W+ C) we find the weight of the load (TT) in 
question. 

We will assume, for simplicity, that a and b are on 
the same level. A slight difference in level will, 
however, have no appreciable effect upon the result. 
The sagging of the cord ab will probably be very 
small, and will be eliminated in the method of differ- 
ence by which the result is calculated. 

The same method may be employed for the meas- 
urement of large weights. If the angle acb is small 
(see Fig. 168), it will be more accurate to calculate 
cd from a measurement of ai, than to measure ed di- 



846 



DYNAMICa 



[Exp. 61. 



rectly. Let us suppose that the cords bB and a A 
have been lengthened or shortened so that the line ab 
is horizontal. The vertical line cd will then be at 
right angles with ab; and since ac^^bc^ ad = bd=^ 
^ab. Knowing aci, we may calculate cd by the Pytho- 
gorean proposition — 

cd = V iac)^—{ad)^, 

and hence find the load (7 or IF as before. 

This method would be adopted in practice if for 
any reason it were inconvenient to obtain a point of 




i 



.not. 



/ \ 

Li .1 

Fig. 169. 



suspension directly above the weight. We should 
prefer, however, to employ a lever long enough to 
reach, as in (1) or (2), between two available points of 
suspension, A and 5, if it were possible to obtain 
one of suitable weight and strength. 

(6) To measure a weight (C, Fig. 169) when sus- 
pended by a cord (a<?) we may pull it one side by a 
spring balance applied horizontally in the direction 
cb. The reading of the balance (corrected by both 
tables of ^ 158) gives the force B acting in the di- 



1 159.] COMPOSITION OF FORCES. 847 

rectioD eh. This with tbe force of the cord acting in 
the direction ha produces a resultant which balances 
the weight of the body (7. The direction in which 
the weight C acts must be parallel to that of the cord 
ac before the weight was disturbed. Since three 
forces in equilibrium are proportional (§ 105) to the 
sides of a triangle to which they are respectively par- 
allel, we have B : (7= he : ac^ or 

C=:^Bxhe-i-ac, 

Instead of measuring he directly, we may pull the 
cord ac first one side to a point i, then in the oppo- 
site direction to a point V at a (nearly) equal distance 
from c. These points may be marked by pins, h 
and V driven into the wall or into some other sup- 
port behind the cord. The distance between h and V 
is then measured and divided by 2 to find the dis- 
tance he. The point e may be found by a thread 
stretched between the pins h and V. In this case 
the distance ae may be directly measured. Or the 
distance ah may be found and ae calculated (since ah 
is known) by the Pythagorean proposition, 



By the use of very small deflections, we may meas- 
ure weights many times exceeding the capacity of the 
spring balances which we employ. 



348 



DYNAMICS. 



[Exp. 62. 



EXPERIMENT LXH, 



CENTRE OP GRAVITY. 



^ 160. Location of the Centre of Gravity. — A flat 
board,^ bcde (Fig. 170), is suspended by a thread 
abb'a' (Fig. 170, 1) passing through a fine hole W in 
the board, and over a peg aa'. A plumb line, a/, is 
also suspended from the same side of this peg, so as to 
hang as close to the board as possible. A projection 
of this line upon the board is to be traced in pencil 
(Fig. 170, 2). The eye must be held in this process 




so as to look perpendicularly upon the board (§ 25). 
The board is then to be hung by another point, d (Fig. 
170, 8), and another line drawn upon it. Then the 
board is to be suspended from a third point, c (Fig. 
170, 4), and a third line traced. All three lines 

^ To lend interest to this experiment the board may be made of 
two thicknesses glued together, with a space («, Fig. 170) between 
them which has been hollowed out and filled with lead. An irregu- 
larly shaped board may also be employed. 



1 161.] CENTRE OF GRAVITY. 849 

should intersect at a point in the surface of the board 
directly in front of the centre of gmvity. If they 
do not, the experiment must be repeated. 

^ 161. Determination of Weight by Displaoement of 
the Centre of Gravity. — A weight (Wy Fig. 171) is at- 
tached at a to one end of a board whose centre of 
gravity (c) has been located (^ 160) ; and the board 
is balanced upon a triangular piece of wood (d) or 
upon a pencil. The line of the support (W Fig. 172) 
is then marked upon the board, and two lines, ab and 
cV are drawn from a and c perpendicular to bb'. 
These lines are then carefully measured. If W is 



t 



*i—i-m 



^^ 



.■•.<••• 



Fig. 171 Fig. 172, 

the weight of the board, which we may consider as if 
concentrated at <? (§ 112), we have WxVc = wXab; 
whence W= w X (aJ) -5- (J'O- 

The experiment should be repeated with different 
weights applied at different parts of the board, and 
with the line bV not always at the same place or in 
the same direction. The different values calculated 
for the weight of the board should be averaged. 
From their agreement we may infer the truth of the 
assumption that the weight of a body acts in all 
cases as if applied at its centre of gravity. 

It is obvious that if IT and w are both known, we 
may calculate the distance (b'c) by the formula 
(6'(?) = wX (ab) -5- W. 



350 



DYNAMICS. 



[Exp. 63. 



To find the distance of the centre of gravity from 
an axis (bb') on whieh a body balances, it is only neces- 
sary to know the weight of the body ( PT), the load 
(w), and its distance (aJ) from this axis. For an ex- 
periment (due to Prof. Hall) in which this principle 
is applied, see Ex. 17 of the Elementary Physical 
Experiments, published by Harvard University. 



EXPERIMENT LXIII. 

BENDING BEAMS. 

^ 162. Determination of the StifihesB of a Beam. — 
A square steel rod, ag (Fig. 173), is mounted on two 
triangular supports with steel edges, i and j\ 1 metre 
J,. 




Fio. 173. 

apart. A screw with a micrometer head ((Z) is ad- 
justed so that its point just touches the middle of 
the beam when a pan, ?n, is suspended from it by the 
wires hk. The micrometer is then read. A load, Z, 
is next placed in the pan, and the micrometer is once 
more adjusted until it touches the beam. The mi- 
crometer is again read. Its point is then withdrawn, 



1 162.] BENDING BEAMS. 851 

SO as not to be injured by the recoil of the beam 
when the weight is removed. A new reading is 
then taken with the pan (m) empty. If this differs 
greatly from the first, the beam has probably been 
permanently bent, and the experiment must be re- 
peated with a smaller load. If the reading is the 
same as before, a larger load may be tried With a 
steel beam 100 cm. long and not over 1 cm. thick, a 
deflection of several centimetres should be possible 
without injury to its power of recoveiy. To dis- 
cover exactly when the point of the micrometer 
touches the beam, we may make use of an electrical 
contact. One pole of a voltaic cell, i, is to be con- 
nected with one end of the beam by a wire soldered to 
it at a. The other pole is connected with one bind- 
ing post of an electrical sounder c. The other bindr 
ing post of this sounder is connected by a wire with 
the metallic nut «, in which the micrometer turns. 
The point of the micrometer and the surface of the 
beam beneath it are scraped bright with a file (or 
better, coated with platinum). When the point of 
the micrometer touches the beam, the electrical cir- 
cuit bceab is thus completed, and the armature of the 
sounder is attracted. A motion of one thousandth 
of a millimetre is sufficient, under favorable circum- 
stances, to make or break the contact. 

Care must be taken to prevent the beam from 
twisting or rocking under the influence of a load. 
The load should not bear more heavily on one side 
of the beam than on the other. Both sides should 
be supported alike at each end of the beam by the 



352 DYNAMICS. [Exp. 63. 

sharp edges i and J. Various deflections under dif- 
ferent loads are now to be determined. Each deflec- 
tion requires two readings of the micrometer, one 
with, the other without the load. The distance be- 
tween the supports i and J should be measured with 
a metre rod, and the breadth and thickness of the 
beams employed should be determined at different 
points with a micrometer gauge (^ 50, II.). 

(1) The deflection of a beam, let us say 1 cm. 
square, is first to be determined with the supports 
(ji and /, Fig. 173) exactly 100 cm. apart, and with 
a load causing the greatest deflection which can be 
employed without permanently bending the beam, 
or exceeding the reach of the micrometer. 

(2) The deflection due to one half this load is 
next to be found. The student should notice that 
this deflection is almost exactly half as great as be- 
fore (see § 115). If it is not, the measurements in 
(1) and (2) should be repeated. The same should 
be done if the zero-reading of the micrometer is 
changed. 

(3) To test the stiffness of the middle portion of 
the beam, the supports i and J are to be placed 50 
cm. apart, — that is, with half the original distance 
between them. The rod is to be mounted upon 
them as before, but with 25 cm. or more at either 
end projecting beyond the supports. The beam is 
to be loaded with 4 times the weight used in (1) 
or 8 times that used in (2). If the beam is equally 
stiff in all parts, the deflection should now be the 
Bame as in (2). (See § 115.) 



1 163.] BENDING BEAMS. 858 

(4) The experiment is next to be repeated with 
the supports 100 cm. apart, with a beam twice as 
broad as the one first employed, but having the same 
thickness and bearing the same load as in (1). If 
the material of the beam is the same as in (1), the 
deflection due to a given weight should be the same 
as in (2), since the breadth and weight have the 
same relative proportion as in (2). 

(5) The beam is now to be turned edgewise, and 
loaded as in (8). The deflection is to be determined 
as before. If the depth of the beam is just twice as 
great as in (2), and the width the same, since the 
force employed is eight times as great as in (2), the 
deflection should be the same as in (2). 

^ 163. Calculations relating to Flexure. — By five 
measurements arranged as above, we are able to 
test (in a single instance in each case) the applica- 
tion of the laws of flexure stated in § 115. These 
laws may be combined in a single formula. If I is 
the length of a beam, b its breadth, t its thickness, 
and d the deflection produced (all in cm.) by the 
force / (in dynes) exerted by the load ; and if F is 
the force necessary to produce a unit deflection in a 
beam of unit length, breadth, and depth (supposing 
such a deflection to be possible), we have — 

bdfi 

The quantity F is sometimes called the modulus 
of transverse elasticity. Knowing this modulus, 
we may evidently compute any one of the five 

2» 



354 • DYNAMICS. [Exp. 64. 

quantities, /, 7, by c2, or ty if the other four are 
known. The student should calculate the value of 
F from at least one set of measurements. He 
should also find, by the rule of simple proportion, 
what force would be required to produce a deflec- 
tion of 1 cm. in the case of each beam which he has 
employed. Thus if, with a given beam, 1 kilogi*am 
produces a deflection of 2 cm.y 500 grams would be 
the force required to produce a deflection of 1 cm. 

The force (500 grams in this case) producing a 
unit deflection may be taken as a measure of the 
stiffness^ of the beam in question. The stiffness of 
a beam is due to the fact that in order to bend it, 
the under part must be stretched and the upper 
part squeezed or compressed. The forces brought 
into play by stretching will be measured directly in 
Experiment 65. 



experimp:nt lxiv. 

TWISTING RODS. 

^ 164. Effect of Conples. — An instrument serving 
both to measure and to illustrate the effect of differ- 
ent " couples " (§ 113) is shown in Fig. 174. It con- 

1 Stiffness must not be confounded with breaking strength. A thin 
beam, though more easily broken than a thick one, is not so in pro- 
portion to its flexibility ; for hy reason of its thinness it can bend 
much farther than a thick beam without breaking. Both the strength 
and stiffness of a beam are proportional to its breadth ; but the former 
depends upon the square of the ratio which the thickness bears to 
the length, while the stiffness depends upon the cube of this ratio. 
(See formula above.) 



^164.] 



TWISTING RODS. 



855 



sists of a rod of ash (ef) i cm. square, driven into 
a square hole in a block (y) which is fastened to the 
floor. The rod passes through a large hole in a table 
to a circular disc of wood (eg) 20 cm. in diameter, at 
the centre of which is a square hole (e), into which 
the upper end of the rod is tightly fitted. Two 
markers, h and g^ measure the rotation of the disc 
by means of a scale of degrees graduated on the 
edge of the disc. At certain points of the disc {abc 
defgh, Fig. 175), small screw-eyes are placed so that 
forces may be applied by cords attached to spring 




Fig. 174. 



Fig. 175. 



balances (a and A, Fig. 174). It is convenient that 
four or more of the points (^cdef^ Fig. 175) should 
be in the same straight line and at equal distances, 
let us say 6 cm. The points a, 6, g^ and h (Fig. 175) 
may be placed so that ad^ dg^ 5e, and eh are at right 
angles to cf^ and each 8 cm. long. This will make 
the diagonal distances ac^ hd^ etc., each 10 cm. 

A very slight force applied at any point of the 
disc will cause the rod cj (Fig. 174) to bend so as to 
touch one side of the hole in the table. To keep 



S56 DYNAMICS. [Exp. 64. 

the rod in the middle of this hole throughout this 
experimeDt,^ equal and opposite forces must be ap- 
plied to the disc. If these forces are applied at the 
same point, no effect will be observed. For instance, 
two equal forces applied at d (Fig. 175) in the direc- 
tions dc and de (or in the directions da and rf^) will 
neutralize each other. Again, if the forces and their 
points of application are all in the same straight line, 
the effect will be zero. Thus a force applied at d in 
the direction dc will offset an equal force applied at 
e in the direction ef. When, however, the lines in 
which the two forces act are parallel but not coin- 
cident, the couple which results (§ 113) will twist 
the rod. The angle through which the rod is twisted 
should be proportional to the magnitude of the couple 
acting upon the disc. The magnitude of the couple 
is equal (see § 113) to the product of either of the 
two forces which constitute it, and the " arm " or 
perpendicular distance between the lines in which 
the forces act. 

The student should satisfy himself that it makes no 
difference where the " arm " is situated. Thus two 
opposite forces of 1 kilogram each applied at a and b 
or at c and ci, at right angles to ef^ will have the same 
effect as if applied in the same manner at d and e, re- 
spectively. The student will notice, moreover, that 
the rod is twisted but never bent by a pair of equal 
and opposite forces, whether these be applied at equal 

1 In trying this experiment, several students should work together. 
One may hold and read one of the spring balances, another the other 
spring balance, while a third observes the deflection of the disc. 



1 164.1 TWISTING RODS. 867 

or unequal distances from the centre of the disc. He 
should also satisfy himself that with a given arm (as 
for instance (2e), the rod is twisted through an angle 
which is proportional to the forces employed (let us 
say 1, 2, or 3 kilograms) ; and that the twists pro- 
duced by given forces (e. ^., 1 kilogram each) are 
proportional to the arms to which they are applied. 
Arms of the following lengths may be most conve- 
niently employed : 6 cm. (a6, cd^ de, «/, or gJi) ; 8 cm. 
(ad, he, dg, or eK) ; 10 cm. (ac, ae, bd. If, gc, ge, hd, or 
^) ; 12 cm. (ce or df) ; 16 cm. (ag or bK) ; and 18 
cm. Cof). Two eqiml forces must be applied in all 
cases in directions at right-angles to the arms, paral- 
lel to the disc, and opposite to each other. They 
should be made to twist the rod sometimes to the 
right and sometimes to the left. 

To measure accurately the angles through which 
the disc rotates, both markers (b and g^ Fig. 174) must 
be observed. It is easy to calculate from a given case 
by simple proportion what couple would be required 
to twist the rod through 1°. This gives us a measure 
of the stiffness of the rod under torsion which may 
be called its coeflScient of torsion.* 

We next employ a rod, e'f, of half the length of ej 
(Fig 174). This rod must be mounted on a block 
(/) much higher than/. We shall find, if the ma- 
terial and the cross-section are the same, twice the 
coeflBcient of torsion. If we use a rod of same 
length, having, however, twice the diameter, we shall 

1 The coefficient of torsion must not be confounded with the 
strength of a rod to resist fracture by torsion. See note 1 163. 



868 DYNAMICS. lExp. 64. 

find a coefficient of torsion 16 times as great as be* 
fore (see Laws of Torsion, § 116). It is therefore 
important to measure and note the length and diam- 
eter of the rods employed. 

We shall apply the principles illustrated in this sec- 
tion to the determination of the coefficient of torsion 
of a wire. 

^ 165. Determination of the Coefficient of Torsion of 
a "Wire by means of a Torsion Balance. — A hard 
drawn brass wire about 2 metres long and 0.25 mm. 
diameter (about No. 81, B.w.G.) is stretched horizon- 
tally between a knitting-needle (bdj Fig. 176) and 
a fixed support (A). The joints should be soldered 
both at c and at i, or made equally firm in any other 
manner. 




Fig. 176. 



The knitting-needle is held in place by a paper 
protractor fixed on the surface of a board (ae). The 
board and protractor are pierced at the centre ((?) so 
that the wire may pass through. A thin strip (/A) of 
some light wood, 20 cm. long, is attached at its central 
point, g^ to the middle of the wire by sealing-wax. 
From the ends of this strip two paper scale-pans are 
suspended by threads. The " torsion " balance thus 
constructed should not weigh more than one or two 
grams. 



^ 165.] TWISTING RODS. 869 

The knitting-needle is first set so that the beam 
(/A) is horizontal. To do this, the beam must be 
sighted with reference to the bars of a window, or 
other horizontal line in the room. The reading of 
the needle is then found by observing both ends. 
This is the zero-reading of the instrument. Then a 
decigram is placed in one of the scale-pans, and the 
needle is turned until the beam is again horizontal. 
The decigram is then removed from the scale-pan, 
and the zero-reading re-determined. If any marked 
change has occurred, the experiment must be re- 
peated. If the zero-reading is again disturbed, a 
weight smaller than 1 decigram should be employed. 

The weight is to be placed first in one scale-pan, 
then in the other. In each case we note the angle 
through which the needle must be turned to the right 
or to the left from its zero position in order that the 
beam may be made horizontal. It is well to observe the 
zero-reading after the experiment, since the constancy 
of this reading is the only safeguard against slipping 
of the joints or permanent straining of the wires. 

Since the balance beam is 20 cm. long, the average 
length of each arm must be 10 cm. Since the weight 
of 1 gram is about 980 dynes, that of 1 decigram will 
be about 98 dynes ; hence the couple exerted by grav- 
ity is 98 X 10 or 980 units. This is balanced by 
twisting a certain portion of the wire (eg) through 
an observed number of degrees ; hence the couple 
due to V is easily calculated. This couple measures 
a coefficient of torsion of the wire (see ^ 164), 
which will be needed in experiments later on. 



360 DYNAMICS. [Exp. 65. 

We notice that the portion of the wire between g 
and h is not twisted at the times of making our read- 
ings, because the beam /A remains horizontal The 
torsion of this part of the wire does not, therefore, 
affect the result. The only use of the wire between 
g and k is to keep the balance in place. The length 
of the wire between e and g should be measured, and 
its diameter should be found in several places by 
means of a micrometer gauge (^ 50, IL), The 
material should also be noted, in order that we may 
utilize our results in certain other experiments later 
on. 



EXPERIMENT LXV. 

STRETCHING YITIRES. 

^ 166. Tonng's Modulus of lUastdcity. — A fine 
steel wire, about 0.25 mm. in diameter (No. 31, 




Fig. 177. 

B. W. G.) and 1 metre long, may, if made of the best 
steel, be stretched 1 cm, without breaking, or losing 
its power of recovery. We will suppose such a wire 
to be held at one end by a small vice (a, Fig. 177) 
and attached at the other end (ft) to a spring bal- 
ance ((?) held in place by a nail (d). Let the read- 



tl66.] STBETCmNG WIRES. 861 

ing of this balance be 0. Now let the wire ab be 
stretched to a point b'j by placing the balance over a 
nail ((2), and let the new i-eading of the balance ^ be 
F. Then if the length of the wire thus stretched is 
ab centimetres and the elongation is bV cm.^ the 
stretching of 1 em. will be bV -^ ai. This is called 
the strain of the wire. When lOO cm. are stretched, 
for instance, 1 cm.y we have a strain of 1 per cent or 
+ .01. 

Now if the diameter of the wire is measured by a 
micrometer gauge, and divided by 2, we have its ra- 
dius, r. From this we can find the cross-sectioa q by 
the ordinary formula (j = tt r*), or 

q = 8.1416 X r^, nearly. I. 

The cros^-section can also be determined by finding 
the weight, w, of a given length (Z) of the wire, if 
its density (c2) is known ; for since the volume of a 
wire is equal to j X Z, we have by definition (§ 154) 
d = w -^ ql^ whence — 

We wilt suppose that by either of these formulae 
the average cross-section of the wire ab has been 
found. Now let the force indicated by the spring 
balance be reduced to dynes by multiplying by the 
appropriate factor.' Let us call this force in dynes/. 

1 Id practice a tmaU force will be required to straighten the wire. 
In this case the force F, below, must be taken as the difference be- 
tween the forces exerted bj the balance at d and (t, 

3 Thus in latitude 60^ 1 kilogram is equal to about 981,000 dynes, 
1 lb. aYoirdupois to 446,000 djrnes, and 1 os. to 27,800 dynes, nearly. 



362 DYNAMICS. [Exp. 65. 

To find the intensity of the force per square centi- 
metre of cross-section of the wire, we divide it by 
the cross-section in question. Thus if the wire had 
a cross-section of one 2,000th of a square centimetre 
(.0005 <?m2), a force of 6,000,000 dynes would repre- 
sent an intensity of 10,000,000,000 dynes per square 
centimetre (since 6,000,000 -r- 0005 = 10,000,000,000). 
The result is called the ^' stress^' exerted upon the 
wire (§ 22). 

It has been stated (§ 114) that for a given mate- 
rial there is always a certain proportion between the 
stress exerted upon it and the strain produced. The 
ratio of the stress to the strain in the stretching of a 
rod or wire is called '' Young's Modulus of Elasticity,^' 
If, for example, a stress of 10,000,000,000 dynes per 
square centimetre produces in a steel wire an elon- 
gation of one half of one per cent, that is, a strain of 
4-. 005, the Modulus of Elasticity of the steel is 10,- 
000,000,000 -^ .005, or 2,000,000,000,000 (two millions 
of millions) dynes per square centimetre. The Modulus 
of Elasticity has also been defined as the force neces- 
sary (under Hooke's law, § 114) to produce a unit 
strain in a rod of unit cross section ; that is, to double 
the length of the rod. Evidently, if 10,000,000,000 
dynes are required as above to increase the length 
of a steel rod, 1 cm, square, by one part in 200, it 
would take 200 times as much force to double its 
length, provided that it kept on stretching at the 
same rate ; hence we find 2 x 10^^ foj. the modulus 
of elasticity, as before. 

Few substances can be stretched one hundredth 



T 167] STBETCHING WlBEa 863 

part of their length without breaking. It is only in 
the case of exceedingly elastic substances, like India 
rubber, that the conditions suggested by the last 
definition can be actually attained. In the case of 
most substances, we can only calculate by the rules 
of simple proportion what stress would double their 
length, provided that fracture or other changes did 
not occur. 

The student may notice that steel (see Table 9) 
has the greatest modulus of elasticity of any known 
substance, because it requires the greatest force to 
produce a given amount of stretching; or because, 
in other words, it yields the least. A substance like 
India rubber, which is in the ordinary sense particu- 
larly elastic, has for this very reason a small modulus 
of elasticity. 

^ 167. Determiiiation of Toung's Modulus of ZUaa- 
ticity. — The data necessary for a determination of 
Young's Modulus are, as will be seen from ^ 166, 
(1) the length, (2) the cross-section of the wire to 
be tested, (3) the elongation produced in it by a 
given force, and (4) the magnitude of this force. 
The length of a wire may be measured, without 
any special diflSculty, by a tape graduated in milli- 
metres. The cross-section requires much greater 
care, whether it be determined (as suggested in 
^ 166) by measurements taken with a micrometer 
gauge at different points, or by its length, weight, 
and density. The principal difficulty consists, how- 
ever, in measuring accurately the elongation of the 
wire, which is usually a very small quantity. To 



364 



DYNAMICS. 



[Exp. 65. 



ij, 



^ 



make the elongation as large as possible, long wires 
are usually employed. 

One of the chief sources of error in measuring the 
elongation of a wire under a given load is due to 
the yielding of the support to which the wire is 
attached. Various devices have been 
suggested by which this effect may be 
eliminated. The simplest is to meas- 
ure the distance between two points 
on the wire. This may be easily done, 
when a double wire is employed, by 
means of two micrometers, a and b 
(Fig. 178, 1), attached to the wall, 
and adjusted so as to touch two cross- 
bars borne by the wires in question.^ 

To avoid the inconvenience of mak- 
ing observations at a considerable 
height above the floor, a wire is some- 
times surrounded by a tube (flJ, Fig. 
178, 2) attached to it at a point a. If 
the point a yields, a point b at the base 
of the tube will yield by an equal 
amount. The height of this point (6) 
and of a point (<?) on the wire may be observed 
(^ 262) accurately by a cathetometer. The in- 
crease of distance between b and e is evidently 
equal to the elongation of ac. In the Physical 
Laboratory of Harvard University the effects due 
to the yielding of the support are avoided by keep- 



1 



b 



^ 



Fio. 178. 



1 This device is due to Mr. Forbes, of the Roxbury Latin School. 



ir 167.J 



STRETCHING WIRES. 



865 



ing the same weight always upon it. The wires 
(which are nearly 6 metres long) are attached to a 
beam by means of a piece of iron (a6rf» Fig. 178, 3) 
shaped like an inverted T. At the middle of the T 
a split plug ((?) driven upwards into a vertical hole 
firmly grasps the wire. Side wires from the arms 
of the T hold a small platform (^) just above the 




Fig. 179. 

floor. The weights to be used in stretching this 
wire are kept on this platform when not in use. 
Obviously the beam and the stem of the T are sub- 
jected to the same strain whether the load be sus- 
pended from the central wire or by the side wires. 

A stout ring (de^ Fig. 179) is attached to the 
central wire (b) by a pplit plug (rf). The stretch- 
ing of the wire is measured by a micrometer, the 



366 DYNAMICS. [Exp. 65. 

point of which touches a small level surface on the 
ring at e. The contact is determined by electrical 
connections, as in ^ 162. Directly below the point 
of contact a platform, /, is suspended, for the pur- 
pose of holding the weights by which the wire is 
to be stretched. There are many theoretical objec- 
tions to this form of apparatus, which being of no 
practical importance have been left out of considera- 
tion. It is obviously necessary that the wire should 
be straight before the stretching forces are applied. 
For this purpose, a small load is always kept on it. 
In the apparatus shown in Fig. 179, the weight of 
the ring (rfe) and platform (/) should be suflScient 
to straighten the wire. In calculating Young's Mod- 
ulus, we consider only the weight which must be 
added to the load already borne by the wire, in 
order to produce the observed elongation. 

To determine the elongation in question, a reading 
of the micrometer must be taken with and without 
the weight. The difference in the readings gives, 
allowing for the pitch of the screw (see ^ 52), the 
distance through which the wire has been stretched 
by the weight in question. 

For a determination of Young's Modulus of Elas- 
ticity, a fine steel wire will answer. Care must be 
taken, however, not to bend the wire sharply over 
the edge of the vices or split plugs to which it is 
fastened. If the wire is 0.25 mm. in diameter, and 
free from kinks or bends, it may be made to bear 
safely a total load of 1 , 2, or even 3 kilograms. 

If / is the force exerted by the weight when re- 



1[ 168.1 BREAKING STRENGTH. 867 

duced to dynes (see ^ 166), e the resultiug elonga- 
tion of the wire in c?it., I the length in cm, of that 
portion of the wire in which the elongation takes 
place, and q its average cross-section in %q. cm.. 
Young's Modulus of Elasticity (-&) is found in 
C. a. S. units (§ 8) by the formula 

qe 
or by the method of reduction explained in ^ 166. 



EXPERIMENT LXVI. 

BREAKING STRENGTH. 

^ 168. Detenninatioii of the Breaking Strength of a 
"Wire. — A steel or spring-brass wire about \ mm. in 
diameter (No. 31, B. w. G.), free from kinks or sud- 




9 



Fig. lao. 

den bends, is to be attached at one end to the eye 
(6, Fig. 180) by which the hook (he) is attached 
to the spring balance (ale). The other end is to 
be fastened to some fixed point, as, for instance, 
a nail (e) driven into a post (d). A bobbin, <?, is to be 
cut out Cas shown in c' and c*' of the cross-sections 
2 and 3), so as to fit over the hook of the balance 
without danger of turning. A few turns of the wire 
are made about the bobbin ; the rest is wound around 



868 DYNAMICS. [Exp. 66. 

a post, d. The indei: of the balance is to be watched 
as a steadily increasing force is applied to the wire.^ 
When the wire breaks, the maximum reading is re- 
corded* The position of the break must now be as- 
certained. If it occurs at 6, or between b and ^, the 
result must be thrown out. If the wire breaks at e 
or at d, the accuracy of the result is doubtful ; be- 
cause a sharp bend in a wire where it passes round 
a corner may cause it to break under forces far less 
than its average breaking strength. If the break 
occurs between c and d, the break is probably a fair 
one. Enough wire will probably remain about the 
post for several repetitions of the experiment. The 
results should agree within five or ten per cent. Sus- 
pected results, much smaller than the average, may 
be discarded. 

The cross-section of the wire must be found both 
by measurements with a micrometer gauge and by 
weighing a known length of the wire, let us say 1 
metre, as accurately as possible. (See ^ 166, for- 
mulae I. and II.) The density of steel may be taken 
as 7.9, of brass 8.4 in this reduction. The student 
should compute by simple proportion the force neces- 
sary to break a wire one sq. cm. in cross-section i he 
may also calculate what length of the given wire 
would break under its own weight. Thus if 100 
cm. of brass weighs 0.42 grams, its cross-section must 
be 0.42 -7- 100 -f- 8.4, or .0005 sq. cm. If it takes 2.94 
kilograms to break such a wire, a wire 1 sq. cm. in 

1 The hand should he held in such a position as not to be injured by the 
hook when the spring recoils. 



1 169.] SURFACE TENSION. 869 

cross-section would require 2.94 -^ .0005 or 5,880 
kilograms to break it. At 0.42 grams per metre, it 
would take 2.94 -i- 0.42 or 7000 metres of the wire 
to break under its own weight. 

Obviously the result of this calculation should be 
the same whether a large or a fine wire is used, pro- 
vided that the quality be the same, because both the 
breaking strength and the weight of a wire increase 
in proportion to its cross-section. 



EXPERIMENT LXVII. 

SURFACE TENSION. 

^ 169. Detennination of the Surface Tension of a 
zaquid. — I. A piece of fine iron wire is bent as in 
Fig. 181, so as to form a fork (Jhg^ with parallel 
prongs (cf and eg^ about 2 cm. apart. The 
fork is then suspended from the hook of 
a balance (a) so as to dip into a beaker 
of water, as in the hydrostatic method 
(Exp. 9). The fork must be entirely cov- 
ered by water when the balance beam 
is lowered see (^ 19) ; but when the latter 
is raised, the prongs only must dip into the water. 

The weight of the fork is first balanced as accu- 
rately as possible ; then the fork is lowered into the 
water, and suddenly raised out of it. A film of water 
will probably be found to fill the space between /<?<?«^ 
and the surface of the water. This film will tend to 
pull the fork back into the water. To balance the 

24 




370 DYNAMICS. [Exp. 67. 

pull which it exerts, an additional weight of about 
3 decigrams must be placed in the opposite scale- 
pan. This weight is to be adjusted, by a number of 
trials, as accurately as possible. As the film gradu- 
ally evaporates, it becomes lighter and lighter; but as 
its weight is, in any case, so small that it may be neg- 
lected, the change of weight will probably have no 
visible effect. The student will notice that the ten- 
sion of the film of water remains sensibly constant as 
it grows thinner and thinner, until it breaks. This 
is entirely unlike the tension of solid substances, 
which depends upon their cross-section. The ten- 
sion which liquids exert depends simply upon the 
breadth of the surface which tends to contract, not 
on the cross-section of the solid contents included 
by that surface. For this reason, the phenomenon 
is called ** surface tension." 

In the case under consideration, the film has two 
surfaces, each let us say 2 cm. broad. The total 
breadth of surface is therefore 4 cm. The student 
is to calculate what force (in dynes) is exerted by a 
Bingle surface 1 cm, broad. 

The surface tension of liquids depends upon tem- 
perature ; hence the temperature should be noted. It 
is greatly affected by impurities in the liquids. An 
invisible quantity of oil, for instance, produces vari- 
ations of ten or twenty per cent. Great care must 
therefore be employed in obtaining the purest dis- 
tilled water. Both the inside of the beaker and the 
lower part of the wire should be cleaned with caustic 
potash, and afterwards rinsed in several changes of 



t 170.] SURFACE TENSION. 871 

distilled water. The parts thus cleaned must not 
afterwards be touched by the finger. 

11. A piece of thermometer tubing with a round 
bore about i to J mm. in diameter is carefully cleaned 
with caustic potash, which may be sucked through 
it with a medicine dropper (of coui'se not by the 
mouth), then cleaned with distilled water. It is now 
dried by heat and filled with mercury. The contents 
are to be placed in a beaker, and weighed. If the 
quantity of mercury is too small to be weighed accu- 
rately, ten tubefuls may be weighed together (§ 39). 
The length of the tube is to be meas- 
ured. The tube is now placed in a 
clean beaker containing pure distilled 
water (see I.). It should be at first 
inclined somewhat, so that the water 
which rises into it through "capil- 
lary attraction" may thoroughly wet 
its inside surface. It is next made 
vertical (see Fig. 182). The height of 
the column of water in the tube above 
the level in the beaker is then meas- 
ured, both when it barely dips into the ^^* ^®^' 
water, and when it dips so deep that the water rises 
nearly (but not quite) to the top of the tube. Other 
measurements should be taken similarly with the 
tube turned end for end. All results should agree 
closely, if the tube is of uniform calibre. 

^170. Calculations relating to CapiUary Attraction. 
— If ^^; is the weight in grams of the mercury which 
fills a tube, 13.6 the density of the mercury, and I 




872 DYNAMICS. [Exp. 67. 

the length of the tube in cm.^ the cross-section is 
(see ^ 166, formula II.) 

w 

The radius of the tube is connected with the cross- 
section by the formula 

J = 7rr2; 

hence, solving, we find 

r = 4/? = 0.564 Vqi nearly. 

^ IT 

If h is the average height of the water in the 
tube above its level in the beaker, 1.00 the den- 
sity of water, the volume of water raised is qh^ or 
TTT^h; the weight in granjs is 1.00 X gA, or 1.00 x Trr^A, 
and the weight in dynes (allowing g dynes to the 
gram) is ghg^ or irgr^h. This weight is sustained by 
the tension of a film lining the inside of the tube. 
The breadth of this film is evidently equal to the 
circumference of the tube (27rr). If a film 27rr cen- 
timetres broad can sustain a force irgt^h dynes, a film 
1 cm. broad would evidently sustain trgr^h -r- 2'7rr, or 
^grh dynes. That is, the "surface tension** of 
water (aS) is given by the formula 

S = igrh = 490 rh dt/nes per centimetre (nearly). 

Obviously, if S is constant, the product, r X A, must 
be constant ; that is, the height to which a liquid will 
rise in a tube is inversely as the radius of that tube. 



IT 171.] COEFFICIENT OF FRICTION. 878 



EXPERIMENT LXVni. 

COEPPICIENT OF FRICTION. 

^ 171. Detennlnation of CoefBoienti of Friction.-*- 
I. A piece of plaued plank ((, Fig. 183) measuring 
let us say 5 X 20 X 40 crw., is drawn horizontally by 
a spring balance, a, over a planed board c. The force 
necessary to maintain a uniform velocity after the 
plank is once started, is observed and noted. Then 
the plank is suspended from the spring balance and 
weighed. The ratio of the force required to draw a 
body to the force required to lift it is called a " co- 




FlO. 183. 

efficient of friction." The coefficient of friction in 
this case is that of wood on wood. If the force of 
traction varies in different parts of the board, the 
average should be calculated ; and from this the 
average coefficient of friction may be found. It is 
instructive to repeat the experiment with the plank 
edgewise, so as to see whether the diminished area of 
the surfaces in contact is or is not compensated for 
by the increased intensity of pressure. For a fair 
comparison, the side and the edge of the plank 



374 



WORK. 



[Exp. 68. 



should of course be equally smooth, and both par- 
allel to the grain of the wood. 

The experiment may also be repeated with the 
plank flatwise, but with a heavy weight upon it as 
in the figure. The value of this weight should be 
found as in ^ 159, and added to that of the board, 
in calculating the coefficient of friction in question. 

The student will notice that it takes considerably 
more force to start a body than to drag it after it is 
once started. This is attributed to the cohesion of 




Fig. 184. 



particles which takes place at various points, partic- 
ularly when two surfaces remain long in contact. 
The ratio of the force required to start a body when 
resting upon a horizontal surface to the force re- 
quired to lift it is sometimes called the '* coefficient 
of starting friction." This must not be confounded 
with the ordinary "coefficient of friction." 

II. The board A C (Fig. 184) already used in I. 
is inclined (by means of a nail, A^ and a block 2>) 
so that the plank a, when once started, slides down 
it with uniform velocity. A measuring rod BC is 
placed at a point 5, 1 metre from A^ and the verti- 



tl72] FBICnON OF FLUIDS. 875 

cal distance BC to the under side of the board is 
then measured. The *^ slope" of the under surface 
(BC-T- AC) is thus found. The slope necessary to 
maintain a uniform velocity may not be the same 
from one end of the board to the other. If it is not 
the same, the average slope should be calculated. 

If we resolve the weight of the block ae into two 
forces, one, a6, perpendicular to the board AC^ the 
other, 6(?, parallel to it, then by definition (see I.) 
the coefficient of friction is 6^ -f- «J ; but, by similar 
triangles, this is equal to the ratio ot BC to AB^ 
which measures the "slope " of the board AC. The 
average slope which must be given to this board in 
order that the plank, when once started, may slide 
down it with uniform velocity, gives accordingly the 
"coefficient of friction" between the two surfaces in 
contact. The result should agree closely with that 
determined as in I. 

^ 172. Fluid Prictlon. — When a well-shaped boat 
moves through water with a velocity of v cm. per 
sec.^ the opposing force (JP) which it encounters is 
approximately equal to the square of this velocity 
multiplied by the area (a) of the surface wet by the 
water, measured in aq. cm.^ and by a certain constant, 
/ (about .003), which is called the coefficient of friction 
oftvater^ that is : — 

F = fav^ dyne9. 

Coefficients of fluid friction must not be con- 
founded with coefficients of friction in the case of 
solids, which are calculated in an entirely different 



876 WORK. [Exp. 68. 

way. The frictional resistance between two solid 
surfaces depends, as we have seen (^ 171), upon 
the pressure between them, but not upon the rela- 
tive velocity of the surfaces. On the other hand, the 
resistance offered by a fluid to the motion of a solid 
does not depend upon the pressure between the sur- 
faces in contact, but does depend upon their relative 
velocity. The nature of the fluid, the shape and 
smoothness of the solid, modify the result ; but the 
material of which the solid is composed is generally 
unimportant. The resistance offered by fluids to the 
motion of solids or the reverse depends upon dis- 
turbances which are wholly confined to the fluid. 
Every fluid has, therefore, its own coeflScient of 
friction. 

When a current of water flows through a large^ 
tube of the length I and radius r (both in cm.), since 
the area of wetted surface is iirrl^ the force oppos- 
ing the flow is 

F=27rrlfv^ (dynes). (1) 

This force is supplied by the pressure (jt?) of the 

water (measured in dynes per sq. cm.) exerted upon 

an area equal to the cross-section (ttt^) of the tube ; 

that is : — 

F=^r^p, (2) 

Equating (1) and (2), we find, — 

1 In capillary tubes, the force encountered is proportional directly 
to the velocity (see IT 260). In tubes from 1 to 6 •nm. in diameter, for 
▼elocities between 10 and 100 cm, per sec., no simple law can be giren. 



1 172.] FRICTION OF FLUIDS. 877 

The velocity (t^) can be estimated from the cross- 
section of the tube and from the volume of water 
which flows through it in a given length of time 
(^ 147, 4) , the pressure may be found by a pressure- 
gauge (see Exp. 69) at the point where the water en- 
ters the tube, provided that there is a free outlet at 
the other end, and that both ends of the tube are on 
the same level. If, as in Fig. 185, one end is higher 
than the other by an amount ac, equal let us say 
to A, then if g is the acceleration of gravity and 
1.00 the density of water, the hydrostatic pressure 

is (see § 63) 

p = 1.00 ^A, nearly. (5) 

The length (?) of the tube may be directly meas- 
ured. The capacity ((?) may be found by measuring, 
or (as in ^ 82) , by weighing the quantity of water 
required to fill it. The cross-section (j) may then 
be calculated by the equation - 

,=i («) 

Hence the radius (r) is given by the formula — 

r = ^J /I (7) 

The coefficient of friction, /, may now be calcu- 
lated by formula (4), since all the quantities are 
known. 

The ** resistance " of a tube to the flow of a given 
liquid may be defined as the pressure in dynes per 
sq. cm. required to maintain through that tube a flow 



378 WORK. [Exp. 68. 

of 1 €u, cm. per sec. Thus if a rubber tube (a J, Fig. 
185) 2 metres long and 3 mm. in diameter is used as 
a siphon to conduct water from a cistern, a, to a 
point J, it will be found that the outlet (J) must 
be about 10 cm. below the level (a) in the cistern in 
order that water may flow through ab at the rate of 
1 cu. cm. per sec. The hydrostatic pressure corres- 
ponding to a difference of level of 10 cm. is nearly 
10 grams per aq. cm.^ that is, 9800 dynes per sq. cm. 
The " resistance " is therefore about 9800 units. 
The resistance of a conduit may also be defined as 




Fig. 185. 

the power (in ergs per second) necessary to maintain 
a unit current (1 cu. cm. per sec.^ through the con- 
duit in question. This definition bears a strong re- 
semblance to the definition of electrical resistance 
(§ 136). The fact that power is required to main- 
tain a current through the tubes and valves of a 
water-motor, together with the friction between the 
solid parts of the motor, will be found to modify the 
" efficiency " of the machine. Tlie next experiment 
relates to determinations of " efficiency." 




1 173.1 EFFICIENCY. 879 

EXPERIMENT LXIX. 

EFFICIENCY. 

% 173. Nature of Bffloienoy. — Let us suppose that 
a 20-kilogram weight is suspended by a tackle (Fig. 
186) consisting of two double blocks, with four cords 
passing between them. Let us first suppose that the 
cords run with absolute freedom round 
the pulleys which the blocks contain. 
The force on each cord must evi- 
dently be 5 kilograms ; and a force of 
5 kilograms, applied ty a spring bal- 
ance to the free end of the cord, as 
in the figure, will just hold the weight 
in place. If the weight were started 
upward by any impulse, no matter 
liow small, the force of 5 kilograms 
constantly applied to the free end of 
the cord would (in the absence of fric- 
tion) continue to raise it with a uni- 
form velocity, until the two blocks 
met together. If the two blocks were 1 metre apart 
in the beginning, we should have 20 kilograms raised 
by the tackle through a height of 1 metre. Each 
of the four cords would be shortened 1 metre in this 
process , hence there would be 4 metres of slack to 
be taken up at the free end of the cord. The spring 
balance must accordingly retreat 4 metres. The work 
spent upon the machine by a force of 5 kilograms re- 






380 WORK. [Ext. 69. 

treating 4 metres (20 kilogram-metres), would be the 
same as that utilized by the machine in raising 20 
kilograms 1 metre high (see § 14). 

Let us now suppose that a slight downward im- 
pulse is given to the weight, so that it descends to 
its original position. The work spent by gravity 
upon the machine, being 20, kilogram -metres as be- 
fore, is utilized in pulling the spring balance forward 
through a distance of 4 metres. In the absence of 
friction, the pull would be 5 kilograms as before. 
The amount of work utilized (20 kilogram-metres) 
would be equal, accordingly, to the amount spent 
upon the machine. 

It is not necessary to consider the magnitude of 
the impulse by which the weight is started upward 
or downward ; for if the weight moves with uniform 
velocity, it is capable of giving back this impulse, 
when it has been raised or lowered to any desired 
point (see § 121), in the act of stopping, when its 
energy of motion is lost. In the absence of all fric- 
tion in the pulley-wheels, stiffness in the cords, and 
resistance in the air, a tackle devoid of weight would 
constitute a theoretically perfect machine, — that is, 
all the work spent upon it would be utilized by it. 
In practice, a considerable part of the work spent 
upon a machine is always transformed by friction 
into heat. That proportion of the work spent upon 
a machine which is utilized by it is called the 
" efl&ciency " of the machine. 

Let ns suppose that, instead of 5 kilograms, a force 
of 10 kilograms is required to raise a 20 kilogram 



IT 173.] EFFICIENCY. 881 

weight by means of the tackle represented in Fig. 
186. Then since, in raising 20 kilograms 1 metre, 
10 kilograms retreat 4 metres, the work spent is 40 
kilogram-metres ; but the work utilized is only 20 
kilogram-metres. The " efficiency *' of the tackle as 
a machine for raising weights is accordingly |^ or 
50%. 

Again, let us suppose that a weight of 20 kilo- 
grams, descending one metre, exerts a force of only 
2 kilograms on the spring balance, which advances 
4 metres. Then the work spent by gravity is 20 
kilogram-metres, but that utilized is only 8 kilo- 
gram-metres ; hence the efficiency of the tackle as a 
machine for utilizing potential energy (§ 122) is ^ 
or 40%. 

Finally, let us consider the tackle as a machine 
for storing and utilizing energy. A force of 10 kilo- 
gi'aras is required to raise the weight, and this force 
must retreat 4 metres to raise the weight 1 metre. 
40 kilogram-metres of work are thus spent upon the 
machine. The free end of the cord is now attached 
to some resistance which it is desired to overcome. 
A force of 2 kilograms is thus applied through a 
distance of 4 metres. The work utilized by the 
machine is only eight kilogram-metres. Evidently 
the efficiency of the tackle as a machine for storing 
and utilizing energy is only ^^^ or 20%. 

When energy is stored in a machine, part of it is 
lost. When this energy is utilized, part of what is 
left is lost. When energy undergoes a series of 
transformations, a certain proportion is lost in each. 



882 



WORK. 



[Exp 69. 



Obviouisly, in stating the efficiency of a machine, 
it is necessary to specify where or how the work is 
spent upon it, and where or how the work is utilized. 




Fig. 187. 

^ 174. Determination of the Efficiency of a "Water- 
Motor. — (1) To find the work utilized ly a water- 
motor, the circumference of the driving-wheel (^, 
Fig. 187) is first measured, then two spring bal- 



IT 174.1 EFFICIENCY. 883 

ances, a and i, are connected by a cord (cd) pass- 
ing round the wheel. The motor is then started, 
and the tension of this cord increased until, through 
the friction which it exerts upon the wheel, the 
velocity of the latter is reduced to about one-half 
of its maximum. The speed of the wlieel is then 
determined by counting the number of revolutions 
made in a given length of time. The reading of 
each spring balance is also found. If it varies, 
several observations must be made, and the mean 
calculated. 

The difference between the two readings is equal 
to the force opposed by friction to the motion of the 
rim of the wheel, and must be reduced to dynes or 
megadynes. If the value of this force in dynes is jP, 
if the number of revolutions in one second is n, and if 
c is the circumference of the wheel in centimetres, 
then in traversing the distance en centimetres against 
the force -P dynes, the work done must be en F ergs. 
If we suppose that the force reduced to megadynes is 
equal to/, then enf represents the work in megergs. 
Since enf megergs of work are performed against 
friction in 1 second, and might be utilized for turn- 
ing machinery (see ^ 175), we infer that the work 
thus utilized would be enf megergs per second. 
This measures, therefore, the power of the machine. 

(2) To find the work spent in driving the motor, 
we must measure the quantity of water which passes 
through it in a given length of time. The water may 
be collected in a stone jar (^, Fig. 187), and weighed 
on a pair of rough platform-scales (Fig. 188). The 



384 WORK. [Exp. 69. 

pressure of the water must also be found bj* means of 
a pressure-gauge connected with the supply pipe (see 
Fig. 187). The gauge should be as nearly as possi- 
ble on a level with the outlet by which water escapes 
from the motor. The pressure must be reduced to 
dynes (or megadynes) per square centimetre. If v ia 
the calculated volume in cubic centimetres of the 
water which flows through the motor in one second, 
and if P is the pressure of this water in dynes per 
square centimetre, then the work spent on the motor 




Fig. 188. 

is vP ergs per second (see § 118). It p is the value 
of this pressure when reduced to megadynes per 
square centimetre,^ then the work spent on the 
machine is vp megergs per second. 

(3) For the accurate determination of efficiency, 
it is desirable to make simultaneotta determinations of 
the power utilized by the motor, and of the power 
spent upon the motor. For this purpose, it is well for 
several students to work together. One may, for in- 

^ The ordinary atmospheric pressure (15 lbs. per sq. in.) is equal 
very nearly to 1 megadyne per square centimetre. See Table 60. 



t 175.] EFFICIENCY. 885 

stance, record the readings of the spring balance, a, 
another those of i; a third those of the pressure- 
gauge ; a fourth may attend to turning the stream of 
water into the stone jar at a given time, and cutting it 
off at a given time ; and a fifth may count the number 
of revolutions made by the wheel of the motor in the 
interval in question. When the experiment is per- 
formed by a single person, the mean readings of the 
balances and pressure-gauge must be inferred from 
observations just before and just after the determi- 
nations of velocity. 

To calculate the eflBciency (e) of the motor, the 
work utilized in one second bi/ the machine is to 
be divided by the work spent in one second on the 
machine. We have, accordingly, — 

e=^. 
vp' 

In repeating the experiment, the tension of the cord 
should be increased or diminished. The maximum 
power of a water-motor is lasually realized when, by 
the resistance which it has to overcome^ the speed 
of the motor is reduced to about half its maximum 
speed. To obtain the maximum efficiency^ the speed 
of the motor must be still further reduced. 

^ 175. The Transmiflsion Dynamometer. — To meas- 
ure the power of a motor actually doing useful work, 
a transmission dynamometer must be employed. One 
of the simplest forms of this instrument is represented 
in Fig. 189. Instead of carrying two cords (c and c?) 
from the driving-wheel (jg) of the motor to two spring 

25 



386 



WORK. 



[Exp. 69. 




balances (a and 5) as in Fig. 187, these cords are 
made to pass around two pulleys (a and 6, Fig. 189) 
to a second wheel (A), to which the motion is thus 
transmitted. The pulleys are suspended by two 
spring balances (J. and E). The 
work done by the motor depends 
as before upon the difference in 
tension of the cords c and d ; but 
if the pulleys run freely, the ten- 
sion of e and / will be the same 
as that of c and d respectively ; 
hence the forces A and B regis- 
ffSi j^rai tered by the spring balances A 
' J WM WneT^ ^^^ -^ (allowing for the weight 
™ ^^ of the pulleys) will be 2c and 2cZ, 
respectively. It follows that 
(c-d) = I QA-E). The difference between the read- 
ings (-4. and jB) must therefore be halved in order to 
find the difference of tension between the cords ^ c 
and d. 

When the wheels move so fast that the revolutions 
cannot be counted, we may find the velocity of the 
cord, cdef^ by measuring its length and counting the 
successive returns of a knot in the cord taking place 
in a given length of time. In other respects the work 
utilized is calculated as in ^ 174, 1. 

^ In practice, if the cord c is approaching g the tension on c will 
be a little greater than on e ; and the tension on d will be a little less 
than on /, hence the difference of tension between c and d will be 
greater than the difference between e and/. That is, the work done 
by g will be a little greater than that received by h. The average 
between these two quantities is measured by the dynamometer. 



If 176.J MECHANICAL EQUIVALENTS. 887 

EXPERIMENT LXX. 

MECHANICAL EQUIVALENTS. 

^ 176. Different Methods for determining the Me- 
chanical Bquivalent of one Unit of Heat — (1) If a 
weight (rf, Fig. 190) is suspended by a cord passing 
over a pulley (a) and round an axle (0, surrounded 
with water in a calorimeter, and made to descend 
slowly to a position d\ by applying a suitable resist- 
ance through a friction-brake, 6, the work done by 
gravity in pulling the weight, let us say w^ 
through the distance I (equal to dd'^ will 
nearly all be converted by friction into 
heat within the calorimeter. Let us sup- 
pose that the total thermal capacity of the 
calorimeter and its contents is c, and that 
its rise in temperature is ^° ; then the i 

quantity of heat developed is ct. If grav- j 

ity exerts a force of g dynes on one gram, i 

it will exert wg dynes on w grams ; and a -r^ 

force of wg dynes acting through the dis- ^"•* 

tance /, must perform a quantity of work ^^^' ^^' 
equal to wgl ergs (§ 14). If wgl ergs are equivalent 
to ct units of heat (§ 16), one unit of heat must be 
equivalent to wgl-^ct ergs. To obtain exact results, 
allowances must be made for the friction of the pul- 
ley, a, for loss of heat by cooling, etc. By a device 
similar in principle to the one described above. Joule 



i 



i 



888 



WORK. 



[Exp. 70. 



found that the mechanical equivalent of one unit of 
heat is about 41,660,000 ergs. 

(2) Two heavy iron bars, A and -B, suspended as 
shown in Fig. 191, may be released simultaneously by 
burning a cord (see ^ 148) or by electrical means, so 
that when the bars meet endwise, a lead bullet (b) 
may be crushed between them. The work done by 
gravity in giving velocity to the bars is thus nearly all 
transformed into heat, through friction of the parti- 
cles of lead against one another. Most of the heat 
will accordingly be found in the bullet. If the bullet 
is immediately lowered into a small calorimeter (c), 
the quantity of heat may be measured in the ordi- 




FiG. 191. 

nary way (see ^ 92). To obtain exact results, an 
allowance must be made for the energy of motion 
which remains in the bars after impact. If Z is the 
difference between the original height of the bars and 
the height attained by them in their rebound, and w 
their combined weight, the work done by gravity is, 
as in (1), wgl. There is no way of allowing accu- 
rately for the energy taken up by the bars in the form 
of vibration, or for the energy of motion directly con- 



1177.] MECHANICAL EQUIVALENTS. 389 

verted within the bars into heat It is said that tlie 
proportion of energy thus lost is small.^ 

(3) By measuring the temperature of a water-fall 
above and below the fall, it would be possible to esti- 
mate the mechanical equivalent of heat. Thus if the 
water is 0°.l warmer at the foot of Niagara Falls than 
above the falls, where the height is 42.5 metres, we 
should infer that to cause a difference of 1®, a water- 
fall must be 425 metres high. Each gram of water 
falling 425 metres, or 42,500 cm. under a force 
of 980 dynes, nearly, must receive from gravity 
980 X 42,600, or nearly 41,660,000 ergs, in the form 
of energy of motion. If the conversion of this energy 
into heat warms it 1°, then the mechanical equivalent 
of 1 unit of heat must be 41,660,000 ergs. 

In practice, the difference of temperature between 
the top and bottom of a water-fall is generally too 
slight to be measured accurately with ordinary in- 
struments. Unless, moreover, the volume of a water- 
fall is very great, evaporation and other causes may 
affect the result. A rough experiment illustrating 
this method of determining mechanical equivalents 
will be described in the next section. 

^177. Determination of Specific Heats by Mechan- 
ical Equivaiento. — A kilogram of lead shot is placed 
in a pasteboard tube (ac, Fig. 192) about 5 cm. in di- 
ameter and 120 cm. long, closed by two corks, a and c. 

1 For an experiment similar in principle, performed by Him, see 
Trowbridge's New Physics, Exp. 105. This modification of Hirn's 
method is due to Professor Gutlirie. The geometrical principles 
connecting arcs and heights have been already considered in the 
case of a ballistic pendulum (see § 109). 




890 WORK. [Exp. 70. 

The free space between the cork, a, and the level of 
the shot, i, is to be measured with a metre rod. The 
cork (a) must be removed for this purpose, and its 
thickness allowed for. A thermometer is now fitted 
through the cork (a\ Fig. 193) so that by inclining the 
tube the bulb may be completely surrounded by the 
shot. The temperature of the shot is to be taken ; 
then the thermometer is removed and the hole closed 

by a wooden plug. The tube is now inverted 

100 times in rapid succession. During each 

inversion the centre of the tube is held at a 

fixed height. The shot are kept at one end 

of the tube by centrifugal 

force until this end comes 

vertically over the other. 

Then the rotation should 

cease, so that the shot may fall through the 

^tm distance ab almost like a solid mass. Care 

^v must be taken, however, not to heat the shot 

J, through agitation which would result from too 

suddenly arresting the motion of the tube. 
The cork, c^ should be supported by a table or other 
solid object so as not to yield under the blow given 
to it by the shot. Under this condition only, the 
energy of motion of the shot will be converted into 
heat within the mass of shot. The temperature of 
the shot is again observed in the same manner as 
before. It should have risen 5 or 6 degrees. 

The experiment is now to be repeated with 1 kilo- 
gram of a substance in the form of shot, but of un- 
known specific heat ; for instance, an alloy of zinc 



f 178.1 MECHANICAL EQUIVALENTS. 391 

and lead. If this substance takes up more space 
than the lead, the distance fallen through in each 
reversal of the tube will not be quite so great. In 
this case more than 100 reversals may be made. The 
total distance fallen through should be as nearly as 
possible the same. Thus, if the distance ab is 100 
cm. in the case of the lead shot, and 98 cm. in the 
case of the alloy, the tube should be revei*sed 102 
times in the latter case, instead of 100 times. 

^ 178. Calcnlatioiui relating to Mechanical Bqtiiva- 
lents. — If « is the specific heat of the lead shot, w its 
weight in grams, g the weight of 1 gram in dynes, d 
the distance in cm, fallen through in each reversal, 
n the number of reversals, and J the mechanical 
equivalent of 1 unit of heat, then the total work 
done by gravity is evidently wg X nd ergs ; and the 
heat into which it is converted is (neglecting all 
corrections) W8t units, which is equivalent to Jwst 
ergs. We have, therefore, — 

Jw8t = wgnd ; 
whence 

It is interesting to compare .the value of J calcu- 
lated by this formula with that found by Joule (see 
^ 176, 1). On account of many large corrections 
which have not been considered, the result will 
probably be too great by some 20 or 30 per cent. 
The principal source of error usually lies in the 
cooling of the shot by contact with the sides of the 



892 WORK. [Exp. 70. 

pasteboard tube. This can be avoided by cooling 
the shot before the experiment to a temperature 
about 6° below that of the tube. Before repeating 
the experiment, the tube must be allowed to return 
to its original temperature. The remaining errors 
have been found in the long run to balance one an- 
other with a probable resultant of about 10 per cent., 
which may be positive or negative according to the 
manner in which the manipulations are performed. 
Instead of computing the mechanical equivalent of 
heat, we may calculate the specific heat of the lead 
shot by the formula — 

where J* may be taken as 41,660,000 ; and if we dis- 
tinguish by a prime (') the qualities of an unknown 
substance, we find similarly, — 






Dividing, we find 



/ n'd't , m'Xt 
- = — r-y, or « = — — -. 
% ndz naif 

In other words, the specific heats of two substances 
are to each other as tl\e distances through which they 
must severally fall in order that each may be raised 
1*^ in temperature. On account of the manner in 
which the two experiments are performed, the values 
of s and «' should be affected by constant errors in the 
same proportion, and hence the ratio between them 
will be affected only by accidental errors (§ 24). The 



T178.1 MECHANICAL EQUIVALENTS. 

last formula is therefore less iDaccurate than the pre- 
ceding formulae. To obtain the most accurate results 
by the aid of mechanical equivalents, as has been de- 
scribed, special devices should be employed to limit 
the fall of the shot to a given distance.. In the ab- 
sence of due precautions in this respect, the results 
must be expected to compare unfavorably with those 
obtained by the ordinary methods (see Exps. 38 and 
34). It is nevertheless considered desirable that a 
student should familiarize himself with a definite 
example of the conversion of work into heat. 



394 MAGNETISM. [Exf. 71. 



MAGNETIC MEASUREMENTS. 

EXPERIMENT LXXI. 
MAGNETIC POLES. 

^179. Determination of the Distance between the 
Poles of a Magnet. — Compound magnets composed of 
thin strips of steel bolted together will be found 

convenient for several experiments in 

: -— r^ magnetism. Such a magnet, formed of 

194 pieces of clockspring, 10 or 15cm, long, 

and 1 or 2cm. broad, is represented in 
Fig. 194. In fitting the strips together it may be 
necessary to soften them by heat ; but their temper 
must be restored (by again heating and suddenly 
cooling them) before they can be thoroughly mag- 
netized. Each strip should be magnetized separately 
by stroking one end of it ten times from the centre 
outward with or upon the south pole of a powerful 
electromagnet. This end will become a north pole 
(§ 126). The other end is then to be magnetized 
similarly by the north pole of the electromagnet. 
The strips are afterward bound together with 
all the north poles turned carefully in the same 
direction. 



IT 179.] MAGNETIC POLES. 895 

A piece of " ferroprussiate paper " ^ prepared for 
making "blue prints" is now to be stretched flat 
over a pane of window-glass, or over a stiff piece of 
pasteboard, with the sensitive surface uppermost, 
it is then to be placed over a powerful bar magnet 
constructed as has been described ; and a few iron- 
filings are to be scattered over it. When the paper 
is jarred the iron-filings will arrange themselves as 
in Fig. 195. The sensitive surface is now to be ex- 









Fig. lyot 

po^ed for about five minutes to direct sunlight, or to 
the light of the sky for a much longer period, until 
the surface not covered by the filings becomes quite 

^ To prepare ferroprussiate paper, take 1 gram citrate of iron and 
ammonia, 1 gram red prussiate of potash, pulverize together and dis- 
solve in 10 grams of water. This quantity should cover 20 or SO 
square decimetres of smooth (not porous) paper. It should be ap- 
plied by lamplight, as rapidly and evenly as possible, with a small 
sponge, in strokes first lengthwise then crosswise, then dried in the 
dark. The student is cautioned that all *' prussiates " are poisonous. 
Ferroprussiate paper, already prepared, may be bought of dealers in 
photographic apparatus. 



396 



MAGNETISM. 



[Exp. 71. 



blue. It is then to be placed in the shade, and the 
iron-filings removed. 

The surface covered by tlie iron-filings should not 
have been affected by the light ; hence the arrange- 
ment shown in Fig. 195 should be represented by a 
white tracing on a blue ground. To make the print 
permanent, it is necessary to soak it in water for 
about ten minutes, after which it may be dried in the 
sun. To avoid delay in waiting for the print to diy, 
the student is advised to defer this " fixing process " 
until the end of the experiment. In the meantime, 




Fig. 196. 

the print should be protected from excessive light 
either from the sun or from the sky. It may be illu- 
minated freely by lamplight or gaslight. 

The magnet is now to be placed over the print, di- 
rectly above its former position. A small compass 
with a needle not more than \cm. long, is to be put 
beside the magnet at different points in the print. 
The direction of the north pole of the compass-needle 
is to be indicated in each case by an arrow drawn in 
pencil upon the paper (see Fig. 196). The direction 
of the arrows should agree closely with the lines of 



f 179.] MAGNETIC POLES. 897 

iron-filings, although the compass-needle is in a 
slightly different plane. The results of this experi- 
ment will be somewhat affected by the earth's magne- 
tism. It is well, therefore, to note the direction («n) 
in which the compass points when the magnet is re- 
moved to a distance. 

A line AB is now drawn so as to bisect as nearly as 
possible the areas N and aS, from which the " lines 
of force " (§ 127) seem to diverge. The line (JLJ5) 
should agree with the general direction of the lines 
of force between iVand /S', whether indicated by the 
compass-needle or by the iron-filings. The areas N 
and iS'are again to be bisected by lines (Ci) and EF) 
perpendicular to AB. These lines should cut the edge 
of the areas (iVand S) at a point where the lines of 
force are also perpendicular to AB, 

The positions of the poles iVand S are determined 
by the intersection of the first line {AB) with the 
perpendiculars ((72> and EF,^ The distance between 
the poles is to be measured. The experiment is to 
be repeated with at least two other magnets as 
nearly as possible like the first. 

The student may be interested to make prints 
showing the arrangement of iron-filings due to two 
parallel magnets, both when their north poles are 
turned in the same direction and when turned in 
opposite directions.^ 

1 See Experiment 40 in the Elementary Physical Experiments 
published by Harvard University. 



398 



MAGNETISM. 



[Exp. 72. 



EXPERIMENT LXXII. 



MAGNETIC FORCES. 



^ 180. Determination of the Strengfth of Magnetic 
Poles. — One of the magnets (e/, Fig. 197), used 
in Experiment 71, is now to be placed horizontally 
in the pan-holder (c) of a balance (the pan being re- 
moved), and counterpoised by an observed weight 




Fig. m. 



in the opposite pan (a). A second magnet (^A) is 
to be placed directly under the first, and parallel 
to it. 

The north poles are at first to be turned in oppo- 
site directions, so that the magnets may attiact each 
other. Small blocks (b and d) are now placed be- 



tl80.] MAGNETIC FORCES. 899 

tween them to keep them apart. The thickness of 
the blocks should be such that when the balance 
beam is raised upon its knife-edges, the index (6) 
may point to zero. The weight in the pan a is then 
gradually increased until the magnets are pulled 
a|)art. Care must be taken to find the greate%t weight 
which the magnets can sustain ; for if .they be once 
separated a much smaller weight can hold them 
apart. In the final adjustment small weights (not 
over \cg^ should be let fall into the scale-pan from a 
height not exceeding \cm. The weight necessary to 
pull the magnets apart is to be noted. 

The magnet gh is now to be turned end for end, 
so as to repel e/, and the weight in the pan a is 
gradually to be diminished until the magnet ef just 
touches the blocks (h and c?). When a small weight 
is added to the pan a the beam will not turn sud- 
denly as in previous observations; but, being in 
stable equilibrium, it may balance in any position. 
Care must therefore be taken to find the smallest 
weight which can cause a separation of the magnets, 
however slight. 

The mean distance between the magnets, from 
centre to centre, is now to be determined by measur- 
ing the thickness of the magnets and the thickness 
of the blocks with a vernier gauge. In setting the 
gauge upon a magnet, if the jaws are of iron or steel 
the blocks of wood {b and d) should be interposed 
between the jaws and the surfaces of the magnet, 
since the strength of the magnet might otherwise be 



400 MAGNETISM. [Exp. 72. 

perceptibly affected. The thickness of the blocks 
may then be found and allowed for. The experiment 
should be repeated with a third magnet, let us say 
ij in place of gh ; then with gh in place of ef. In 
this way the forces of attraction and repulsion be- 
tween each pair which can be formed out of the three 
magnets will be determined. 

The student may be interested to prove that it 
makes no difference which of two magnets is the one 
suspended. This fact is an illustration of the gen- 
eral principle that action and reaction are equal 
and opposite. It will be noticed that the attrac- 
tion between two magnets when close together, 
is much greater than their repulsion. This is 
due to the effects of induction (see § 129, foot- 
note). 

^181. Calculatioiui relating to Magnetic Forces. — 
If w be the weight in grams necessary to counterpoise 
a magnet ; w^ the weight of the counterpoise neces- 
sary to lift the magnet and at the same time to pull 
it away from the attraction of a parallel magnet at 
the distance d ; and w^ the weight similarly required 
when the two magnets repel each other ; then if 1 
gram =^ dynes, the force of repulsion which we call 
positive is -+- (wg — w^g^ dynes, and the force of 
attraction, which we call negative, is — {w^ g — wg^ 
dynes. The numerical sum, or algebraic difference, 
J, between these forces is accordingly (w^g — w^g^ 
dynes. Substituting this value in the formula of 
§ 129, we have, if any two of the magnets are equal 



IT 181: ] MAGNETIC FORCES. 401 

in respect to the strengths (9 and «') of their poles^^ 
88^ = 9^ = -J- ; or « = 2 V(wi — w,) g. 

Thus if the attraction between two nearly equal 
magnets at a distance of 2 cm. is 6C0 dynes, and the 
repulsion 800 dynes, a force of 900 dynes (0.92 ^r., 
nearly) will be required to offset the effect of revers- 
ing one of the magnets, the mean strength of their 
poles is, accordingly, about J >v/.92 X 980, or 30 units 
each. 

The results of this experiment are subject to errors 
which are sometimes (though rarely) almost as great 
as the quantities measured. They are nevertheless 
valuable in enabling us to form an immediate e8timate 
of the strength of magnetic poles, which, though 
rough, may guide us in the less direct but more ac- 
curate methods which follow. 

^ If no two of the magnets are equal, we must form three equa- 
tions from observations made with each pair of magnets ; thus — 

s^ = ^ (1); "" = ^ (2); and ^,- = e__ (g). 

Multiplying (1) and (2) together and dividing bj (3) we have— 

^^rsr^ 1P^'*'"""2P' "A^ 



402 



MAGNETISM. 



[Exp. 73. 



EXPERIMENT LXXIII. 



MAGNETIC MOMENTS. 



^ 182. Determination of the Couple exerted by the 
Earth's Magnetism on a Suspended Magnet. — A mag- 
net (^gh^ Fig. 198) used in Experiment 72 is to be 
suspended horizontally by a wire cf. The coefficient 
cr^^????;^;^;^ of torsion of the wire has been 
^^^^^^^V found in Exp. 64. The wire is at- 
tached at c to a knitting-needle 
(6<i) revolving on a graduated cir- 
cle (ae) as in the torsion balance 
(Fig. 176, ^ 165). The wire is, 
however, vertical, and the circle 
horizontal in this experiment. A 
short piece of wire should be at- 
tached vertically by wax to each 
Fio. 198. g^^j Qf ^YiQ magnet to serve as a 

sight. The needle is first turned so that the north 
pole of the magnet points north, and its reading is 
taken. Then it is turned until the magnet points 
east, and the reading again taken. A distant object 
should now be sighted in the direction indicated by 
the sights. The needle is then turned so that the 
magnet points west. The same distant object should 
be in line with the sights. The reading of the needle 
is again observed. The experiment should be re- 
peated with the other magnets employed in Experi- 
ment 72. 



1 182] MAGNETIC MOMENTS. 403 

If the poles of the magnet are I centimetres apart, 
if they contain 8 units of magnetism each, and if 
the earth exerts on each unit of magnetism a force 
which has a horizontal component equal to ^ dynes, 
then the 8 units of magnetism in the north pole must 
be urged northward with a force of Hs dynes, and 
the south pole will be urged southward with an 
equal force. The two forces will constitute a couple* 
(§ 113) C, with an arm equal to the distance Z, be- 
tween the poles ; since the magnet is at right-angles 
to the forces in question. We have, therefore, 

0= mi, or e:=^ 

8L 

This couple must be balanced by an equal and 
opposite couple due to torsion in the wire. It is 
obvious that in turning the magnet end for end it 
must be made to revolve through 180® so as to make 
an angle of 90"* (on the average) with its original 
(north and south) direction. To produce torsion in 
the wire the needle must be turned through more 
than 180° in all, or more than 90° from its original 
setting. 

Let us suppose that the needle has revolved through 
a total angle a, or an average angle of ^ a from its 
original position ; if the magnet had remained point- 
ing to the north the twist in the wire would be | d ; 
but the revolution of the magnet through 90° causes 
the wire to untwist through 90° at its lower end. 
The angle of torsion is therefore \ a — 90®. It is 
now easj' to calculate the couple exerted by the 



404 MAGNETISM. [Exp. 73. 

earth. If it requires a couple of t dyne-centimetres 
to twist the wire through 1° (see Experiment 64) it 
must require {\a — 90) X t dyne-centimetres to 
twist it through the angle in question* Substituting 
this value for c in the formula above we have — 

^_ Ga-90)e 

It is interesting to estimate the value of H by the 
rough values of « and I already determined in Experi- 
ments 71 and 72. If, for instance, the distance be- 
tween the poles is 10 <?m., and the strength of each 30 
units, and if the couple produced is 60 dyne-centi- 
metres, then the earth must exert a force of \ of a 
dyne on each unit of magnetism when free to move 
only in a horizontal plane. This is what is meant by 
the statement that the ^^ horizontal intensity " of the 
earth's magnetism is \ or 0.17, nearly. In practice 
large errors would be committed in estimating the 
horizontal intensity in this way, on account of the 
uncertainty of the factor « (see ^ 181). A much 
more exact method will be considered in connection 
with Experiment 74. 

The student should note that the couples acting on 
suspended magnets are proportional to the products 
of the distance between the poles and the strength of 
the poles, both of which have been already deter- 
mined. These products («Z, «7', «"r) are called the 
magnetic moments of the magnets to which they re- 
spectively belong. 



T183.] 



THE MAGNETOMETEB. 



406 



EXPERIMENT LXXIV. 



MAGNETIC DEFLECTIONS. 



^ 183. Detennination of Magnetio Deflections by 
means of a Magnetometer. — A Surveying-Compass 
(Fig. 199) is placed in the middle of a wooden table, 
in the construction of which 
no iron has been employed 
even in the form of nails. All 
iron or steel objects are to be 
removed from the immediate 
neighborhood. The directions 
of the magnetic north, south, 
east, and west are to be deter- 
mined by this compass, and 
marked by pencil lines upon ^^^ ^^^• 

the table. In all experiments in magnetism the mag- 
netic points of the compass will be those referred to, 
unless otherwise stated. A magnet already tested in 
Experiment 71, considerably longer than the compass 
needle, is now placed at the east of the compass with 
its north pole toward the compass (see Fig. 200, 1). 
The distance of the magnet from the compass must 




j^ 



&d 



FlQ. 200. 

be noted. It should be small enough to cause a 
measurable deflection of the compass, let us say 5 or 



406 MAGNETIC DEFLECTIONS. [Exp. 74. 

10 degrees, but at least twice the length of the mag- 
net.^ The position of each end of the magnet is then 
marked in pencil on the table, and the deflection of 
the compass observed by the reading of two pointers, 
attached one to each end of the needle. 

The magnet is now turned end for end (as in Fig. 
200,' 2) and the deflection again observed. The ex- 
periment is to be repeated with the magnet at an 
equal distance from the compass, but at the west of 
it, as in Fig. 200, 3 and 4. There will thus be 8 read- 
ing in all, from which the average deflection of the 
needle may be calculated. The mean distance of the 
centre of the magnet from the centre of the needle 
may be found quite accurately by measuring the dis- 
tance between the outer and between the inner pencil 
marks on opposite sides of the needle, adding, and 
dividing by 4. The experiment is to be repeated 
with the other magnets employed in Experiment 71. 

The results of this experiment are to be reduced as 
will be explained in ^ 185. 

^ 184. Theory of the Magnetometer. — When a 
magnet is placed near a compass-needle, and at the 
east or west of it, as in Fig. 200, so that one of its 
poles is nearer than the other, the needle is deflected 
under the influence of the nearer pole. The lines of 
force due to a magnet at any point nearly in line with 
the two poles are (see Fig. 195) nearly parallel to 
the magnet ; and hence in the case which we have 

^ For very accurate measurenients the distance of the magnet 
from the compass should be at least 4 times the length of the magnet 
and 12 times the length of the needle. 



t 184.1 



THE MAGNETOMETER. 



407 



supposed they are nearly east and west. That is, 
the magnet tends to make the compass-needle point 
east and west. 

Let us suppose that the magnet is at the east of the 
compass, and that its south pole is (as in Fig. 200, 2) 
nearer than the north pole. Then the north pole of 
the compass-needle (<?, Fig. 201) will be attracted by 
the south pole of the magnet more than it is repelled 
by the north pole. The resultant force will there- 




A Z>*Jff<tiit9 fhfttm^ 







Fio. 201 



Fig. 202. 



fore be an attraction toward the east, which we will 
represent by the line cd (Fig. 201). At the same 
time the earth pulls the north pole of the compass- 
needle northward, with a force represented let us say 
by the line ca. The resultant of these two pulls is 
a force c6, easily found by geometrical construction 
(§ 105). 

On the other hand the south pole of the compass- 
needle (O will be repelled by the south pole of the 



408 MAGNETIC DEFLECTIONS. [Exp. 74. 

magnet more than it is attracted by the north pole. 
It will accordingly be urged westward with a force 
</df. At the same time it is drawn southward by the 
earth's magnetism with a force doC . The resultant 
force, <!h\ may be found as before. Assuming that 
the forces acting upon the south pole of the needle 
are equal and opposite to those acting upon the north 
pole, it follows that dV must be equal and opposite 
to ch. If the needle cd is free to turn, it will obvi- 
ously take the direction of the two resultants. 

The relation between the forces exerted by the 
earth and by the magnet upon the north pole of the 
compass-needle is shown in Fig. 202. The magnetic 
force is represented by ^S; the earth's force by 
by CA\ the resultant by GB. The angle BAQ is 
called the angle of deflection. The tangent of this 
angle is by definition equal to AB -7- CA; since AB 
and CA are at right-angles. Obviously, the magni- 
tude of a deflecting force bears to that of a directive 
force at right-angles to it a ratio equal to the tangent 
of the angle of deflection produced. 

It has been stated that when the two poles of a 
magnet are at unequal distances from a compass- 
needle, the nearer pole has the greater effect. Since 
the two poles are always equal and opposite, the ac* 
tion of a magnet as a whole evidently depends not 
only upon the strength of its poles, but also upon the 
difference of their distances from a given point. We 
must accordingly consider the length of a magnet, as 
well as the strength of its poles, in calculating the effect 
which it will produce. It is found, in fact, that the 



IF 185.] THE MAGNETOMETER. 409 

forces produced by different magnets at a given dis- 
tance are very nearly proportional to the " moments " 
of the magnets in question, that is (see ^ 182), to the 
products of the strength of the poles and the distance 
between them. The moments of the magnets («/, 9fl\ 
etc.) employed in this experiment have been ab*eady 
determined (^ 182). If a, a', etc., are the deflections 
produced, we should have — 

r =^ — ^, etc., nearly. 

tan a tan a *" 

The student should satisfy himself that this is the 
case before proceeding to the calculations of the next 
section. 

A compass, having on each side of it a pair of re- 
volving supports, capable of holding several magnets, 
successively at a given distance from the needle, 
affords one of the most direct and accurate methods 
of comparing magnetic moments together, and is 
properly called a magnetometer. 

^ 185. CaloulationB relating to Magnetic Deflections. 
— Example. Let us suppose that in Fig. 200 the 
average distance between the centre of the magnet 
NS and the centre of the needle n« is 25 cw., and 
that the distance between the poles of the magnet 
(^ 179) is 10 cm. so that as in (2) the south pole is 
20 cm. from the needle and the north pole 30 cm. from 
it. Assuming that each pole has a strength of 80 
units (see ^ 181) the attraction of the south pole for 
a unit of positive magnetism at the centre of the needle 
(see § 129) must be 80 -*- (20)2 q^ ^ dyne. The 



410 MAGNETIC DEFLECTIONS. [Exp. 74. 

opposite pole must exert a repulsion on the same unit 
of magnetism equal to 30 -f- (30)^ or ^ dyne. The 
resultant of these two forces is evidently -^ — ^ o^* 
^ dyne acting in an easterly direction parallel to 
AB (Fig. 202). The earth's magnetism acts in a 
northerly direction parallel to CA (Fig. 202). 

AB 

Now since --^ = tan CAB. 

CA 

we have CA = — ^, _ 

tan CAB 

If, for example, CAB = 14°, the tangent of CAB is 
.249 (see Table 5) or J, nearly ; then CA is evidently 
4 times as great as AB; hence if .^6 = ^ dyne 
per unit of magnetism, CA = J dyne per unit of 
magnetism. 

In practice an estimate of the earth's magnetism 
made in this way will be found, to differ greatly from 
that made as in the last experiment, on account of a 
tendency to underestimate the strength of the mag- 
netic poles in Experiment 71. 

Let us suppose that this strength were estimated at 
15 units instead of 30 units. Then in the calculation 
above we should have estimated the earth's field at 
^ dyne per unit of magnetism (instead of J). In 
^ 182, however, we should have estimated the earth's 
field at J dyne per unit of magnetism. That is, our 
estimate in Experiment 73 would be too great, and 
that in Experiment 74 too small in proportion to the 
error originally made in estimating the strength of 
the poles. Now when one of two estimates is too 



1186] DISTRIBUTION OF MAGNETISM. 411 

great, and the other too small in a given proportion, 
the geometric mean between them must be equal to 
the quantity which we seek. Hence to find the true 
value of the horizontal component of the earth's mag- 
netism, we multiply together the estimate of Experi- 
ments 73 and 74, and extract the square root of the 
result. Thus v^i ^ i^ =^ h '^^^® result is inde- 
pendent of the value provisionally adopted for the 
strength of the magnetic poles. If the two esti- 
mates agree closely the arithmetic mean may be sub- 
stituted for the geometric mean (§ 57). 

Knowing now the true value of J?", we may re- 
calculate the moment (ilf ) of the magnet and the 
strength of the poles by formulae derived from ^ 182 : 

H HL 



EXPERIMENT LXXV. 

DISTRIBUTION OP MAGNETISM, I. 

^ 186. Determination of the Distribution of Magne- 
tism on a Rod by the Method of Vibrations. — A steel 
rod {aj^ Fig. 203) one metre long, and about 1 cm. in 

^ 'S 1 2 11 g 1 2 1 V ^ 

Fig. 203. 

diameter, is marked with a file at ten points (a . . ./) 
10 cm. apart, beginning with a point a, 5 cm. from one 



412 



MAGNETIC VIBRATIONS. 



[Exp. 75. 






Q 



Fig 204. 



end of the rod. It is then magnetized by stroking it 
from e to a 10 times with the south pole of a power- 
fm m . ful electro-magnet, and by stroking 
r it 10 times from/ toy with the north 

pole of this magnet. A small piece 
of a sewing-needle (/, Fig. 204) 
about 1 cm. long, and highly magne- 
tized is attached horizontally by seal- 
ing-wax to a bullet e, and suspended 
by a fine fibre (erf) of untwisted silk 
from a cord (a) in a test tube (^^). 

The torsion of the fibre (cd) should be so slight 
that the cork (a) may be twisted through 860 °, with- 
out deflecting the needle (/) more than a few de- 
grees from the magnetic north, toward which one end 
should point. The needle is then to be deflected by 
a magnet ; and when the magnet is suddenly taken 
away the needle should make a series of vibrations 
in a horizontal plane. The weight of the bullet 
should be so proportioned to the magnetic strength 
of the needle that there may be about 10 vibrations 
completed in one minute. The exact time required 
for 10 vibrations of the needle is to be determined 
when it is vibrating in an arc not exceeding 30 ° or 
40° (see Table 3, ^). The north pole of the needle 
should be distinctly marked. 

The test tube is now to be placed opposite the end 
of the rod, then held successively on each side of each 
of the ten points (a — j\ Fig. 203). The direction 
indicated by the north pole in each position is to be 
represented by arrows (drawn as in Fig. 203) the 



IT 186.] 



DISTRIBUTION OF MAGNETISM. 



413 



direction of which may be compared with that of tlie 
lines of force issuing close to the magnet in Fig. 196. 
In addition, the rate of vibrations of the needle is to 
be determined by counting the number of vibrations 
completed in 1 minute, or in whatever time may 
have been required for 10 vibrations under the influ- 
ence of the earth's magnetism alone. In all cases the 
arc of vibration should be limited to C0° or 40° (see 
Table 3, g). 

The number of vibrations made in the given time 
on one side of a is to be averaged with that made on 
the other side ; and in the same way the average 
number of vibrations for each of the ten points is to 
be found. These numbers are 
then all to be squared (see 
Table 2). The results are to 
be plotted on co-ordinate pa- 
per (see § 59). Distances in 
centimetres are represented 
by a horizontal scale at the 
top of the figure, and the 
square of the number of oscil- 
lations is shown by the verti- Fig. 20.5. 
cal scale at the left of the figure. Thus, if opposite 
the point 6, 15 cm, from the end of the magnet, the 
needle makes 60 vibrations per minute, we place a 
cross at the right of the square of 60 (8600) and 
under 15 cm. The vertical distances are measured 
upward if the north pole of the needle is repelled by 
the bar, and downward if it is attracted by it. In the 
same way other points may be found through which 





*^ 2^^ ^ 


^"^ ITT t 


^« '^ ' J 




™" :'4C t 


™ . ,-^ 




fk ^ i 1; 


-:::::::!s:t:S 




^ — ^- 


i* \ltz 


' itz 


- _ . £^ 



414 MAGNETO-ELECTRIC INDUCTION. [Exp. 76. 

a curve is to be drawn as in Fig. 206. Evidently, in 
this figure, N represents the " positive " or " north " 
end of the magnet. 

This method of representing the distribution of 
magnetism depends upon the general principle that 
forces are proportional to the squares of the rates of 
oscillation which they produce (see § 110). The 
curve represents accordingly the strength of the mag- 
net at different points as compared with the strength 
of the earth's magnetism. We should strictly allow 
for the effect of the earth on all the rates of oscilla- 
tion ; but as it is represented only by 100 units 
on the vertical scale, this effect would be hardly 
perceptible.^ 

The student should draw by the eye two vertical 
lines iV3r' and SS\ dividing each area enclosed by 
the curve as nearly as possible into two equal parts. 
The distance between these lines indicates approxi- 
mately the distance between the poles of the mag- 
nets. This latter may therefore be found by the scale 
at the top of the paper. 



EXPERIMENT LXXVI. 

DISTRIBUTION OP MAGNETISM, II. 

^ 187. Magneto-Electric Induction. We have seen 
that when iron-filings are brought into the neighbor- 

1 The effects of '* induced magnetism " may introduce errors of 
6 or 10 per cent in this experiment (see 1 207). The shape of the 
curve in Fig. 208 will not, however, be materially altered. 



t 187.] DISTRIBUTION OF MAGNETISM. 415 

hood of a powerful magnet, they tend to' arrange 
themselves along certain lines called " lines of force." 
These lines of force are not, like the meridians upon 
the surface of the globe, purely geometrical concep- 
tions. According to Tyndall, the apparently empty 
space between the poles of a powerful electro-magnet 
" cuts like cheese." The most surprising fact con- 
nected with this phenomenon is that a knife with 
which such a magnetic field is cut becomes tempora- 
rily electrified. The point and the handle of the 
knife resemble, for the time being, the two poles of a 
voltaic cell, from which a current of electricity can be 
derived by making the proper connections. It is not 
necessary to use a knife ; any piece of metal, a wire 
for instance, will do as well. All tendency to pro- 
duce a current ceases when the knife or wire stops 
moving, or as soon as all tlie lines of force have been 
cut. The effect of a sudden motion upon a galvan- 
ometer may accordingly be almost instantaneous. In 
such cases it is measured by the " throw " of the 
needle (§ 109). It is found that the " throw " is 
proportional, other things being equal, to the inten- 
sity and extent of that part of the magnetic field 
which has been cut through, or, according to a system 
of representation universally adopted, it is propor- 
tional to the number of lines of force which have been 
cut. 

If a loop of wire is placed around the middle of a 
long bar-magnet (Fig. 206) and suddenly made to 
slip off one end of the magnet, it will evidently cut 
nearly all the lines of force on that end of the magnet. 



416 MAGNETO-ELECTRIC INDUCTION. [Exp. 76. 

A delicate galyanometer connected with the ends 
of the loop will be affected. This affords a conven- 
ient method of comparing the strengths of different 
magnetic poles. In practice we employ a coil of wire 
instead of a simple loop ; for when each turn cuts all 
the lines of force, the effect is found to be propor- 
tional to the number of turns which the wire makes 
about the magnet. It is not necessary to slide the 
coil completely off the magnet. A motion of a few 
centimetres may affect the galvanometer. When the 
motion is confined to one end of the magnet it will 
be found to deflect the needle in opposite ways ac- 
cording to which way the coil is moved. In other 
words the direction of the electrical current depends 

Fig. 206. 

upon the direction of the motion. Let us suppose 
the direction of the motion to be always the same, 
that is, from left to right, or from the north toward 
the south end of the magnet. Then the galvan- 
ometer will be deflected one way when the motion 
of the coil takes place near one end of the magnet, 
and the other way when it takes place near the other 
end of the magnet. That is, the direction of the 
electrical current depends on the direction of the 
lines of force. Near the middle of the magnet a 
neutral point will generally be found. If the coil be 
moved from this neutral point toward either end of 
the magnet, it follows from the statements made 



1 187.] THBOW OF A NEEDLB. 417 

above that the direction of the current will always 
be the same. This direction is with the hands of a 
watch, as seen from the south pole of the magnet. 

The throw of the needle is proportional, other 
things being equal, to the distance through which 
the coil is moved ; hence it is important in comparing 
results that this distance should be always the same. 
If the coil is moved always through a given distance, 
the effect will be found to be greatest when the mo- 
tion takes place near the ends of the magnet, where 
the lines of force are the thickest. In other words 
the magnitude of the electrical current depends upon 
the closeness of the lines of force. The effect is very 
nearly the same whether the coil moves more or leBi 
iwiftly ^ through a given distance. In the first case 
we have a rapid motion, and hence a comparatively 
strong current lasting for a short time ; in the second 
case we have a weaker current lasting for a propor- 
tionately long time. The forces exerted upon the gal- 
vanometer needle are proportional to the current ; 
hence, by the fundamental law of motion (§ 106), 

ft = 7WV, 

since the product (//) of the force and the time of 
its action is the same in both cases, the momentum 
given to the needle must be the same. 

We shall make use of these facts to estimate the 
relative strength of the magnetism of a rod in differ- 

^ In order that this may be true, the duration of the motion mu8t 
be several times less than the time occupied by one vibration of the 
galvanometer needle. 

27 



418 



MAGNETO-ELECTIC INDUCTION. [Exp. 76. 



ent parts, and to distinguish positive from negative 
magnetism. 

^ 188. Constmction of an Astatic Qalvanometer. — 
A delicate galvanometer, such as has been already 
employed for the detection of currents created by a 
thermopile (Exp. 39), is repre- 
sented in Fig. 207, and may be 
constructed as follows : — 

Two magnetized needles, c and 
A (Fig. 208), of nearly, equal 
strength are connected by a ver- 
tical piece of wire, with their 
north poles in opposite directions^ 
and suspended horizontally, by a 
fine thread {be) of untwisted 
silk, from a screw a. This screw 
Fio. 207. is held by a nut 5, itself capable 

of rotation, so that the thread may be raised or 
twisted at pleasure. The two needles c and h should 




(^ 



.*-^tR 





Fig. 208. 



form a nearly " astatic " combination (a privative and 
iarrjfiCy to stand) ; that is, one which, owing to the 



ir 188.] ASTATIC NEEDLE GALVANOMETER. 419 

equal and opposite forces exerted upon it by the earth, 
has no strong tendency to stand in any particular 
position. 

The strength of either magnet may generally be 
increased by stroking one of the poles, as in ^ 179, 
with the dissimilar pole of a powerful magnet, or di- 
minished by touching similar poles together. A very 
light touch is usually sufficient to produce a percep- 
tible change in a magnet. The delicacy of the in- 
strument depends upon the delicacy of the balance 
which can be established between the two needles. 
It is generally possible to make the combination point 
permanently east and west. In practice, however, 
the needles are magnetized so that the time occu- 
pied by one oscillation is 5 or 10 times as great as 
that of either needle by itself. The needle is then 
sufficiently astatic for most purposes. It may 
be remarked that the rate of oscillation of an 
astatic needle is the best test of its adjustment (see 
f 193, 4). 

100 metres of insulated copper wire about } mm. 
in diameter are now to be wound on the two rectan- 
gular bobbins/ and i (Fig. 208, 1 and 2) .i The bob- 
bins are shaped so that the lower needle (A) may hang 
inside of them, and the upper needle (c) just above 

1 If it is desired to use the instrument later on (Exp. 86, II. and 
Exp. 96) as a differential galvanometer, the 100 metres of wire 
should be cut in two, and the two parts twisted together before wind- 
ing them on the bobbins. The galvanometer will thus have four ter- 
minals instead of two. If two of the terminals are temporarily joined 
together, the other two may be connected with binding-posts in the 
ordinary manner. 



420 MAGNETO-ELECTRIC INDUCTION. [Exp. 76. 

them. Two indices of aluminum wire, d and e (Fig. 
208, 1 and 3), are then attached to the upper needle, 
and a cardboard protractor (/) is set beneath them. 
The instrument is usually mounted on w^ooden sup- 
ports, with levelling screws k and Z, and covered with 
a glass shade to cut off currents of air. The galvan- 
ometer thus constructed should be sensitive to a few 
millionths of an ampere. 

^ 189. Determination of the Distribution of Magnet- 
ism on a Rod by the Method of Induction. — A coil 
(6, Fig. 209) consisting of about 100 turns of No. 20 
insulated copper wire, wound on a brass bobbin, is 
fitted to a brass tube ad so as to slide freely between 




Fig. 209. 

the stops a and e, through a distance of about 10 cen- 
timetres. The tube must be large enough to admit 
the long magnet employed in Experiment 75. It is 
first to be fastened near one end of this magnet by 
means of the clamp rf, so that a point (a. Fig. 203) 
6 cm. from the end of the magnet may come half-way 
between the stops a and c (Fig. 209). 

The needle of a delicate galvanometer (Pig. 207), 
such as has been already employed for the detection 
of electrical currents (Exp. 89), is now to be loaded, 
if necessary, by att ching small bits of lead with 
sealing-wax to each end of the needle, so that its 
time of oscillation may be at least 10 seconds. The 



1189.] BALLISTIC GALVANOMETER. 421 

instrument is to be set up with the plane of its coils 
approximately north and south. The nut I is then 
turned so that, by the torsion of the thread bc^ the 
needle of the galvanometer is made to point to 0°. 
The terminals of the coil b (Fig. 209), are then to 
be connected with the terminals of the galvanometer. 

The coil (6) is then suddenly made to slide from a 
to c (Fig. 209), and the throw of the galvanometer 
is noted. When the oscillation of the needle has 
ceased ^ the coil is made to slide back suddenly from 
c to a, and the throw of the galvanometer is again 
noted. 

The experiment is to be repeated with the tube 
clamped so that other points (6, (?, d, e, etc.. Fig. 208) 
may come successively half-way between the stops a 
and c (Fig. 209). 

In each case two throws of the galvanometer are 
to be observed. The direction of each throw is to be 
noted, and the average deflection calculated. 

The positions of the centre of the tube with re- 
spect to the magnet are also to be noted. The results 
are to be plotted on co-ordinate paper as in Fig. 205, 

1 The student should learn to stop the yibrations of a magnetic 
needle. If a magnet is directed toward a needle as in Fig. 200, ^ 183, 
a deflection in either direction may be produced. If the magnet be 
turned so as to tend to cause a deflection at every instant opposite 
to the motions of the needle, the latter will come very quickly to rest 
To stop a wide oscillation, the magnet must be brought near the 
needle, but when the oscillation becomes feeble, the process should 
be continued from a greater distance. To affect an ordinary astatic 
needle, the magnet should be held not only at right-angles with it, but 
also considerably above or below it. A perfectly astatic needle 
should not be affected by a magnet in the same horizontal plane. 



422 MAGNETIC DIP. [Exp. 77. 

^ 186, except that the vertical distances are to rep- 
resent throws ^ of the galvanometer needle, instead of 
squares of the rates of oscillation. If the throw in 
a given case is in the same direction as at the north 
end of the magnet when the coil is stopped in a given 
direction, the distances are to be measured upward ; 
otherwise downward. From the curves thus ob- 
tained the poles of the magnet are to be located as 
in ^ 186, and the distance between them is to be 
estimated. The result should agree closely with that 
obtained in the last experiment. 



EXPERIMENT LXXVII. 

MAGNETIC DIP. 

^ 190. The Earth^B BCagnetism. — If fine iron-filings 
are sprinkled over a horizontal pane of glass, they 
will show a slight tendency to arrange themselves in 
lines parallel to the magnetic meridian, particularly 
if the glass be jarred. One might infer that the lines 
of force due to the earth's magnetism are horizontal. 
This is not, however, the case; the direction in 
which the lines are inclined is from north to south, 
according to the compass, but the lines make any 
angle with the horizon (§ 128) ; 70*^ or 80** for instance 
in the United States. We have already made use of 

^ If the throws exceed 80^ the student should plot the chords of the 
angles in question (Table 8), instead of the angles themselves (see 
5109). 



t 190.] 



THE DIPPING NEEDLE. 



423 




the surveying-compass to find the magnetic meridian 
(^ 183). The compass affords, however, little or no 
idea of the angle which the lines of force make with 
the horizon, because a com- 
pass-needle is suspended so 
as to move approximately 
in a horizontal plane.^ To 
find the magnetic dip (§ 
128), we may make use of 
an instrument known as 
the " dipping-needle." A , 
simple form of this instru- 
ment consists of a knitting- 
needle ad (Fig. 210), with 
an axis be soldered to it a 
right-angles and resting on two glass surfaces b and c?, 
attached by sealing-wax to wooden supports (be and 
<?f ), and made horizontal by means of a spirit level. 

In practice the needle must be balanced by bend- 
ing the axis ic?, or by adding bits of sealing-wax or 
solder to it, so that it will stay, when unmagnetized, in 
any position, as ad. Then the needle is magnetized by 
stroking the end a ten times from the centre outward 
with the north pole of a powerful magnet, and by 
stroking the end d similarly with the south pole of 
the magnet. The needle will no longer balance in 
any position ; but the north pole will, in north lati- 



Fio. 210. 



1 The needles of surveying compasses intended for use in widely 
different latitudes are frequently provided with a small sliding 
weight by which variations in the magnetic dip and intensity may be 
counterpoised. 



424 MAGNETIC DIP. [Exp. 77. 

tudes, dip downward as in Fig. 210. To measure 
the angle of the dip, a cardboard protractor, cut out 
at the centre so as not to interfere with the axis of 
the needle bcj is attached vertically to one of the 
wooden "supports (6«), and turned round so as to be 
north and south according to the compass. The axis 
be is made to point horizontally east and west, and to 
coincide a^ nearly as possible with the axis of the 
graduated circle. The mean reading of the two 
ends (a and d) of the needle should then give cor- 
rectly the angle of the dip. Errors of parallax 
must of course be guarded against (§ 25). Various 
other sources of error may be eliminated by a series 
of experiments. In some of these the axis be should 
be turned end for end, in some the whole instrument 
should be turned end for end, and in some the mag- 
netism of the needle should be reversed by stroking 
the end d upon the north pole, and the end a upon 
the south pole of a magnet. By averaging the va- 
rious results, the angle of the magnetic dip may be 
determined within a few degrees. 

^191. The Earth Inductor. — If a hoUow square of 
wire CDEF is laid upon the floor with the side 
CD magnetically east and west, and rotated about 
CD as an axis into the position ABCD^ it is evident 
that the wire EF must cut all the lines of force due 
to the earth's magnetism which pass through the 
areas ABCD and CDEF, The line CD will cut no 
lines of force, because it is stationary ; and the wires 
CE and DF will cut none, because their motion is 
in a plane parallel to the lines in question. AH 



H 191.] 



THE EARTH INDUCTOR. 



425 



the lines cut will therefore be included in the area 
ABUF. 

If the square is now held against the west wall of 
the room, in the position C'VEF, and rotated as be- 
fore about an axis (^C'l/^ perpendicular to the lines 
of force, into the position A!SG'iy^ the number of 
lines cut will be as before included in the area 
AB'E'Fi and similarly if the square is rotated 
about an axis C"D'\ in the north wall of the room 




Fig 211. 

perpendicular to the lines of force, the lines cut 
will all be included in the area A''B"E"F". Now 
the areas iABEF, A!BEF, M'B'E'F") are all 
equal, — each being twice the area included by the 
square. If, therefore, we connect the terminals of 
the square with a galvanometer, and observe the 
throws of the needle which take place when the 
square is suddenly turned over, we shall have a 
means of comparing the relative numbers of the liaes 



426 MAGNETIC DIP. [Exp. 77. 

of force which pass through the square in its three 
different positions. 

From these data we may infer the direction of the 
lines of mi^netic force. If, for instance, the throw 
of the needle is much greater when the square is 
tiu*ned over on the north wall of the room than on 
the west wall, we may infer that more lines of force 
pass through the square in the former position ; and 
that, accordingly, these lines are more northerly than 
westerly. If, again, the throw is much greater when 




Fig. 212. 

the square is turned over on the floor than on either 
wall, we may infer that the lines of force are more 
nearly vertical than horizontal. We will suppose, for 
simplicity, that the walls of the room face exactly 
north and west by the compass, so that no lines of 
force pass through the loop when held against the 
west wall of the room. 

Let A BUD and AB^ED (Fig. 212) represent re- 
spectively the square in its horizontal and in its ver- 



1 191.] THE EARTH INDUCTOR. 427 

tical position, AD being magnetically east and west ; 
let the plane ADF'FCO^A be drawn perpendicular 
to the lines of force, and the planes BEFQ and B'E' 
F'C parallel to the lines. Then the areas ABIC 
dkxAABF'C include respectively the lines which pass 
through the square in its two positions. Since the 
lines are equally spaced, their numbers are as the 
areas which include them. These areas are to each 
other as ACiAC\ or since by construction B0=^ 
AC\ they are to each other sls AC: BO. This ratio 
(A (7: 5(7) is by definition the tangent of the angle 
ABC^ which measures the magnetic dip. 

Now if a' is the angle through which the needle is 
thrown when a loop of wire is turned over on the 
floor, and if a'' is the same for the north wall of a 
room, the impulses given to the needle are to each 
other as the chord of a' is to the chord of a'^ (see 
§ 109), or approximately as a' is to a'\ It follows that 
the angle of the dip a is given by the formula — 

- chord a' a' , 

tan a = — — — -. = —. nearly. 
chord a" a" -^ 

The same proportion will be found to hold for a 
round loop of wire. In practice we employ a coil of 
wire, containing, let us say 100 turns, since the effect 
upon the galvanometer increases with the number of 
turns. 

The student should note that a sliding motion 
given to such a coil either along the floor or along 
the wall causes no deflection of the galvanometer. 
This is because the lines of force are cut by the two 



428 



MAGNETIC DIP. 



[Exp. 77. 



halves of the coil in opposite ways. . It will be found 
to make no difference whether the coil is rotated 
about an axis passing through its centre, or on one 
side of it. We need to consider only the angle 
through which rotation has taken place. A coil ca- 
pable of being thus rotated 180° about a horizontal 
and about a vertical axis constitutes what is called 
an " earth inductor," because of the currents of elec- 
tricity which by the action of the earth's magnetism, 
may be " induced " in it. 



h. ilp'T ;:-:5:;:; -::::::: ::-:^v.-: 

i - ]r"' 

h w ]/ " 



ir:-\ 



a'i}::::::iiv:":-:" 




Fio. 213. 



^ 192. Determination of the Magnetic Dip by means 
of an Earth Inductor. — A convenient form of earth 
inductor is represented in Fig. 213.^ It consists of a 
coil of wire A, mounted on a wooden axle diy with a 
head i, through which the coil may be set in rotation 

1 The instrument may he greatly simplified if it is Intended only 
to be turned by hand. This generally requires the co-operation of 
two students, one to turn the earth mductor properly, the other to 
observe the4hrows of the galvanometer. 



1 192.] THE EARTH INDUCTOR. 439 

by the spring cbd. An auxiliary spring* ai may also 
be employed to hasten the rotation through the first 
right-angle, and to slacken it in the second right- 
angle, so that the coil may be arrested by the catch 
/, when it has rotated through exactly ISO"*. By 
winding the spring abd round the head of the axle in 
the other direction, the coil may be made to return 
to its original position. The apparatus is per- 
manently attached to the floor by means of two 
hinges y and A, the axes of which are east and west. 
If the coil is properly counterpoised, it will operate 
also when the whole instrument is tipped on its side, 
as represented by the dotted lines in Fig. 213. 

Wedges are to be placed beneath the frame so that 
the axis of the coil may be exactly vertical in one 
position, and exactly horizontal in the other position. 
The catch /must be adjusted if necessary, so that the 
coil may be horizontal in the second position. If the 
hinges are properly placed the plane of the coil will 
be at right-angles to the magnetic meridian in both 
positions. 

The axis of the coil is first to be made horizontal, 
and the terminals of the coil are to be connected (see 
^ 193, 11) with a galvanometer (Fig. 207, ^ 188), 
placed at a considerable distance from the earth in- 
ductor so as to avoid jarring, and adjusted as in ^ 189. 
The catch/ is then to be lifted by pulling a string 
attached at g. The throw of the needle is to be noted. 
When the needle has come to rest (see ^ 189, foot- 
note) the coil is made to return suddenly to its origi- 
nal position by the same mechanism. The throw of 



430 MAGNETIC DIP. [Exy. 77. 

the needle is again observed, and the mean throw 
(a^) calculated. 

The experiment is to be repeated with the axis of 
the coil vertical. The mean throw (a'') is to be 
found. The angle of the dip (a) is then to be calcu- 
lated by the formula (see ^ 191), 

a' 
tan a = ---, nearly. 



1[ 193.J GENERAL PRECAUTIONS. 431 



ELECTRICAL MEASUREMENTS. 

CURRENT STRENGTH. 

^ 193. General Precautions in the Measurement of 
Electric Currents. — Nearl}*^ all measurements of elec- 
tric currents involye the use of galvanometers de- 
pending upon the deflection of a magnetic needle. 
The same precautions must accordingly be observed in 
electrical as in magnetic measurements. 

(1) Delicacy of Suspensions. A needle weigh- 
ing less than 10 grams may be safely suspended by a 
single fibre of the best cocoon silk. When several 
fibres are employed they should be fastened together 
with wax, but not twisted together. If great delicacy 
is desired, the finest possible thread should be 
employed. 

When a needle is hung on a pivot, as in an ordi- 
nary compass, great care must be taken to preserve 
the sharpness of the steel point upon which it turns. 
A lever should be arranged so as to lift the needle 
from the pivot when the instrument is not in use ; 
and when in use, care should be taken not to jar the 
compass. A slight jariing may be used as a last re- 
sort to relieve the friction between the needle and its 
pivot when the latter has been already dulled. It is 
preferable, when possible, to observe the turning- 
points of the needle while oscillating in a small arc. 



432 ELECTRICAL MEASUREMENTS. 

and from these to infer its position of equilibrium 
(see ^ 20). 

(2) Preservation of Magnetism. The needle 
of a galvanometer should be carefully protected from 
strong magnetic forces, whether due to permanent 
magnets or to electric currents, since such forces 
are apt to affect the magnetism of the needle. This 
precaution is especially important in the case of 
" astatic " needles (^ 188), since the slightest change 
in either of the two parts of which such needles are 
composed may completely destroy the balance be- 
tween them, and thus seriously injure the delicacy of 
the combination. 

Strong currents should never be sent through deli- 
cate galvanometers. The terminals of such gal- 
vanometers (a and 6, Fig. 214) should be 
joined together with a wire or "shunt" 
({?), forming a cross-connection between 
the wires (d and e) which convey the cur- 
rent to and from the galvanometer. An 
Fig. 214. electric current of unknown strength should 
be first tested by the galvanometer with the shunt. If 
the galvanometer shows little or no deflection, the 
shunt may be safely removed. 

(3) Magnetic Surroundings. All iron, steel, or 
other magnetic substances should be removed, if pos- 
sible, from the neighborhood in which magnetic 
measurements are to be performed. The positions of 
magnetic bodies which cannot be moved should be 
accurately noted. Especial care must be taken to 
guard against changes in the position of magnetic 




t 193.] GENERAL PRECAUTIONS. 433 

bodies in a course of experimeDts.^ The position of a 
galvanometer should be accurately located, since con- 
siderable variations, both in the direction and in the 
strength of the earth's magnetism, often occur in 
different parts of the same building, unless special 
care has been taken to avoid the use of iron in its 
construction. When there is no simpler way of de- 
scribing the place of an instrument, its distances may 
be found from the floor and from two walls of the 
room. 

(4) Rate of Oscillation. Any change in the 
strength of the magnetic forces acting upon a needle, 
in the magnetism of the needle itself, or in the free- 
dom of its suspension will be found to affect its rate 
of oscillation. It is well, therefore, to determine this 
rate before and after every experiment in which such 
changes are likely to occur. This precaution is par- 
ticularly important in the case of astatic needles and 
in the method of vibrations (Exp. 82). 

(5) ExcENTRiciTY. When a compass-needle is 
suspended at a point not exactly in the centre of the 
graduated circle by which its position is determined, 
errors due to " excentricity " may be introduced. Such 
errors are avoided by reading both ends of the needle. 

(6) Zero-Reading. A galvanometer is always to 
be adjusted (except in the method of vibrations, Exp. 
82) with the plane of its coil vertical, and parallel to 
the needle in its zero position, — that is, the position 
which the needle takes when no current is flowing 

^ Students should be cautioned against carrying small objects 
made of iron or steel about their person. 

28 



434 ELECTRICAL MEASUREMENTS. 

through the coil. In the case of a galvanometer pro- 
vided with an ordinary compass-needle, the plane of 
the coil is accordingly to be made parallel to the mag- 
netic meridian. In this position the reading of the 
needle should be zero. It is well to make sure (§ 32) 
that the zero-reading is not disturbed in the course 
of an experiment, either by dislocation of the galvan- 
ometer or by changes in the position of magnetic 
bodies in the vicinity (see 3). 

(7) Mutual Induction. To prevent the coils 
of one instrument from affecting the needle of an- 
other instrument, these instruments should be sepa- 
rated as widely as may be practicable. In certain 





Fig. 215. 

delicate experiments the effects of magnetism pro- 
duced in one building are measured by electrical 
wires carried to an entirely separate building. Coils 
of wire are in general made horizontal if possible ; 
magnets vertical ; since in these positions minimum 
magnetic effects are usually produced on galvanome- 
ters in their vicinity. 

(8) Connecting Wires. The wires conveying 
an electric current to and from an instrument should 
be parallel and close together, so that the equal and 
opposite currents in these wires may neutralize each 
other as far as magnetic effects are concerned. A 
typical case is represented in Fig. 215, where by the 
parallel wires 6c, de^ and a/, a battery B is connected 



1 193.] 



GENERAL PRECAUTIONS. 



435 



through a rheostat R with a galvanometer G (see 
Exp. 92). It will be found convenient in practice to 
twist the wires together. In rheostats the wires are 
Avound double (see Fig. 240, Exp. 86) to avoid mag- 
netic effects. 

(9) Reversal of Currents (§ 44). Every in- 
strument capable of being affected by magnetic influ- 
ences from outside should be provided with means of 
reversing the current through it, without changing 
its direction in other parts of the circuit. Any such 
instrument is called a " commutator." A convenient 
form of " commutator " is represented in Fig. 216.^ 




Fio. 216. 



(10.) Waste of Power. The commutator may 
be made also to serve as a "key," — that is, to cut off 

1 Tins commutator consists of a square block of mahogany or 
ebonite, with four holes abed (Fig. 216) bored half-way through it. 
The screws of four binding-posts are driven horizontally into these 
holes, which are then filled with mercury. Two copper rods (Fig. 
216, 3), bound together by a handle of mahogany or ebonite, are bent 
so as to reach respectively either from a to 6 and from c to rf, or from 
a to c and from hiod (see Figs. 216, 2 and 4). • The wires (^4 and B) 
from the positive and negative poles of a battery are connected with 
two opposite mercury cups, as a and d\ the wires C and D, leading to 
the instrument in whicli the current is to be reversed, are connected 
with tlie other pair of opposite cups (as h and c). It will be seen 
that in one position of the commutator (Fig. 216, 1 and 2), the wire 
A is connected with C, while B is connected with D ; in the other 
position (Fig. 216, 4 and 5) A is connected with 2), while B is con- 
nected with C. 



436 ELECTRICAL MEASUREMENTS. 

the current from the battery. This is done by simply 
removing the rods (Fig. 216, 8) from the mercury 
cups. In the absence of a commutator or key, one 
of the battery wires should be disconnected when the 
battery is not in use, not only to prevent unnecessary 
waste of power, but also to avoid serious errors 
which may result either from the deterioration of the 
battery or from heating the wires. 

When a battery is not required for several days it 
is well to empty out the fluids which it contains, each 
into a separate vessel, in which it may be preserved 
for future use, if not already exhausted. The zincs 
and coppers or carbons should be placed in pure 
Avaterj the porous cups left to soak in a solution of 
dilute sulphuric acid so as to be ready for immediate 
use ; the clamps, being disconnected from the poles of 
the battery, should be carefully cleaned and dried.^ 

(11) Electrical Connections. All electrical 
connections depending upon metallic contact should 
be carefully examined. The metallic surfaces should 
be scraped bright and bound together with consider- 
able pressure. A good electrical connection between 
two copper wires may generally be made by twisting 
them together. A soldered joint is to be preferred if 
the connection must remain good for an indefinite 
length of time. A liberal supply of binding-posts, 
screw-cups, and couplings, will be found of value in 
electrical measurements. 

1 These remarks apply particularly to cells of the Daniell or 
Bunsen type (Figs. 284 and 236, Exp. 84). With a Ledanch^ cell 
(Fig. 236), these precautions are unnecessary. 



IT 194] TANGENT GALVANOMETER. 437 

The best temporary connection is undoubtedly 
made by dipping copper into mercury (see 9). The 
surface of the copper should first be amalgamated by 
dipping it into nitrate of mercury and rubbing it with 
a cloth. 

(12) Insulation. Care must be taken that elec- 
trical connections are not made when they are not 
wanted. The student should carefully examine the 
insulating material with which his wires are wound, 
particularly when the wires are to be twisted to- 
gether. He should make sure that there is no cur- 
rent between any two of the binding-posts of a 
commutator or rheostat which can be detected by a 
galvanometer when the metallic connections are 
broken. The outside of battery cells should be dry 
for if they are not, electrical leakage is apt to take 
place. There is in fact more or less leakage in all 
experiments; but if the apparatus be perfectly dry 
this will probably not be enough to affect the accu- 
racy of any of the measurements which follow. 



EXPERIMENT LXXVIII. 

CONSTANTS OF GALVANOMETERS. 

% 194. Construction of a Single-Ring Tangent Gal- 
vanometer. — A form of galvanometer frequently em- 
ployed, because of its simplicity of construction, is 
represented in Fig. 217. A horizontal cross section 



438 ELECTRICAL CURRENT MEASURE. [Exp. 73. 

is given also in Fig. 218. The instrument consists 
of a compass (a, Fig. 217, and dgif. Fig. 218) mounted 
on a wooden support in the middle of a coil of insu- 
lated wire. The compass needle {eK) is made very 
sliort 1 so that tljQ whole of it may be virtually at the 
centre of the coil. To assist in reading the deflec- 
tions of the needle, two long light pointers (/ and ^) 





Fig. 217. 



Fig. 218. 



are attached to it at right angles. The wire is wound 
on a grooved brass ring in a single layer. The ends 
of the wire are carried to binding-posts (e, Fig. 217) 
at the base of the instrument as close together as pos- 
sible. Levelling screws (h and e. Pig. 217) are usu- 
ally added. In the construction of the instrument 

the JJ' 'km' "^ 'f'r'^" "'""''' ""' ^""^'^ A the diameter of 
the coil. Kohlrausch, Physical Measurement, Art. 68. 



t 195] LAW OF TANGENTS. 439 

neither iron nor steel must be used (^ 214, 3) ex- 
cept in the magnet itself, and in the steel pivot upon 
which it turns. The compass sliould have a lever to 
lift the needle from the pivot when the instrument is 
not in use (^ 214, l).i 

^ 195. Law of Tangents. — When an electrical 
current of sufficient strength is sent through the coils 
of a galvanometer, lines of magnetic force due to the 
current may be recognized by the 
the aid of iron-filings scattered upon ^-^C^ 
a horizontal piece of glass. We will '-"if r^^^HS i 
suppose that the plane of the coil is ^AV^i^^^^Vr^; 
parallel to the magnetic meridian :;^^5^i^^^7 
(that is, vertical, and magnetically "'^,rKM^r^^.A 
north and south ^ 214, 6), and that ''''^'^^%r\ 
the glass passes through the centre of 
the coil. Lines of force will then be 
formed in a direction which, if the current is suffici- 
ently powerful, may differ imperceptibly from east 
and west near the centre of the coil. 

When a compass-needle is placed at the centre oi 
the coil, it takes a direction, as might be expected, 
parallel to the lines of force passing through that 
point. If we suppose the current to be ascending on 

1 Single-ring galvanometers in the Jefferson Physical Laboratory 
have been constructed with 10 turns of No. 16 insulated copper wire, 
wound on a brass ring 36 cm. in diameter. The supports are made of 
wood. The needle is 2J cm. long. The pointers are of aluminum, 
and each about 5 cm. long. The circle is divided into degrees and 
half-degrees. The coil is arranged in sections of 1, 2, 3, and 4 turns, 
with connections so that any number of turns can be employed from 
1 to 10. By sending the current through these sections in different 
directions the sections may be tested against one another. 



440 ELECTRICAL CURRENT MEASURE. [Exp. 78. 

the north side of the coil, and descending on the 
south side, the north pole of the needle will point 
nearly to the east. The electric current tends in 
fact to deflect the compass-needle due east and west, 
but the earth's magnetism combined with it always 
gives to the needle a more or less northerly direction. • 
The actual direction of the compass-needle is de- 
termined (see % 184) by two forces : one, JJ, due to 
the horizontal component of the earth's magnetism 
acting in a northerly direction ; the other, F^ due in 
this case, not (as in % 184) to a magnet, but to the 
magnetic effect of the electrical current acting in an 
easterly or westerly direction. The angle (a) of 
deflection is given accordingly, as in % 184, by the 
formula, 

= tan a. (1) 

The units of current now in use have been defined 
(§ 132) with reference to the magnetic field which a 
current produces in a coil of wire. If L is the length 
of the wire, B its mean radius, and c the current in 
absolute units, we have 

F=^. ■ (2) 

Or if (7 is the current in amperes (§ 19), we have — 

Substituting this value in (1) we have — 

CL ^ , . ^ 



1 196.] CALIBBATION OF GALVANOMETERS. 441 

Let US suppose that two currents (7' and C" produce 
the deflections a' and ci' respectively ; then 

\^mH = '''" '''' (^> 

and 

^^ = ea«a'. (6) 

Dividing (5) by (6) we find — 

C : C" : : tan o! : tan a" ; (7) 

that is, in a given galvanometer two currents are pro- 
portional to the tangents of the angles of deflection 
which they respectively produce. This is known as 
the Law of Tangents. 

^196. Calibration of a Tangent Galvanometer. — 
The single-ring galvanometer described in ^ 194 

Fig. 220. 

may approximate more or less closely to the condi- 
tions required of a perfect tangent galvanometer. 
To test the accuracy with which the " Law of Tan- 
gents " (^ 195) is fulfilled, a battery of six small 
Daniell cells may be employed. The cells should be 
as nearly as possible of the same size and composition. 
The plane of the galvanometer coil is to be made 
parallel to the magnet meridian (^ 193, 6) so that 
the compass-needle points to 0** at both ends; then the 
two terminals are to be connected, with the poles of 
the battery arranged in series, as in Fig. 220, and in 



442 



ELECTRICAL CURRENT MEASURE. [Exp. 78- 



Fig. 221, 1, so that the cells may all act together. 
The connecting wires should be well insulated (^ 193, 
12) and twisted together (1[ 193, 8). The deflection 
of the galvanometer is to be found by reading both 
ends of the needle (^ 193, 5). 

The connections of the poles of the first cell (^) 
are now to be interchanged (Fig. 221, 2) so that it 
acts against the other five. The deflection is to be 
found as before. Then the original connections of A 
are to be restored, but those of the second cell (-B) 
reversed (as in 3), and the deflection again noted ; 




Fig. 221. 



and so in turn each cell is to be opposed to the rest 
(as in 4, 5, 6, and 7). Then A and B are both to be 
reversed (as in 8), then C and D (as in 9), then E 
and F (as in 10). The student may be interested to 
test the equality of the cells by opposing J., B^ and C 
against i>, E^ and F (as in 11, or as in 12). In re- 
peating the measurements, the connections of the 
galvanometer should be interchanged (^ 193, 9), and 
the measurements should be repeated in the inverse 
order, to eliminate variations in the strength of the 



1 197.] CALIBRATION OF GALVANOMETERS. 443 

cells. The results are to be reduced as in % 197, 
below. 

^ 197. Reduction of Results of Calibrating a Tan- 
gent Galvanometer. — 111 (1) we have six cells in 
series; in (2), (3), (4), (5), (6), and (7), we have 
in each case one cell opposed to five others or the 
equivalent of four cells. The average deflection gives, 
therefore, the effect of four cells of the same average 
strength as the six cells in (1). In (8), (9), and 
(10), we have in each case two cells opposed to four 
others, or the equivalent of two cells in all ; the aver- 
age deflection corresponds accordingly to two cells of 
the average strength. 

In 11 and 12 there should be little or no de- 
flection. Since the galvanometer is sensitive to the 
direction as well as to the magnitude of the current, 
the deflections in 11 and 12 should be equal and 
opposite. 

The results are arranged in tabular form below : 

1. No. of cells ax;ting. 2. Average deflection. 8. Tangent of deflection. 4. Ratio of 8 to 1. 
6 660.5 1.511 .252 

4 450.3 1.011 .263 

2 270.1 .612 .256 

We notice that the path of the electrical current is 
the same in all the arrangements, except that in 
some cases it passes through a given cell in one di- 
rection, in other cases in the opposite direction. It 
is stated that the electrical resistance of a cell is the 
satme, regardless of the direction of the current.^ 

1 Work is required to drive a current backward through a cell, 
whereas if a current passes through it in the ordinary direction, the 
cell is a source of power (see § 137). In calculating the electrical re- 



444 ELECTBICAL CURRENT MEASURE. [Exp. 78. 

The total electrical resistance 'is accordingly the 
same in each of the tweive arrangements shown 
in Fig. 221. It is also stated that the electro-motive 
force of a battery is proportional to the number of 
cells acting , hence by Ohm's law (§ 138) the ratio of 
the numbers in the third column to those in the 
second column should be nearly constant. If it is 
not, the galvanometer should be discarded for accu- 
rate purposes. The experiment should be repeated 
with a galvanometer in which the Law of Tangents 
is at least approximately fulfilled. 

^ 198. Determination of the Constant of a Single- 
Ring Gkdvanometer. — It is evident from formula 4, 
^ 195, that the deflection of a galvanometer depends 



rp 



Fig. 222. 



not only upon the electrical current, but also upon 
the length and radius of the coil of wire through 
which it flows. In order to measure currents with a 
galvanometer, it is therefore necessary to determine 

sistancB of a cell we do not consider the gain or loss of power doe to 
chemical agency, but only the loss ci power due to conversion into 
heat. The statement that the resistance of a cell is the same without 
regard to the direction of the current does not mean, therefore, that it 
is as easy to drive a current backward through it as to drive it for- 
ward, but that the cell would be equally heated in both cases. The 
truth of this statement has recently been called into question, but 
the method of calibration described above has been found practically 
to yield accurate results. 




iri98.J GALVANOMETER CONSTANTS. 446 

accurately the dimensious of the coil of wire. To find 
the diameter of a coil, we measure with a long Vernier 
gauge (Fig. 222) the distance between the flanges of 
a bobbin (a?, Fig. 223) upon which «- 
the coil is wound. Then we find the 
thickness of two blocks ab and kl 
which fill the space between the wires 
and the edges of the flanges. Sub- 
tracting ab and kl from al we have 
the outside diameter (6i) of the coil. 
We now measure the width of the i J^B" 
bobbin and the width of the flanges. ^^^' ^^• 

Subtracting the latter from the former, we have the 
width of the coil of wire. The whole number of 
turns of wire is now to be counted. Usually the 
groove is broad enough for one more turn of wire 
than that actually wound upon it, since this amount 
of space is necessary for turning the wire. The width 
of the groove is to be divided by the number of turns 
which would fill it, to find the average diameter (bc^ 
or jk^ of the wire. Subtracting this from the out- 
side diameter (6A) we have the mean diameter (6;, or 

ce) of the coil. Dividing 
by 2 we have the mean ra- 
dius of the coil. 

Instead of measuring the 
diameter of the coil, we 
may find its circumference 
Fia. 224 by passing a thin steel tape 

graduated in mm. around the outside of the coil. 
If € is the circumferencei, the outside diameter is 




446 ELECTRICAL CURRENT MEASURE. [Exp. 78. 

(? -T- TT. From this the mean diameter and radius 
may be calculated as before. The results are to be 
still further reduced as in ^ 199. 

^ 199. Calculation of the Constant and Reduction 
Factor of a Tangent Galvanometer. — The constant 
(Z") of a coil of wire is equal to the ratio of its 
length to the square of its radius (§ 133). That is, 
in the notation of % 195, 

Substituting this value in formula 4, ^ 195, we liave 
CK 



10 H 
or solving for (7, 



= tan a, (2) 



C = 10 ^ tan a. (3) 

The constant, K, of a given galvanometer is therefore 
an important factor in the calculation of a current 
from the deflection which it produces in that gal- 
vanometer. 

If n is the number of turns in the coil,^ we have 

L=27rnB, (4) 

which substituted in (1) gives 

zr 27rnR 27rn ,rx 

1 The student must remember that when a coil is made in two 
parts, so that half the current flows through each, the effect is the 
same as if the whole current flowed through one half. The total 
number of turns must therefore be halved in order to find the effec- 
tive number n. 



1199.] GALVANOMETER CONSTANTS. 447 

By this formula the constant of the tangent gal- 
vanometer is to be calculated. Thus for 6 turns of 
radius 18 cm. we have a constant 2 x 3f X 5 -^ 18, or 
1.75, nearly. With such a galvanometer, assuming 
that the horizontal intensity of the earth's magnetism 
is 0.175, nearly, we should have from (3) — 

175 
(7 = 10 X ^pp tan a = tan a (nearly) ; 

that is, the current in ampdres would be numerically 
equal to the tangent of the angle of deflection 
produced. 

In most galvanometers this is not the case. To 
find the current, we have to multiply the tangent of 
the angle of deflection by some factor, which may be 
greater or less than unity. This is called the reduc- 
tion factor of the galvanometer.^ 

Denoting it by J, we have from (3) — 

7=10 J. (6) 

It is important to find the reduction factor of a gal- 
vanometer which is to be used often, since it greatly 
shortens the reduction of results. 

Substituting from (6) in (3) we have simply — 

C= Itan a. (7) 

It may be observed that if a = 45% so that tan a 

1 Some writers call the reduction factor " the constant *' of a gal- 
vanometer. Since the reduction factor depends upon the earth's 
magnetism (see 6), it is evidently not constant. The effect of changes 
in the earth's magnetism in a short course of experiments may, how- 
ever, generally be disregarded. 



448 ELECTRICAL CURRENT MEASURE. [Exp. 79. 

= 1, we have 0=^ I. The reduction factor of a 
galvanometer is therefore numerically equal to the 
current which deflects it 45*" ; that is, the current 
which produces a field of force at the centre of the 
coil equal to the horizontal component of the earth's 
magnetism. 



EXPERIMENT LXXIX. 

COMPARISON OP GALVANOMETERS. 

^ 200. Construction of a Double-Ring Tangent Gkd- 
vanometer. — A " double-ring " tangent galvanometer 
is represented in Fig. 225, also in horizontal section in 




Fig. 225. 



Fig. 226. It consists of two parallel coils of wire wound 
on brass or wooden rings a and &, with a surveying- 



ir20l.] COMPARISON OF GALVANOMETERS. 449 

compass cd between them (see also Fig. 199, ^f 183). 
In the case of a single-ring galvanometer, it has been 
stated that the length of the needle should not ex- 
ceed -j^ the diameter of the coils. In the ija OJ 
double-ring galvanometer, it may be J of j • 

this diameter without introducing any •-. 
serious error into the results (Kohlrausch, if 
Art 93). For measuring battery cur- \ 
rents, each coil should contain about sii^ : 
turns of No. 12 insulated copper wire, pie /jpi 
It is recommended that the average di- ^*^- ^26. 
ameter of the coils should be 32 cm. and the mean 
distance between them 16 cm,^ The needle of the 
surveying-compass should be not more than 8 cm. 
long. When a current is made to divide in such an 
instrument into two parts, so that half flows through 
each coil, it is found that the tangent of the angle of 
deflection is approximately equal to the magnitude 
of the current in ampdres. 

^ 201. Determination of the Redaction Factor of a 
Galvanometer by the Method of Comparison. — The 
siilgle-ring galvanometer (Fig. 217) is to be adjusted 
with its coil north and south (^ 19Si, ®), as near as 
possible to the place (^ 193, 3) where the horizontal 
intensity of the earth's magnetism was determined 
(^ 183). The double ring galvanometer (Fig. 225) 
is to be similarly adjusted in some position conven- 

^ These dimensions have been calculated for places where the hor- 
izontal component of the earth's magnetism is .169 or .17 nearly. In 
places where this horizontal component is nearly .18 the dimen- 
sions should be 80 and 15 em. respectively. 

29 



450 ELECTRICAL CURRENT MEASURE. [Exp. 79. 

lent for future measurements. This position should 
be accurately noted. The two instruments (J. and 
(7, Fig. 227) are then to be connected in series with 
a constant battery (B) capable of yielding a current 
of one or two ampdres. The deflection of each gal- 
vanometer is to be found by reading both ends of 
each needle (^ 193, 5). The connections of (7 are 
then reversed (see ^ 193, 9), and both deflections 
again noted. The connections of A are next re- 
versed and new readings taken. Finally the connec- 





FiG. 227. 

tions of C are again reversed, so as to be the same as 
at the start, — the needles being read as before. 

The observations of the two galvanometers should 
be made at the same time, as nearly as possible. Let 
a be the average angle through which A is deflected ; 
a' that through which is deflected ; then if the re- 
duction factors (^ 199) of A and C are I and I' re- 
spectively, the current C which traverses both gal- 
vanometers must be (see ^ 199, formula 7) — 

C ^= I tan a = I' tan a' ; 

hence the reduction factor (/') of may be found by 
the equation — 

Tf J- tan a 

tan al ' 

We notice that the reduction factors of two galvan- 
ometers are to each other inversely as the tangents. 



f 202.] 



THE DYNAMOMETER. 



451 



of the angles of deflection produced by a given 
current. 

The student should be cautioned not to connect the 
two galvanometers in multiple arc (§ 140) ; for in 
this case the current divides into two parts, which 
may or may not be equal. Not knowing the ratio be- 
tween the two parts, we can draw no conclusion as 
to the relative sensitiveness of the two galvanometers. 

When the instruments are connected as above in 
aerieSj the same current (if there is no leakage) must 
traverse the coils of both. 



EXPERIMENT LXXX. 



THE DYNAMOMETER. 

% 202. Construction of a Dynamometer. — A form 
of dynamometer useful for measuring battery currents 
is represented in Pig. 228. It consists of a wooden 
bobbin, fgpn^ with two grooves, 
in each of which are wound 50 
turns of No. 16 insulated copper 
wire. Small holes are bored 
through the bobbin at/,^, w, and 
/?, so that it is possible to measure 
directly the inner and outer di- 
ameters of the coil. The average 
diameter is about 25 cm. 
A small hollow wooden cube *'^<^- ^^®- 

(y^O' nieasuring 5 cm. each way, is now wound with 
80J turns of No. 24 copper wire, the ends of which 







^^ 


^^BJl ■-.■■'■ iiTT^i, ■''™ 


■i^bB 






J^^J^a^^ 


Jt 


m-Xf^j J V 


5\ 




1^ 


^^^ 


m 




^ 



452 ELECTBICAL CURRENT MEASURE. Exp. 80. 

are connected by No. 31 spring brass wires (jch and 
md) to a fixed point beneath, o, and to the centre (c) 
of a knitting nee41e (bd\ as in the torsion balance (see 
Fig. 176, ^ 165). The length of the wire should be 
taken so that the coefficient of torsion of the wire 
ch may be some round number, let us say 10 dyne- 
centimetres, per degree (see ^ 165). Thus if 100 cm. 
of the wire has been found (Exp. 64) to have a co- 
efficient of torsion of 2 dyne-centimetres per degree, 
we may make ch just 20 cm. long, so that it may 
exert a couple of ^^ x 2 := 10 units per degree. 

It will be observed that the constant of the large 
coil, having in all 100 turns, and a mean radius of 
12-5 em., is (see ^133) — 

^_2xim6xl00__ 50, nearly, (1) 

1^.0 

while the magnetic area of the smaller coil is (see 
§134)- 

^ = 80i X 5 X 5 = 2000, nearly. (2) 

The constant of the dynamometer is accordingly 
(§136)- 

i) = 50 X 2,000 = 100,000 absolute units, nearly. (3) 

In other words, a current of 1 absolute unit would cre- 
ate a couple of 100,000 units, tending to twist the 
wire. A current of 1 ampere (being -^ of the ab- 
solute unit) will have ^ the effect, not only in the 
cube (ijkl)^ but also in the large coil (^fgpn)> The 
couple produced, depending upon the product of 
these two effects (see §§ 133, 134), will be accord- 



IT 203. THE DYNAMOMETER. 458 

ingly less than 2> (in formula 3), in the proportion of 
100 to 1. It follows that 1 ampdre will exert in this 
instrument a couple of about 1000 dyne-centimetres ; 
and that it will require a twist of 100° in the wire ch 
to balance it if, as has been supposed, 1° corresponds 
to 10 dyne-centimetres. Since the couple produced 
is proportional to the square of the current (§ 135), 
the current must be proportional to the square root 
of the angle of torsion which is required to balance 
this couple. 

The proportions of the dynamometer have been 
chosen above so that the square root of the number 
of degrees indicated by the needle hd may give at 
once (approximately at least) the current in tenths 
of an ampdre. 

^ 203. Determination of the Constants of a Dyna- 
mometer. — Before making use of a dynamometer to 
measure electrical currents, it is necessary to find 

(1) the constant of the large coil (^fgpn^ Fig. 228), 

(2) the magnetic area (§ 134) of the small coil (ijkV)^ 
and (3) the coefficient of torsion of the wire. 

(1) The diameter of the large coil may be deter- 
mined as in ^ 198 ; but as the coils of the dynamome- 
ter contain several layers of wire, it is more accurate 
to measure directly the outside and inside diameters. 
For this purpose holes are made at/,^, w, and p, in the 
side of the bobbin. The number of turns, if unknown, 
may be estimated by counting the layers and the 
number of turns in each. From the whole number 
of turns and from the mean diameter of tlie coil, the 
constant (A') is to be calculated as in ^ 199. 



454 ELECTRICAL CURRENT MEASURE. [Exp. 80. 

(2) To find the mean diameter of the square coil, 
the outside diameters jk and kl are to be measured 
by a Vernier gauge. The diameter of the wire is to 
be found by measuring the width of the 80 or more 
turns between i andy, then dividing by the number 
of turns. Subtracting this diameter from the outside 
diameters y* and Ar?, we have the mean diameter of 
the coil. Unless a wire passes through the middle of 
the cube in the direction co, it is obvious that there 
must be a whole number of turns plus one half turn 
on the cube ijkl. To avoid making a mistake, the 
turns should be counted on both sides of the cube. 
The magnetic area, -4, of the square coil is then calcu- 
lated as in § 134. 

(3) The instrument is now to be laid upon its 
side, and a light balance-arm is to be attached to the 
cube (see Fig. 176, ^ 165). The wire ch will prob- 
ably have to be supported near h to prevent it from 
sagging under the weight of the cube. The wire 
should, however, rest freely upon the support, so as 
not to affect the toraion. The coefficient of torsion 
of the wire ch is then to be found as in ^ 165. 

^ 204. Determinatdon of Reduction Factors by means 
of a Dynamometer. — The Dynamometer is now to be 
set upright with the plane of the large ring north and 
south, and adjusted by twisting the needle bd so that 
the planes of the large and small coils are at right- 
angles. A fixed mark should be placed on the wall 
of the room so as to be in line with two sights jk on 
the small coil, when the coil is at right-angles to the 
large coil. The reading of the needle is to be ob- 



1204.] THE DYNAMOMETER. 455 

served. The instrument is then to be connected (as 
in ^ 201) in series with a single-ring tangent galvan- 
ometer, and with a battery of several Bunsen cells, 
capable of sending a current of about 1 ampdre 
through the circuit. The needle bd is to be turned 
until the sights ^ and k on the small coil come in line 
with the same mark as before. The reading of the 
needle is to be again observed, and also that of the 
tangent galvanometer. 

The current is now to be reversed in the large coil, 
but not in the small coil of the dynamometer ; then 
reversed in the battery; then the original connec- 
tions of the dynamometer are to be restored. In each 
case readings of the dynamometer and of ithe gal- 
vanometer are to be made.^ 

If t is the coefficient and a the angle of torsion of 
the wire, the couple is ta. If K is the constant of 
the large coil, A the magnetic area of the small coil, 
we have for the current c, by § 135 — 



= Y -T^i ^^ absolute units ; 
KA 



or in ampfires, (7 = 10 y -=^ , 

since an ampere is one tenth of an absolute unit. 

From the current, (7, and the mean deflection, d[, 
which it produces in the tangent galvanometer, we 



1 The couple produced by a current may also be measured by turn- 
ing the instrument on its side as in 1 208, 8, and directly counter- 
poising the current with weights placed in one pan of the balance. 



456 ELECTRICAL CURRENT MEASURE. [Exp. 81. 

may find the reduction factor of the latter by the 
formula — 

tan a 

We may also find the horizontal component (JT) of 
the earth's magnetism by the formula — 

derived from ^ 199, 6, using the new value of I. 

If the values of I and H found by means of the 
dynamometer differ from those previously determined 
(Exps. 74 and 78) by more than 5 or 10 %,^ the stu- 
dent 8h()uld repeat all the measurements upon which 
these values depend. 



EXPERIMENT LXXXL 

ELECTRO-CHEMICAL METHOD. 

^ 205. Determination of the Reduction Factor of a 
Galvanometer by the Electro-Chemioal Method. — The 
galvanometer is to be adjusted with the plane of its 
coil parallel to the magnetic needle (^ 193, 5), and 
its exact position noted (^ 193, 3). The terminals 

1 The use of a small square coil in a dynamometer is simply for 
convenience in the explanation of the instrument to students. For 
accurate measurements, a round coil is to be preferred. In any case 
there are certain corrections to be applied to the dynamometer on ac- 
count of the size and shape of its coils (unless these be carefully pro- 
portioned) which if neglected may account for errors of 3 or 4 %. 



1205. ELECTRO-CHEMICAL METHOD. 467 

of tbe galvauometer (h and i) are to be connected 
with the poles of a Daniell cell, a and b (Fig. 229, 2), 
through a commutator defy (see ^ 193, 9). The 
ordinary copper (or positive) pole is replaced by a 
spiral of copper wire (J, Fig. 229, 1 and 2) with a 
coupling c, provided for convenience in weighing. 
The spiral should have been 'cleaned with nitric acid 
before the experiment. The solution of sulphate of 
copper with which it is surrounded should be satu- 
rated and free from all impurities, especially acid, 
ammoniacal, and oxidizing or reducing agents. The 
deflection of the galvanometer should be about 46, 




M^ 




Fig. 229. 

— more rather than less. If it is less than SO^ the 
porous cup should be changed, or another cell substi- 
tuted. When the spiral has been freshly coated with 
copper by the action of the battery, it should be dis- 
connected from the coupling ((?), dipped in three 
changes of fresh water, then in alcohol, and dried 
in a temperature not exceeding 100^ to avoid oxida- 
tion of the copper. Its weight is then to be found 
within a milligram, if possible, by a series of double 
weighings (Exp. 8). 

The spiral is now to be replaced in the cell, and 
connected with the galvanometer as before. The time 



458 ELECTRICAL CURRENT MEASURE. [Exp. 81. 

when the connection is made must be accurately noted* 
The deflection of the galvanometer is to be recorded 
at intervals of one minute. Each end of the needle 
should be alternately observed (^ 198, 5). At the 
end of 25^ minutes the commutator defg is to be 
suddenly turned (see ^ 193, 9) so that the current 
through the galvanometer may be reversed. Obser- 
vations of the galvanometer needle are to be contin- 
ued, at intervals of one minute, for another 25 
minutes. There will thus be 50 observations in all. 
At the end of 50 minutes and 50 seconds, exactly, 
the current is to be suddenly cut off. The copper 
spiral is to be cleansed in three changes of water, with 
care not to dislodge any of the fresh deposit, then 
dipped in alcohol, dried, and reweighed accurately 
as before. The results are to be reduced as in 
IF 280. 

^ 206. Theory of the Sleotro-Cheinioal Method. — 
It has been found that a current of 1 ampere deposits 
1 gram of copper in the course of 50 minutes and 
about 50 seconds (the total duration of the experi- 
ment). The strength of the solution has little or no 
effect upon the result, always provided that enoriffh 
copper is present in it (§§ 142, 148). The amount 
of copper deposited varies only with the strength 
and duration of the current. 

If C is the strength of the current in ampfires, t 
the time in seconds, and to the weight of copper 
deposited, we have accordingly — 

Ct 



1206.] ELECTRCM3HEMICAL BIETHOD. 459 

and e=—,-"'. nearly. (2) 

If, as in the experiment, t = 50 minutes and 50 seo- 
onds, that is, 3050 seconds, we find simply — 

0=.w. (3) 

That is, the average value of a current in ampdres is 
numerically equal to the weight in grams of copper 
deposited by it in 3050 seconds. 

Now from ^ 199, 7, we have, at any point of time, 

C=Itana^ (4) 

where a is the angle of deflection produced by the 
current in a tangent galvanometer, and / is the re- 
duction factor of the galvanometer. Hence, averag- 
ing the different results from the 50 observations of 
the needle, we find, comparing (8) and (4) — 

w = average of I tan a. (4) 

In practice, if the angles do not differ by more than 
10 %, the same result (nearly) may be obtained much 
more easily by averaging the angles themselves, then 
finding the tangent of this average. That is, if A is 
the average angle of deflection — 

w = I tan A, nearly. (6) 

The reduction factor may now be calculated by the 
i'ormula — 

tan A ^ ^ 

Having found the constant, Kj of the galvanometer 
(^ 199, 1), we may calculate the horizontal com- 



460 ELECTRICAL CURRENT MEASURE. [Exp. 82. 

ponent (iT) of the earth's magnetism, as in ^ 204, 
by the formula (derived from ^ 199, 6) — 

S= f • (8) 

If the value of H obtained by the electro-chemical 
method does not agree with previous determinations 
(Exps. 74, and 80), the last experiment (Exp. 81) 
should be repeated until at least 3 results, obtained 
either by the same or by different methods, agree 
within let us say 5 %. All previous measurements 
leading to a different result should now be repeated.^ 



EXPERIMENT LXXXII. 

METHOD OP VIBRATIONS. 

^ 207. Construction of a Vibration Galvanometer. — 
A form of galvanometer easily constructed is repre- 
sented in Fig. 230. It consists of a coil cfg (made by 
winding 14 turns of No. 18 insulated copper wire 
upon a hoop of wood, brass, or pasteboard, 10 cm. in 
diameter) with a short magnetized needle e, attached 
to a bullet d and suspended at the centre of the coil 
by a fine waxed fibre (cd) of untwisted silk (see 
^ 186). The strength of the magnet and the weight 
of the bullet should be proportioned so that the 

* The student will do well to examine his calculations before re- 
peating the measurements upon which they depend. A common 
error is a miscount or misconception of the number of turns of wire 
utilized in the coil of a galvanometer or dynamometer, particularly 
when the coils are connected in multiple arc. See footnote, 1 199. 



1207. METHOD OF VIBRATIONS. 461 

needle may complete 10 vibrations in about 1 minute. 
A short test-tube may be employed to cut oflf cur- 
rents of air (see Fig. 204, ^ 186). 

The ends of the coil may be carried to binding- 
posts, / and g. Connections at / and g may also be 
made by simply twisting the wires together (^ 193, 

11). 

When an ordinary battery current is sent through 
the coil, the magnetic field of force created by the 
current will greatly increase the rate of vibration of 
the needle. We have seen (^ 186 and § 110) that a 
field of magnetic force is proportional to the %quare 




Fig. 230. 

of the number of vibrations which it produces in a 
magnetic needle. In accordance with this law, the 
dimensions of the instrument have been chosen so 
that the square of the number of vibrations com- 
pleted in 1 minute may represent approximately the 
strength of the current in thousandths of an ampdre. 
In calculating these proportions, it was assumed 
that the needle made exactly 10 vibrations per minute 
under the influence of the earth's magnetism, the 
strength of which was taken as 0.176 dynes per unit 



462 ELECTRICAL CURRENT MEASURE. [Exp. 82. 

of magnetism (see Exps. 72, 73, 74, 80, and 81). 
No allowance was made for the effects of magnetism 
induced in the needle, which (unless the needle be of 
the best steel and highly magnetized) may account 
for errors of 5 or 10 per cent with currents of 1 or 
2 amperes. To obtain accurate results with a vibra- 
tion galvanometer, it would be necessary both to 
calibrate it (see ^ 196) and to compare it (as in Exp. 
79) with a galvanometer of known reduction factor. 
When, however, as in this experiment, the instru- 
ment is to be used for rough work and for relative 
indications only, such tests need hardly be applied. 

The influence of the earth's magnetism upon the 
vibration galvanometer must be allowed for, as will be 
explained in % 209. 

^ 208. Determination of the Relative Strength of 
Battery Currents by means of a Vibration Galvanometer. 
— A vibration galvanometer (^ 207) is to be set up 
with the plane of its coil vertical, but (contrary to 
the usual custom, ^ 193, 5) at right-angles with the 
magnetic meridian. The time required for 10 vibra- 
tions of the needle (which should be about 1 minute) 
is now to be accurately determined. The needle may 
be set in vibration by bringing a magnet near it, then 
suddenly taking the magnet away. The arc of vi- 
bration should not exceed 30 or 40 degrees (see 
Table 3, ^). 

The terminals of the galvanometer, /and g, are now 
to be connected respectively with the poles, a and 5, 
of a battery constructed as will be described below. 
The student must notice carefully whether the needle 



1^208.1 METHOD OF VIBRATIONS. 463 

points in the same direction as before, or whether the 
needle is reversed. In the latter case the connec- 
tions of the galvanometer with the battery should be 
interchanged; that is, /should be connected with i, 
and g with a. 

The number of vibrations made in 1 minute (or 
whatever time was required for 10 vibrations under 
the earth's magnetism) is now to be accurately deter- 
mined. In no case should the arc of vibration ex- 
ceed 80 or 40 degrees. 

The battery to be employed in this experiment con- 
sists of a glass tumbler, half-filled with dilute sulphu- 
ric acid^ (10 % by weight), a porous cup with an 
internal diameter not less than 5 cm,^ containing a 
solution of sulphate of copper, and two strips, one of 
sheet zinc, the other of sheet copper, each 5 by 10 
cm. Connecting wires should be soldered to both 
strips. The current from this battery is to be tested 
under the following conditions : 

(1) When the zinc and copper strips are placed 
side by side in the sulphuric acid, but not touching 
each other. 

(2) The same after the zinc has been amalga- 
mated by rubbing it with mercury. 

(3) (4) (6) The same after the current has been 
allowed to flow for five, ten, and fifteen minutes 
respectively. 

(6) The same except that the bubbles gathered 

^ To avoid accidents in mixing sulphuric acid with water, the acid 
should be poured in a fine stream into the water, so that the heat 
generated may be quiclclj dissipated. 



464 ELECTRICAL CURRENT MEASURE. [Exp. 82. 

on the copper strip have been removed by a camel's- 
hair brush, without exposing the copper to the air. 

(7) The same, except that the copper has been 
exposed for a few minutes to the air. 

(8) The same except that the copper has been 
amalgamated by being rubbed with nitrate of mer- 
cury.^ 

(9) The zinc and copper strips are now to be care- 
fully weighed ; the zinc is to be replaced in the sul- 
phuric acid, but the copper is to be immersed in th« 
solution of sulphate of copper contained in the porous 
cup, and the latter is to be placed in the tumbler conr 
taining the acid.* 

(10) (11) (12) The same after the current has 
been allowed to run for five, ten, and fifteen minutes 
respectively. The zinc and copper strips are now to 
be reweighed. The results are to be reduced as will 
be explained in the next section. 

^ 209. Reduction of Results obtained with the Vi- 
bration Galvanometer. — It has been stated that the 
square of the number of vibrations completed in one 
minute by a vibration galvanometer constructed as in 
^ 207, gives approximately the current to which 
these vibrations are due in thousandths of an ampere. 
To find, accordingly, the current in amperes, we 
square the number of vibrations produced in the 
given length of time, and divide by 1000. 

^ Copper may also be amalgamated bj dipping it into nitric acid, 
then rubbing it with mercury by means of a cloth. Care must be 
taken not to let nitric acid come in contact with the hand. . 

3 This combination constitutes a Danlell cell. See also Fig. 285, 

t2a 



1209.] METHOD OF VIBRATIONS. 465 

It must not, however, be forgotten that the earth's 
magnetism alone accounts for about 10 vibrations per 
minute. The earth's field is accordingly equivalent 
to that produced in the vibration galvanometer by 
-^^ or 0.1 ampere. Caje should have been taken in 
the experiment to have the earth's magnetism and 
the current acting always in the same direction. 
In this case all the results will be too great by 
0.1 ampere. By subtracting this amount in each 
case, the effect of the earth's magnetism will be 
eliminated. 

The strength of each current in ^ 208, (1) to (12), 
should be calculated roughly in this way. 

The student will notice that the visible action of 
the sulphuric acid on the zinc is arrested by amalga- 
mating the zinc with mercury ; that the action begins 
again when the zinc is connected with the copper 
strip, but that the bubbles of gas are then set free 
from the copper instead of from the zinc ; that the 
amalgamation of the zinc does not impair the useful- 
ness of the battery ; that the current steadily de- 
creases when both strips are in sulphuric acid, though 
it is temporarily increased by removing the bubbles 
from the copper, and by exposing the copper to the 
air ; that amalgamation of the copper does not pre- 
vent the formation of bubbles upon it, nor improve 
in any way the action of the battery; that the for- 
mation of bubbles is arrested by placing the copper 
in the solution of sulphate of copper, and that in 
this case the battery. furnishes a steady current ; that 

the zinc plate loses in weight, but that the copper 

30 



466 ELECTRICAL CURRENT MEASURE. [Exp. 83. 

plate gains in weight by a nearly equal amount,^ 
owing to fresh copper deposited upon it. We have 
already made use (in Exp. 81) of the quantity of 
copper thus deposited to measure an electrical 
current 



EXPERIMENT LXXXIII. 

THE AMMETER, I. 

% 210. Testing an Ammeter. — The name ^^ ammeter '* 
(an abbreviation of ampSre-meter) is given to any 
instrument indicating directly the strength of elec- 
trical currents in ampdres. Ammeters are manufac- 
tured in various forms. Most of them depend upon 
the attraction which an electrical current, circulating 
in a coil of wire (5, Fig. 231), exerts upon a perma- 
nent magnet or upon a core of soft iron. In some in- 
struments this electro-magnetic attraction is balanced 
by a spring, in others by gravity ; in others again it is 
balanced by the attraction of a permanent magnet (c). 

Such instruments depend 
for their accuracy upon the 
constancy of the magnet. 
Fig. 231. g^j^^ gy^^ jf correctly grad- 

uated at the start, are subject to errors which may be 
indefinitely great. Recently instruments have been 

1 If we assnine that there is no wasteful action of the battery, the 
quantities of zinc dissolved and of copper deposited should be to 
each other as the atomic weights of zinc and copper, 64.9 and 63.1 
respcctlTely. 




1210.] TESTING AMMETER& 467 

manufactured in which currents are measured by the 
attraction between two coils of wire traversed by the 
same electrical current. Such instruments are prop- 
erly called electro-dynamometers (see Exp. 84). If 
carefully graduated, they may serve as standards for 
the determination of electrical currents. 

Ammeters are usually intended to measure cur- 
rents of at least 10 ampdres, and being generally sen- 




FiG. 232. 

sitive only to about -^ ampSre, they cannot measure 
small currents very precisely. On the other hand, 
the tangent galvanometers described in ^ 194 and 
^ 200 are intended to measure currents of a few 
amperes only. To compare an ammeter with such 
instruments, it must be connected with two or more 
of them in multiple arc (§ 140). A powerful battery 



Fig. 233. 

of three or four Bunsen cells is then included in the 
circuit. A diagram of connections is given in Fig. 282, 
where A represents the ammeter, BB the battery, 
and D two galvanometers. To avoid the influence 
of the connecting wires upon the instrument (^ 193, 
8), the arrangement would practically be made as in 



468 ELECTBICAL CUBBENT MEASUBE. [Exp. 84. 

Fig. 233. The battery cells are represented in both 
diagrams (Figs. 232 and 233) as being connected in 
multiple arc (§ 140), since in this way they usually 
yield the greatest current through instruments of low 
resistance (§ 146). 

If a, a\ &c., are the deflections of the galvanome- 
ters ; J, I\ &c., their reduction factors, the currents 
through them are respectively / tan a, J' tan a\ &c. 
Hence the total current C is — 

C = J tan a + ^' *«w a' + ^c. 

The experiment should be repeated with batteries 
containing different numbers of cells, or the same 
number differently arranged, so as to produce cur- 
rents of from 1 to 10 amperes. 

The results should be tabulated in the ordinary 
manner, in three columns, containing respectively, 
(1) the current calculated from the galvanometer de- 
flections ; (2) the current indicated by the ammeter, 
and (3) the corresponding correction of the ammeter. 



EXPERIMENT LXXXIV. 

THE AMMETER, II. 

^211. Determination of Battery Currents by means 
of an Ammeter. — The electrical resistance (§ 136) of 
ammeters is usually so slight that it may be neglected. 
To measure the maximum current which a battery 
can produce, the screw-cups of the ammeter are to be 
connected by short thick copper wires with the pole- 



ir2ii.] 



BATTERY CURRENTS. 



469 



cups of the battery in question. The wires should 
be parallel or twisted together, as in the last experi- 
ment (see Fig. 233), and scraped bright at both ends 
(^ 198, 11). The indication of the instrument is to 
be noted. 

With any instrument of the class known as amme- 
ters, the student is to determine the maximum cur- 
rent which can be derived from various well-known 
forms of voltaic battery, as, for instance, the Bunsen 






Fig. 234. 



Fig. 235. 



Fig. 236. 



cell (Fig. 284), the Daniell cell (Fig. 285) and the 
Leclanch^ cell (Fig. 286). The observations may 
be continued in each case at intervals of five minutes 
for half an hour.^ The material employed in each 
cell, and the dimensions of every part,^ should be 

1 An old Leclanch^ cell may be employed for this experiment. It 
may serve subsequently for experiments with Wheatstone's Bridge, 
but for other purposes it will be rendered nearly useless. 

^ If a sufficient current cannot be obtained from a single cell of a 
given sort, two or more cells should be employed. The student 
should notice that with instruments like the ammeter having a very 
low resistance, it is more effective to arrange batteries in multiple 
arc than in series. See § 136, also Figs. 232 and 233, 1 210. 



470 



ELECTBICAL CUBBENT MEASURE. [Exp. 84. 



carefully noted. The corrections for various cur- 
rents indicated by the ammeter have been found in 
the last experiment. The proper correction should 
be applied to each reading. The results are to be 
represented by a series of curves (Fig. 287) plotted 

on the same sheet of co- 
ordinate paper. A scale 
at the top of the paper 
indicates the time in min- 
utes, and a scale at the 
left of the paper repre- 
sents the current in am- 
FiG. 237. pSres. Each curve should 

be marked with the name of the cell or battery to 
which it belongs. 




IT 212.] 



METHOD OF HEATING. 



471 



ELECTRICAL RESISTANCE. 



EXPERIMENT LXXXV. 



METHOD OP HEATING. 



^ 212. Determination of Resiatanoes by the Method 
of Heating. — A short spiral (a, Fig. 238) of fine 
German silver wire, .01 cm. diameter (about No. 36) 
and 16 cm. long, is soldered to the two 
terminals I and c of two insulated cop- 
per wires, d and e, passing through a 
cork fitting the inner cup of a calori- 
meter (5, Fig. 239). The wires (Id 
and ce) should be so thick that their 
electrical resistance may be neglected in 
comparison with that of the spiral. The 
cork and wires are then inverted and 
placed in the calorimeter (JB, Fig. 239) 
containing a sufiScient quantity of dis- 
tilled water to cover the spiral. The 
temperature of the water, which should 
be slightly below that of the room, 
is found by a series of observations (^ 92, 10) made 
with a thermometer passing through the cork as in 
Fig. 239. The thermometer is provided with a stirrer 
(see Tf 66, Fig. 50) so that a uniform temperature 
may be maintained. 

The instrument thus constructed (J8, Fig. 239) is 




Fig. 238. 



472 ELECTRICAL RESISTANCE. [Exp. 86. 

to be connected in series with a Bunsen cell (J.) and 
with a tangent galvanometer ((7) adjusted in the 
same place and manner as in Exp. 83. 

The time when the connection is made must be 
accurately noted. The tangent galvanometer is to be 
observed at intervals of one minute. Between the ob- 
servations, the water in the calorimeter is to be stirred 
by twisting the stem of the thermometer. When 
the temperature reaches that of the room, the direc- 
tion of the electrical current is to be suddenly re- 
versed by interchanging the battery connections (see 
^ 198, 9). The observations of the galvanometer are 



fH 




Fig. 239. 

to be continued until the temperature of the water 
rises as high above that of the room as it was origi- 
nally below it. Then the circuit is to be broken. 
The time when the current is interrupted must be ac- 
curately recorded. Several more observations of the 
temperature within the calorimeter are to be made at 
intervals of one minute, so that the resulting temper- 
ature may be accurately determined. 

The weight of the calorimeter and of the water 
which it contains are finally to be found by weighing 
the calorimeter with and without the water. 

^ 213. Calciilation of Resistance by the Method of 
Heating. — Let w be the weight of water, and TTthat 
of the calorimeter from which its thermal capacity 



ir213.] METHOD OF HEATING. 473 

c is to be calculated,^ and let t^, and t^ be the temper- 
atures of the water at the moment when the circuit 
was first made and finally broken. These tempera- 
tures are to be inferred from the observations made 
before and after the experiment (see % 93, 2). Since 
the average temperature of the water agrees with 
that of the room, no allowance need be made for cool- 
ing in the mean time (^ 93, 3). The quantity of 
heat, Hy generated by the electrical current is there- 
fore — 

ir=(«; + c) X a— O. 

Now let T be the time in seconds during which 
this heat was generated; then the average rate at 
which the heat was generated must have been ^ units 
per second. Since 1 unit of heat per second corres- 
ponds to a power of 4166 watts (§ 16), the power, P, 
spent by the electrical current, in watts, is — 

P ^ use ^- ^'^^^ (^+^) O2-O I, 

We now calculate the average current, C7, in am- 
peres, from the angles of deflection (a) averaged as 
in ^ 206, and from the reduction factor of the galvan- 
ometer, 7, already determined (Exps. 78-81) by the 

formula — 

C = I tan a. 11. 

We have finally, by Joule's Law (§ 136) for the 

resistance, £, of the conductor in ohms — 

^ If the calorimeter is of brass, its thermal capacity is .094 W, 
nearly. To this should be added about 0.6 units for the thermal ca- 
pacity of the thermometer and stirrer. See f 90 (2). 



474 ELECTBICAL BESISTANCE. [Exp. 86. 

If the experiment were varied so as to make the 
current just 1 ampere, then, since (7 = 7, -B would be 
equal to P. This is in accordance with 'the defi- 
nition of resistance (§ 136). The student should 
bear in mind that the resistance of a conductor in 
ohms is nothing more or less than the power in watts 
required to maintain in that conductor a current of 
1 ampdre. 



EXPERIMENT LXXXVL 

COMPARISON OP RESISTANCES. 

^ 214. Construction of a Rheostat. — A rheostat 
may be constructed as in Fig. 240. A series of brass 
blocks (/«/) is firmly attached to a plate of ebonite 




Fig. 240. 

(ZX), which is a non-conductor of electricity. The 
brass blocks are connected by coils of German-silver 
wire, which should be well insulated with silk. Each 
wire should be doubled in the middle (see Fig. 240), 
and the double wire should be coiled up or wound 
on a bobbin. The equal and opposite currents in any 
part of the coil thus neutralize each other as far as 



1^214.] COMPARISON OF RESISTANCES. 475 

external magnetic effects are concerned.^ Brass 
plugs B^ (7, &c., are fitted into hollows between the 
blocks, so as to make good electrical connections. 
When all the plugs are in place, a current flowing 
through the blocks in series from the binding-post 
A to the binding-post fi", should meet with a hardly 
appreciable resistance. If, however, one of the plugs 
(as D) is removed, the current is obliged to pass 
through one of the coils. It meets therefore, with a 
certain electrical resistance. 

The resistance of the first coil in the series is usu- 
ally 1 ohm (§ 20) ; that of the second is 2 ohms ; the 
third and fourth are either 2 and 5 or 3 and 4 ohms. 
It is thus possible, by taking out one or more plugs 
at the same time, to introduce resistances from 1 
to 10 ohms into the path of a current. The series of 
resistances may be extended by adding three new 
coils of 20, 20, and 50 ohms' resistance. With seven 
coils, we may thus obtain any resistance from 1 to 
100 ohms. With three more coils of 200, 200, and 
500 ohms resistance, we may extend the limit to 1000 
ohms. With additional coils of 0.1, 0.2, 0.2, and 0.5 
ohms, the resistance may be adjusted to a tenth of an 
ohm, &c. For convenience, extra coils of 1, 10, 100, 
and 1000 ohms are usually provided. The same re- 
sults may be obtained by the series 1, 2, 3, 4, 10, 20, 
80, &c. The line of resistances is usually bent, as in 
Fig. 241, so as to occupy as little space as possible. 
Connections with the two ends of the series are made 

^ The effects of '' self induction " should also be to a great ex- 
tent eliminated by this method of winding the coils. 



476 



ELECTRICAL RESISTANCE. 



[Exp. 86. 



by means of the binding-posts a and d. It is con- 
venient for many purposes to include an entirely 
separate line of resistances, hefc^ in the arrangement. 
In the first part of this experiment the inner line will 
not be required. It should therefore be entirely dis- 
connected from the outer line by the removal of the 
plugs which join the two lines together. 






Fig. 241. 



U'Et'dCi 



Fig. 242. 



Both series of resistances are usually packed in a 
box (Fig. 242), variously called a *' box of coils," a 
" resistance box," or simply a " rheostat." 

^ 215. Detennination of ResiatanceB by the Method 
of SubBtitation. — To find the electrical resistance of 
any conductor, as for instance the coil of the dyna- 




Ql 



Fig. 243. 

mometer employed in the last experiment, the coil 
((7, Fig. 243) is to be connected in series with a 
battery (J?) and a tangent galvanometer (^G^)- The 
deflection of the galvanometer is to be carefully ob- 
served. The dynamometer is now to be disconnected. 



1215.] COMPARISON OF RESISTANCES. 477 

and in its place a rheostat, B (Fig. 244), is to be in- 
troduced into the circuit by means of the binding- 
posts c and d. The plugs* connecting the inner 
and outer lines of resistance are to be removed, 
so that the current can circulate only through the 
outer line. The plugs along this line should all be 
driven lightly into place, and turned round in their 
sockets, so as to make good electrical connections. 
Enough plugs are now to be removed to reduce the de- 
flection of the galvanometer to its former magnitude. 
The resistance in ohms brought into play by the 
removal of each plug is indicated by the number op- 




li 



Fig. 244. 

posite its socket (Fig. 241). If the first resistance 
tried is too small, that is, if it fails to reduce the cur- 
rent sufficiently, one about twice as great is tried ; 
if the first resistance is too large, we try one about 
half as great. In fact we use with a set of resistances 
the same method of approximation as with a set of 
weights (TT 2). 

In the process of trying the several resistances, the 
current from the battery is liable to change. It is well, 
therefore, to replace the dynamometer in the circuit, 
and having observed the galvanometer, to substitute 
immediately the box of resistances (as previously ad- 
justed) for the dynamometer. When two conductors 



478 ELECTRICAL RESISTANCE. [Exp. 86. 

can be thus substituted one for the other in an elec- 
trical circuit without affecting the current, their elec- 
trical resistances are evidently equal according to the 
general principle of substitution (see § 43). We 
have only, therefore, to add together the resistances 
of those coils in the box through which the cur- 
rent flows, in order* to find the resistance of the 
dynamometer. 

To save time in making connections, the terminals 
of the coil G may be carried to the binding-posts a 
and e of the rheostat (Fig. 246). One of the battery 
wires is then carried to d, the other to the galvanom- 
eter (?, and back to /. Plugs connecting h with e, 





Fig. 245. • 

h with e, and c with /, are to be removed ; the others 
are to remain. The binding-posts e and / are thus 
insulated from the rest of the instrument. The bat- 
tery current then flows from d to a through the outer 
line of resistances, then from a to e through the coil 
(7, then through/ to the galvanometer Q- and back to 
the battery. If h and c be now connected by the in- 
sertion of a plug, the current will flow directly from 
d to a, and thus the rheostat resistance will be "cut 
out of the circuit." If the plug connecting h and c 
be removed and inserted between J and «, the current, 
after flowing through the outer line of resistances. 



1 216.] COMPARISON OF RESISTANCES. 479 

will make a short circuit from ( to e, instead of pass- 
ing through the coil C. The coil will therefore be 
" cut out of the circuit." By moving a single plug, 
accordingly, from one place to another, the rheostat 
may be substituted in the circuit for the dynamom- 
eter, and viee versa. The accuracy of the units indi- 
cated by the box of resistances may be provisionally 
taken for granted. 

^216. Determination of ReaistanceB by the Method 
of Interchange. — A battery, JB, (Fig. 246), is to be con- 
nected with a coil, (7, of unknown resistance, and with 
a rheostat, -B, of variable resistance in multiple arc 
(§ 140). The wiies from the coil and from the rheo- 




FiG. 246. 

Stat are to be carried back to the battery, each through 
one half of a differential galvanometer, GO-. The re- 
sistance of the rheostat is to be adjusted if possible, 
by the removal of plugs, so that the deflection of the 
galvanometer may be reduced to zero. Since this 
occurs when the currents through the two halves of 
the galvanometer are equal, the total resistance in 
the two branches of the circuit containing C and B 
must be equal. Assuming therefore that the two 
halves of the galvanometer and the connecting wires 
have equal resistances, the resistance of the coil C 
must be equal to that of the rheostat R. 



480 ELECTRICAL RESISTANCE. [Exp. 87. 

To make sure that the two halves of the galvan- 
ometer are exactly alike, the positions of the coil ((7) 
and rheostat (jB) should now be interchanged, and 
the resistance of the rheostat readjusted if necessary. 

In the absence of a set of resistances by which the 
rheostat may be adjusted within, let us say, ^ of an 
ohm, two adjustments must be made. In one, the 
resistance (jBi) of the rheostat will be too small, and 
the galvanometer will be deflected x^ in one direc- 
tion. In the other adjustment the resistance (^j) of 
the rheostat will be too great, and the galvanometer 
will be deflected y° in the opposite direction. 

The resistance (jB) sought can evidently be found 
by the ordinary method of interpolation (§ 41, ^ 26), 
that is — 

iJ = iJi + ^_ (iZ, — iJ,), nearly. 

^-T y 

In the absence of a differential galvanometer, the 
student should make by the method of substitution 
(^ 215) as many determinations of resistance as time 
will allow. Other methods of comparison will be 
considered in experiments which follow. 



EXPERIMENT LXXXVIL 

wheatstone's bridge. 

^ 217. Determination of Eleotrical ReBiatances by a 
Wheatstone'B Bridge. — A form of Wheatstone's Bridge 
used by the British Association and ordinarily known 



t 217.] 



WHEATSTONE'S BRIDGE. 



481 



as the "B. A. Bridge," is represented, with slight 
modifications, in Fig. 247, which gives 
a view of the apparatus from above. 
Three strips of copper, a6, ce^ and fg^ 
are arranged in a line on a piece of 
wood, with small spaces between them. 
A fine German-silver or platinum wire 
Ay, often called the "Bridge wire"^ is 
stretched over a rail 1 metre long, 
graduated in mm. The wire is soldered 
at both ends to corners of the strips 
(ah and^), which are turned up so as 
to be on a level with the wire. A 
cross-wire is attached to a slider (f. Fig. 
248) so that it may be made to touch 
the wire hj at any point. Binding-posts 
are usually added at a, J, c, d, e, /, ^, 
and i. The latter serves to connect any 
conductor (as Gi) with the cross-wire, 
and thus to make an electrical connec- 
tion between it and any point of the 
wire ij. 

The terminals of a delicate galvan- 
ometer a, (see also ^ 188, Fig. 207) 
are to be connected with the binding- 
posts d and i. The resistance coil (7, 
tested in Exp. 85, is to connect h and c. 
Two binding-posts (a and d, Fig. 242) fio. 247. 



1 To avoid misconceptions arising from this name, it may be well 
to point oat to the stadent at the start that the '* Bridge wire '' is not 
the •* Wheatstone*s Bridge " (§ 141). 

81 



482 ELECTRICAL RESISTANCE. [Exp. 87. 

of the rheostat used in Exp. 86 (^, Fig. 248) are to 
be connected by thick copper wires with e and /(Fig. 
248). One of the plugs is to be removed from the 
rheostat, so as to give a resistance of 1 ohm. The 
poles of a battery (.B) are than to be connected with 
the binding-posts, a and g. 

The current from the battery is thus made to 
divide into two parts. One part flows from a to d 
through the coil (7, then from d to ^ through the re- 
sistance R (or the reverse); the other part flows 
from a to i, through the resistance of the wire hi ; 
then from i to ^ through the resistance of the wire 




Fig. 248. 

ij (or the reverse). The resistance of all other con- 
ductors may be neglected. The galvanometer cir- 
cuit forms a cross-connection or " Wheatstone's 
Bridge " (§ 141) between the points d and i of the 
parallel circuits adg and aig. The points a, d,^, and i 
correspond accordingly to A^ B^ (7, and D in Fig. 18, 
§ 141. The slider i is to be moved from one end of 
hj to the other until a point i is found having the 
same potential as d (§ 141), so that the galvanometer 
shows no deflection. The distances hi and ij are to 
be carefully measured. The poles of the battery are 
next to be interchanged and the experiment repeated. 



t2J7.] WHEATSTONE'S BRIDGE. 488 

The average of the distances hi and ij is to be found. 
Assuming that the wire is uniform, the resistance of 
these portions A and B will be to each other as their 
lengths, hi and ij. That is — 

A_hi 
B if 

The resistance C is now calculated from the resist- 
ance R in the box of coils (1 ohm in this case) by 
the formula (§ 141) — 

C=Rx^. I. 

The experiment is to be repeated with the places of 
C and B interchanged. In this case the formula will 
become — 

C=Bx% 11. 

hi 

By removing from the box of coils different plugs, 
other measurements of the resistance C may be made. 
The student should satisfy himself that with various 
values of B^ the same value of C is always obtained. 
The most accurate value is usually that which is 
found when B is nearly equal to (7. 

If the value of O thus determined differs by more 
than 10 % from that found in the last experiment, 
the latter should be repeated. By this means ^ross 
errors in the box of coils may be found out. It 
should be remembered that the British Association 
Unit which is copied in many boxes of coils is only 
about 987 thousandths of a true ohm. 



484 ELECTRICAL RESISTANCE. [Exp. 88. 



EXPERIMENT LXXXVIIL 

SPECIFIC RESISTANCE. 

^ 218. Speoifio Resistance. — The specific electri- 
cal resistance of a given material may be defined as 
the resistance of a conductor made of that material, 
1 cm. long and 1 9q, cm. in cross-section. In the prac- 
tical units of the volt-ohm-ampdre series, the specific 
resistance, 8^ is equal accordingly to the electromotive 
force in volts (see § 138) required to maintain a cur- 
rent of 1 ampdre between two opposite faces of a 
centimetre cube cut out of a given substance ; or 
again, it is equal to the power in watts (see § 137) 
required to do the same thing. The power required 
to maintain a current of 1 ampere through L centi- 
metre-cubes of the substance, arranged in series, so 
that the same current traverses each, is obviously LS 
watts. If we place Q rows of centimetre-cubes side 
by side, each row containing L of the cubes, it is ob- 
vious that to maintain a current of 1 ampere in each 
row will require LS watts ; hence the total power 
required for all the rows will be QLS watts. 

Since each row is traversed by a current of 1 am- 
pere, the compound conductor, consisting of Q rows, 
must carry a current of Q ampdres. 

The resistance of this conductor may now be 
calculated by Joule's Law (P= 0^ E, see § 136); 



ir219.] SPECIFIC BESISTANCE. 485 

for substituting QLS for P, and Q for (7, we 
have — 

j._ P _ QLS_LS J 

We notice that in the formula L represents the 
length and Q the cross-section of the compound con- 
ductor. The resistance of any conductor is accord- 
ingly proportional to its length, and inversely as its 
cross-section. To find it, we multiply the specific 
resistance by the length and divide the product by 
the cross-section. Obviously, specific resistances of 
different materials are important factors in calcula- 
tions relating to electrical resistance. 

To calculate specific resistance (S), we must first 
find the actual resistance (-B) of a conductor of 
known length (i) and cross-section (0; we then 
have, from I., — 

8 = ^ II. 

It will be found convenient to express the result in 
terms of microhms (§2) instead of ohms. This is 
done by moving the decimal point six places to the 
right (i. e., multiplying by 1,000,000). 

^ 219. Determination of Specific Resistance. — A 
fine German-silver wire (not insulated), about 1 metre 
long, is soldered (near a and 5, Fig. 249) to two cop- 
per strips. These strips are to be so thick that their 
electrical resistance may be neglected. They are to 
be scraped bright (^ 193, 11), and connected with 
the binding-posts h and c? of a Wheatstone's bridge 



48G ELECTEICAL RESISTANCE. [Exp. 88. 

apparatus, in place of the coil used in the last experi- 
ment (see Fig. 248, ^ 217). To prevent the wire 
from crossing itself at any point, it may be looped 
round a glass jar a (Fig. 249). The resistance (^R) 
of the wire is to be found as in the last experiment. 

The wire is now to be straightened, and the dis- 
tance between the copper strips accurately determined. 
This gives the length {L) of the conductor spoken 

OA* 

(^^ ^c 

Fio. 249. 

of in the last section. The diameter (d) of the wire 
is to be measured at let us say ten different points 
with a micrometer gauge (^ 60, II.), and the results 
averaged. The cross-section (0 of the wire is then 
calculated by the ordinary formula — 

The specific resistance of the German silver of 
which the wire is composed is finally to be calculated 
by formula II. of the last section. 

The experiment may be repeated with wires of dif- 
ferent lengths, diameters and materials. 



% 220.] 



THOMSON'S METHOD. 



487 



EXPERIMENT LXXXIX. 

THOMSON'S METHOD. 

^ 220. Determination of the Resistance of a Galvan- 
ometer by Thomson's Method. — The terminals of a 
galvanometer, Cr (Fig. 250), and of a rheostat, -B, are 
to be connected with a Wheatstone's Bridge appara- 
tus in the same manner as any other resistances would 
be connected, when it is desired to compare them 




Fig. 250. 

together (see Exp. 87). A battery, 5, is also to be 
connected in the same manner. Instead, however, of 
putting a second galvanometer in the circuit di, to 
tell when the current in that circuit is reduced to 
zero, a simple key, -BT, is placed there. 

The galvanometer needle will probably be strongly 
deflected by the current passing through the instru- 
ment. It must be brought back nearly to zero by a 
powerful magnet, M, properly placed. If the battery 
is too strong for the magnet, a weaker battery may 
be substituted, or the same result may be obtained by 
connecting the poles of the battery with a cross-wire 
or shunt of suflBciently low resistance. The key is 



488 ELECTRICAL RESISTANCE. [Exp. 89. 

now to be closed. If the effect is to increase the de- 
flection of the needle, the slider (i) is to be moved 
toward that end of the ** Bridge wire " (hf) nearest 
the galvanometer. If the effect is to diminish the de- 
flection, the slider is to be moved toward the rheos- 
tat. Finally a point (i) is found where the closing 
of the key has no effect upon the galvanometer. The 
resistance of the latter is then calculated as in the 
last experiment. 

The experiment is to be repeated with a rheostat 
resistance as nearly as possible equal to that of the 
galvanometer. The current should be reversed, and 
the resistances interchanged as in Experiment 87. 

The resistance of the galvanometer is to be calcu- 
lated by one of the foi-mulae of H 217. 

^ 221. Explanatioii of Thomson's Method. — Thom- 
son's method of measuring the resistance of a galvan- 
ometer depends upon the fact that when the circuit 
di (Fig. 250) is closed through JT, more or less cur- 
rent will ordinarily pass from i to i, or the reverse. 

The electrical potential (§ 139) of the point d will 
therefore be affected, just as the pressure at a given 
point in a water pipe would be affected by connecting 
that point with one in another pipe where the pressure 
was different. Since the current from a to (2 depends 
(according to Ohm's Law, § 138) upon the difference 
of potential between those points, it is evident that 
if a retains the same potential as before, any change 
in the potential at d must affect the current. The 
deflection of the galvanometer is accordingly in- 
creased or diminished. The object of nearly neutral- 



1221.] THOMSON'S METHOD. 489 

izing the deflection is that any change in it may be 
made perceptible ; for if the needle were already de- 
flected for instance 89°, since 90° is the maximum 
possible deflection, it would be hard to detect an in- 
crease in the current. We have seen that the elec- 
trical potential at d is changed when it is connected 
with a point c at a different potential ; obviously if d 
and i are at the same potential, there will be no change 
in the potential of c?, and hence no change in the de- 
flection of the galvanometer. The student should 
note that we may find a point 2, having the same 
potential as a point d, either (1) by observing the 
deflection of a galvanometer in the circuit di (see 
Exp. 87), or (2) by observing the change in the de- 
flection of a galvanometer in any other branch of the 
compound circuit. 

The chief diflBculty in this experiment lies in the 
arrangement of a permanent magnet so as to neutral- 
ize the deflection of a galvanometer needle without 
destroying temporarily the sensitiveness of the in- 
strument. The advantage of this method, aside from 
its theoretical interest, is chiefly in cases where it is 
impossible to obtain a second galvanometer suflB- 
ciently sensitive to measure the resistance of the 
first. 



490 



ELECTRICAL RESISTANCE. 



[Exp. 90. 



EXPERIMENT XC. 

mange's method. 

% 222. Determination of the Internal Resistance of a 
Battery by Mance's Method. — A rheostat (^, Fig. 
251) and a galvanometer (^Cr) are to be connected 
with a Wheatstone*s Bridge apparatus as in Experi- 
ment 87 ; and a battery cell (-B) is to be put in place 
of the unknown resistance ((7, Fig. 248). Instead, 
however, of placing a second battery in the circuit 
ag^ a simple key (-BT) is put there. 




The needle of the galvanometer will probably be 
strongly deflected by the current passing from d to i, 
or the reverse. As in the last experiment, this de- 
flection must be nearly reduced to zero, by bringing 
a powerful magnet (M) near the galvanometer. A 
shunt may be introduced if necessary between the ter- 
minals of the galvanometer (see ^ 193, 2). The key 
is now to be closed. If the deflection of the galvan- 
ometer is increased, the slide (i) is to be moved toward 
the battery. If the deflection is diminished, it should 
be moved toward the rheostat. The change in the 



IT 223.] MANGE'S METHOD. 491 

position of the slider will probably throw the galvan- 
ometer and magnet out of adjustment. The posi- 
tion of the magnet must therefore be changed. After 
a series of trials the slider may be placed at a point t, 
where no sudden effect is produced upon the galvan- 
ometer by closing the key. 

If the galvanometer is affected one way when the 
key is first closed, then the other way, the first 
effect is the one by which the adjustment of the 
slider is to be made. 

The experiment is to be repeated with a resistance 
in the rheostat as nearly as possible equal to that of 
the battery ; but the methods of reversal and inter- 
change employed in Exp. 87 will hardly be justified 
by the accuracy of the experiment. The resistance 
of the battery is to be calculated by one of the for- 
mulae of ^ 217. 

^ 223. Zbeplanation of Mance's Method. — The ef- 
fect in Mance's method of the battery current upon 
the galvanometer has generally to be diminished by 
shunting the galvanometer. The opposite difficulty 
however, sometimes arises. When it is desired to 
measure the resistance of a battery composed of two 
nearly equal cells, opposed to one another, the cur- 
rent from these cells may be insufficient to affect the 
galvanometer. In this case an auxiliary battery 
must be introduced into the circuit akff. We will 
first suppose that such an auxiliary battery is em- 
ployed. If the two cells of which the resistance is 
to be measured exactly neutralize each other, the case 
differs from that of an ordinary Wheatstone's Bridge 



492 ELECTRICAL RESISTANCE. [Exr. 90. 

only in the nature of the resistance which is to be 
measured. The theory is therefore the same. 

If, however, one of the two cells is stronger than 
the other, an allowance must be made for the current 
which flows from the battery (B) through the gal- 
vanometer, whether the auxiliary battery is connected 
or not. This is done by neutralizing the deflection 
of the galvanometer due to the battery B. 

The fundamental principle upon which Mance's 
method depends is that two batteries in any system 
of conductors, however complicated, produce each 
the same effect as if the other were not present. 
The current in any part of the circuit is in fact the 
algebraic sum of the two currents which the batteries 
would separately produce. We have seen that a 
battery in the circuit akff affects a galvanometer in 
the circuit di, unless the resistances ai and ij are pro- 
portional to ad and dg respectively. If a current 
already exists in the galvanometer a change in that 
current must be produced by a battery in the circuit 
akg^ unless the proportion above is fulfilled. 

Let us now suppose that the battery in the circuit 
akg is just strong enough to neutralize the current 
from the battery -B, which would naturally flow 
through the circuit akg. Then the effect of introduc- 
ing this battery into the circuit may be simply to 
arrest the current in akg. The same effect is pro- 
duced by breaking the circuit by means of the key 
K. Evidently the act of opening or closing the key 
in a circuit is equivalent to connecting or disconnect- 
ing a battery of considerable strength. 



ir223.] USB OP A SHUNT. 498 

When the circuit is made the resistance between 
the poles of the battery is much less than when the 
circuit is broken. The result is an increased cuiTcnt 
from the battery, and in a very short time a change 
in its electromotive force. The observations should, 
therefore, be taken the moment that the circuit is 
closed. The galvanometer needle sometimes first 
jumps in one direction, then slowly changes to the 
other direction. The slow movement in the needle 
may be explained as the result of a gradual change in 
the electromotive force of the battery. The first 
effect indicates which of the resistances is too great 
or too small. 

The chief advantage of Mance's method is that it 
enables us to measure the resistance of batteries at a 
given instant while furnishing a current. Concordant 
results must not be expected between Mance's and 
other methods. It is now thought tliat there is some- 
thing not yet understood in the nature of battery re- 
sistances which causes these resistances to appear 
to be greater or less according to the manner in 
which they are determined. 



EXPERIMENT XCI. 

USE OP A SHUNT. 

^ 223. Determination of the Resistance of a Galvan- 
ometer by means of a Shunt. — I. Two tangent gal- 
vanometers (ab and gh^ Fig. 252) already employed 



494 ELECTRICAL RESISTANCE. [Exp. 91. 

in Exp. 79, are to be set up in the same places 
as in that experiment, and connected in series with 
a battery (B) capable of causing deflections of from 
50® to 60°. The connecting wires hcdeg and qfh are 
to be made bare at a point between the two galvan- 
ometers and at a point (e) between the galvanometer 
(^A) and the battery. The wires are to be clamped 
at these points by the binding-posts of a rheostat 
QR). All the plugs are now to be put into their 
places. The galvanometer gh will then be short cii- 
cuited through the rheostat (iZ). The deflection of 
the galvanometer should accordingly fall to 0°. If it 
does not, the plugs in the rheostat should be turned 




Fig. 252. 

round in their sockets with light pressure until at 
least a minimum deflection is obtained. ^ 

When plugs are removed from the box of coils, a 
part only of the current willflow through the rheo- 
stat. The galvanometer (gh) will then be deflected. 
Plugs are to V>e removed from the box until the de- 
flection of the galvanometer (gh) reaches about 30° 
or a little more than half the deflection of ab. The 
resistance of the rheostat is to be noted, and the de- 
flections of the two galvanometers are to be simul- 
taneously determined as in Exp. 82. This method 

^ The plugs should be carefully cleaned if necessary by rubbing 
them with paper. 



IF 224 ] USE OF A SHUNT. 495 

is applicable to galvanometers of low resistance. The 
results are to be reduced by ^ 224, I., formula (6). 

II. Instead of the galvanometer aJ, a second rheo- 
stat resistance may be introduced into the circuit 
edcbaf. The value of this resistance is to be noted. 
The deflections of ihe galvanometer gh must be ob- 
served (as in I.) with and without the shunt ef. The 
resistance of the shunt must also be noted. 

This method requires a constant battery (see Exp. 
84), with an internal resistance which is either known 
(see Exps. 92 and 93) or so small that it may be neg- 
lected in comparison with the resistj^nce in the circuit 
edcbaf. The method is used in practice only in the 
case of high-resistance galvanometers. On account 
of the extreme sensitiveness of such instruments, the 
current from an ordinary voltaic cell must be re- 
duced by the use of a very large resistance in the 
circuit edcbaf. In comparison with this resistance, 
that of the voltaic cell may usually be neglected. 
The resistance of the shunt should be such that 
when connections are made through it, the deflection 
of the galvanometer may be about half as great as 
when these connections are broken. The results 
are to be reduced by ^ 224, IL, formula (12). 

^ 224. Calculations of Resistance depending upon 
the Use of a Shunt. — I. If J and i are the reduction 
factors of the two galvanometers, A and a their de- 
flections, then since the whole (furrent, C, passes 
through the first galvanometer (a6. Fig. 252), it must" 
be given T3y the equation (see formula 7, ^ 199) — 
C=zItanA. (1) 



496 ELECTRICAL RESISTANCE. [Exp. 91. 

Only a portion ((?) of this current passes through the 
second galvanometer (^gK) ; this portion is — 

(? = i tan a. (2) 

The remainder (c') of the current flows through the 
rheostat. Evidently — 

c' = C — c = I tan A — i tan a, (8) 

Now the current (c) through the galvanometer {gK) 
must be to that (c*') through the shunt inversely as the 
resistances (let us say G and 8') in question (§ 140). 
That is — 

c:c'::S:G. (4) 

The resistance of the galvanometer ( 0^) may there- 
fore be found by the formula — 

yy e' S a I tan A — i tan a ^^\ 

c i tan a 

It should be remembered that the resistance of the 
galvanometer (gh. Fig. 262), calculated by this for- 
mula, includes that of the wires, eg and fh^ connecting 
it with the rheostat. The result is rendered inaccu- 
rate by any bad connection within the rheostat. A 
minimum deflection of 1° in the galvanometer (^^A), 
produced with all the plugs in place in the rheostat 
(B), indicates an under estimate of both the galvan- 
ometer and rheostat resistances not far from 1 or 
2%. 

II. If E is the electromotive force of the battery 
(5, Fig. 252), li the resistance in the circuit edchaf 
(including strictly the internal resistance of the bat- 
tery), and if fl^ is the resistance of the galvanometer. 



1224.] USE OF A SHUNT. 497 

the current, C> produced (when the connection be- 
tween e and/ is broken) must be (see § 138) — 

C= ^ . (1) 

If now a connection is made between e and/ through 
a shunt of the resistance S^ so that the current flows 
partly through Q- and partly through S^ the resistance 
(r) of this multiple circuit will be (solving the equa- 
tion in § 140) — 

r = ^^^. (2) 

The current C now becomes — 

or, substituting the value of r and reducing, — 

O = ^C(^ + S^ (4) 

The portion (c) of this current which flows through 
the galvanometer is to the whole current ( (7') as iS is 
to a + SQ 140) ; that is — 

Substituting the value of 0' from (4) we have — 

„_ US 



498 ELECTRICAL RESISTANCE. [Exp. 92. 

hence e=2M±^^+I^. (7) 

But from (1) E= CB+CGi 

hence ^^(^^^^^^^(^S ^ CB+ CG, (8) 

eBa + cBS +caS= CBS + CGS, (9) 

cBa + cGH— CGS= CBS— cBSy (10) 

and G (cB + cS — (7^) = ii-S ( C— c), (H) 

whence, finally, (? = J^ig^, (12) 

In the use of this formula it is necessary to know- 
only the relative values of the currents C and c. With 
nearly all instruments, when the deflections are small, 
the currents are proportional to these deflections. 
We may accordingly substitute the deflections pro- 
duced in such cases for the currents which they 
represent. 



EXPERIMENT XCII. 

ohm's method. 

^ 225. Determination of the Resistance of a Battery 
by Ohm's Method. — A tangent galvanometer (ff,Fig. 
253) and a rheostat ( K) are to be connected in series 
by the wires hc^ rfe, and of, with a Daniell cell (JB) 
capable of deflecting the galvanometer needle 50° or 
60° when all the plugs of the rheostat are in their 



f 225.] OHM'S METHOD. 499 

places. The deflection of the galvanometer is to be 
accurately observed. The 1-ohm plug is now to be 
removed from the rheostat, and the deflection again 
noted. The resistance of the rheostat is then grad- 
ually increased until the deflection of the galvanom- 
eter is reduced to less than half of its original magni- 
tude. In each case, the deflection is to be carefully 
observed, and the resistance noted. 

The connections at b and/ being now interchanged 
(^ 193, 9) so that the direction of the current 
through the galvanometer is reversed, the experiment 
is to be repeated. If any diflferences are observed in 
the deflections corresponding to a given resistance, 




Hi 



Fig. 253. 

the mean angle of deflection is to be calculated in 
each case. 

If «! and 02 are the mean angles of deflection in 
any two cases, R^ and R^ the corresponding rheostat 
resistances, C^ and C^ the currents through the gal- 
vanometer, I the reduction factor of the galvan- 
ometer (Exps. 78, 80, 81), B the resistance of the 
battery, galvanometer, and connecting wires, then 
we have (see ^ 199, 7) — 

Ci = I tan «! (1) ; C^ = I tan a^, (2) 

Now by Ohm's law (§ 138) these currents are in- 
versely as the corresponding resistances, that is — 



600 ELECTRICAL RESISTANCE. [Exp. 92. 

Ci : (7, : : iZ, + 5 : iJi + 5, (3) 

hence we find — 

A C, + BC\ = Ii, C,-\-BCu (5) 

BC, — BC, = B, C, — B, Ci, (6) 

B(C\-CO = E,C,-R,C,, (7) 

J5 = A^^^t, (8) 

and finally, substituting the value of Oi and Ca, and 
cancelling J, we have — 

n B2 tan aj — Ri tan a^ ^q>. 

tan a^ — tan a^ 

The student may thus calculate several values of B. 
The best value for J3^ is 0; that is, we obtain the 
most accurate results by utilizing the observation of 
the galvanometer when all the plugs are in place. 
Evidently if B^ = 0, the value of B becomes simply 

B = ^ ^"^^ ^' . (10) 

tan ai — tan a^ 

The best value for R^ is one nearly equal to B, 
The simplest way to find this value is to calculate 
the value of B from any two of the observations. 
It must be remembered that the battery resistance 
thus calculated includes that of the galvanometer 
and connecting wires. Having found the resistance 



1 226.] 



BEETZ' METHOD. 



601 



of the galvanometer, &c. from the last experiment, we 
may find by subtraction the internal resistance of the 
battery. The results with a tolerably constant bat- 
tery should agree with those obtained by Mance's 
Method (Exp. 90) within 5 or 10 %. 

The calculation of the electromotive force of a 
battery from the results of Ohm's Method will be 
considered in ^ 230. It may be remarked that if 
this electromotive force is not constant, formula (3) 
is not justified. In this case the succeeding formulae 
which depend upon (3) may give false or even absurd 
results. 



EXPERIMENT XCIII. 

BEETZ' METHOD. 

^ 226. Explanation of Beetz' Method. — In Beetz* 
method two batteries, H and E' (Fig. 254) are 
placed in the same circuit (abcda)h\xt so as to be op- 

12 8 4 5 




Fig. -^64. 

posed to each other ; and the circuit is divided into 
two lobes, like a figure 8, by means of a wire ac^ act- 
ing as a shunt to both batteries. A known resistance 
Bf is placed between b and c ; another known rej-ist- 
ance (^) is introduced between a and c\ a delicate 



502 ELECTRICAL RESISTANCE. [Exp. 93. 

galvanometer (6?) is placed between c and d. We 
will suppose that the two positive poles of the bat- 
teries are connected at c. 

Let us now consider what effect the battery -B' 
would produce if 5" were not acting. The current 
descending in the branch be would divide into two 
parts (Fig. 254, 2) ; one flowing directly from c to a, 
the other indirectly from c to a through d. These 
two parts would unite at a, and thence return to the 
battery. 

Let us next consider what effect B' would pro- 
duce if jB' were not acting. The current ascending 
in dc (Fig. 254, 8) would divide into two parts ; one 
flowing directly from c to a, the other indirectly 
from <? to a through 6. Both parts uniting at a would 
return to the battery. 

When both batteries act together, each may be con- 
sidered to produce the same effect as if the other 
were not acting. The result is represented in Fig. 
254, 4. We notice that in the diagrams the portion 
of the current from B' which flows through d is as 
great as the whole current from B'L To produce this 
effect it is evident that the batterj'^ B must be stronger 
than B'\ It is also evident that two equal and op- 
posite currents through d must neutralize each other; 
hence the result of combining two batteries as in 
Fig. 254 may be such as is represented in Fig. 254, 
5 ; namely, a current entirely confined to the circuit 
6(?, containing the stronger battery, no current what- 
ever flowing through the weaker battery. 

In practice we employ a battery, 5', more than Bvffi- 



11227.] BEETZ' METHOD. 503 

dent to reverse jB''; then we weaken the current 
which it sends through the circuit d, either by in- 
creasing the resistance jB', so that the whole current 
from B^ is reduced, or by diminishing the resistance 
i2, so that a greater portion of the current may flow 
directly from e to a, without passing through the 
battery B'\ The use of the galvanometer, (7, is 
simply to tell when an exact balance has been estab- 
lished between the two opposing currents through d 
(see Fig. 254, 4). No current is then indicated by 
the galvanometer. 

It is possible to calculate by Ohm's Law (§ 138) 
and by the principle of divided circuits (§ 140) the 
magnitude bf each of the currents represented in 
Fig. 254, 4, and thus to find under what conditions 
the currents through d are equal and opposite. The 
expressions become, however, more or less compli- 
cated. The final solution, which is simple, may be ob- 
tained much more easily by the method which 
follows. 

^ 227. Principle of Electromotive Forces in Equilib- 
rium. — Let M be the electromotive force, and jB' the 
resistance of the first battery ; let E'^ be the electro- 
motive force of the second battery (-B'O' ^^^ '®^ ^ 
be the current through the rheostat iZ. Then if, ac- 
cording ^to the diagram (Fig. 254, 6) the current 
through B" has been reduced to zero, the current C, 
having no choice of circuits must flow through B' 
and R as well as through jB. The result is the same 
as if the circuit through B" did not exist. We have 
accordingly an electromotive force E\ causing a cur- 



504 ELECTRICAL RESISTANCE. [Exp. 93. 

rent C through a total resistance jB + -B' + J2'« 
Hence, by Ohm's Law (§ 188),— 

E= CCE + B' + R). (1) 

The power of the battery is spent in heating the sev- 
eral resistances iZ, -B', and Bf. We need to consider 
only the power (P) spent in heating the resistance 
R. We have (see § 136) — 

P=C^R. (2) 

The ratio of this power (P) to the current ((7)deter- 
mines that part {E) of the whole electromotive force 
{E') which is required to maintain the current ((7) 
through the resistance (i2) in question; Since in 
passing through the resistance R the loss of potential 
is E, we have (see §§ 187, 138, and 139) — 

E=^^^=CR. (8) 

The power spent by the battery jB" upon a small cur- 
rent C' flowing through it in the ordinary direction 
(from a to c) will be (7" E' (§ 187) ; but the power 
required to take electricity from a point a to a point 
<?, where the electrical potential is higher than at a by 
the amount E^ is C" E. Evidently such a current 
through the battery can exist only on condition that 
E^ is greater than E. 

On the other hand, a current (7" flowing from c to 
a would represent an expenditure of power equal to 
C^^ E. The power required to drive the current 
backward through the battery £" is, however, (7" E'\ 



1228] BEETZ' METHOD. 606 

Evidently a reversed current can exist only if ^is 
greater than II'\ It follows that if JS and J?" are 
equal, the current through B" will be reduced to zero. 
It is evident, conversely, that if the galvanometer in 
the diagram (Fig. 254, 1) shows no deflection, H and 
E'' must be equal ; that is (from 3), — 

I!'=CB; (4) 

from which we find — 

rr// 

a formula by which we may calculate the current 
from a bmttery (^) which, flowing through a known 
resistance, i2, neutralizes a known electromotive 
force, I!\ 

^ 228. Calculation of Battery Resistancea in Beetz' 
Method. — For the determination of the resistance of 
a battery by Beetz' method, two experiments are 
necessary. Let r^ and r/ be the values of B and B 
(^ 226) in the first experiment, and let r^ and r^ be 
the corresponding values in the second experiment. 
Then from ^ 227 we have, dividing (1) by (4),— 

W 7, "' ^^^ 

and ^ = -^ + ^^ + ^^ (2) 

Assuming that the proportion between U' and ^" is 
the same in both experiments, we have, equating (1) 
and (2),— 



506 ELECTRICAL RESISTANCE. [Exp. 93. 



(3) 

Bn + n r, H- r/ r, = JS^ + r^ r^ + n r,' (4) 

Bra — 5r, = r,r; — r/r, (6) 

B=^jI^j:Zlil3. (6) 

The same result may be obtained from formula (8), 
% 226, namely, — 

^ c\-o, ' ^^^ 

by substituting for the total external resistances iZ^ 
and iZj their values, rj + r/ and ra + ^2' respec- 
tively, and also substituting for the two correspond- 
ing currents 0^ and C^ their values '(from ^ 227, 

7^// 7^// 

formula 6) — and — respectively. The factor H" 

is cancelled in the reduction. 

Beetz' method differs from Ohm's method chieflj'' 
in the manner in which we estimate the relative 
strength of two currents. In Ohm's method the ratio 
between the currents is determined by the angles of 
deflection produced in a tangent galvanometer. In 
Beetz' method, it is determined by the resistance 
between the poles of a constant battery, enabling 
the current to neutralize the effect of that battery. 
Beetz' method is essentially a null method (§ 42). 

Beetz' method may be used not only to measure 
the resistance of a battery (see 6), but also, when 
that resistance has been found, to determine the rela- 



1229.] BEETZ' METHOD. 607 

tive magnitude ^ of two electromotive forces (see 1 
and 2, also % 230, 8). 

When the electromotive force of a battery is 
known, it furnishes us with the means of measuring 
currents with great precision (see formula 6, % 227). 
^ 229. Determination of Battery Reaistancea by 
Beetz' Method. — The copper or positive pole (P, Fig. 
255) of a battery (-6), consisting of two Daniell cells 
in series, is to be connected by a wire (^PKK'J^} with 
the positive pole (P') of a weaker battery (-6'). 
The circuit is to be completed between the negative 
poles (iV' and iV) of the batteries through a delicate 
galvanometer ( ff) provided with a shunt (*S') to pre- 




FiG. 265. 

vent it from being injured by the battery currents 
(^ 193, 2) and through the inner line of resistances, 
6c, of a box of coils. The inner and outer lines, be 
and da^ are to be connected with a plug between 
c and d, but separated at a and b throughout the ex- 
periment. The wire PKKP" is to be made bare at a 
and connected at that point with the binding-post of 

^ If a tangent galvanometer be introduced into the circuit of the . 
stronger battery (B"), for instance between a and &(Fig. 254), so that 
the current C becomes known, we may calculate also the absolute 
▼alues of the electromotive forces by formulae (1) and (4) of If 227. 
This important modification of Beetz' method is due to Foggendor£f. 
See t 230, 3, and Exp. 99. 



608 ELECTRICAL RESISTANCE. [Exp. 93. 

the outer line of resistances. Keys {K and K'^ are 
to be placed one on each side of a. When all the 
plugs are in place, and the keys closed, the circuit of 
the battery (jB) is completed through the lines of re- 
sistance be and da^ the course of the current being 
PKadcbN, The circuit of B is also completed 
through the outer line da^ thus: P'K'adcQN', The 
student should note the direction in which the gal- 
vanometer is deflected. 

When the connection between a and d is broken 
by removing the " infinity plug,"^ both of the circuits 
named above ai*e interrupted. If the keys K and K' 
are closed, the batteries will be opposed to one an- 
other. Neither battery can furnish a current unless 
it is strong enough to force it backward against the 
other battery. If the battery B is stronger than B\ 
the current will follow the course PKaK'P'N'QchN. 
Since the current in S is reversed, the galvanometer 
will be deflected in the opposite direction. The stu- 
dent should make sure that this is the case. If^it is 
not, there is probably some error in the connections, 
which must be corrected. 

The infinity plug is now to be returned to its place, 
and other plugs removed between a and d. 

It will be seen that when the resistance of the 

1 Two of the brass blocks in each chain of resistances should have 
no metallic connection between them, except that furnished by the 
plug. When the plug is removed there should be no perceptible cur- 
rent from one block to the other. In other words, the resistance be- 
tween the blocks should be practically infinite. The plug in question 
is called accordingly the ** infinity plug." It is usually marked oo or 
INF. 



t229.] BEETZ' METHOD. 609 

outer line a(2, common to the two battery circuits, is 
very small, the galvanometer is deflected one way ; 
when the resistance is very large the galvanometer 
is deflected the other way. The next step is to find, 
by gradually increasing the resistance, at what point 
the change in the deflection takes place. 

To avoid using up the batteries (^ 198, 10), the 
keys K and K' should be left open, except at the 
moment when it is desired to test the deflection of 
the galvanometer. The key K in the circuit of the 
stronger battery is always to be closed first, then the 
other key, K\ immediately after it. As soon as the 
direction «of the deflection has been recognized, the 
keys are opened in the inverse order.^ 

If the galvanometer is deflected in the same way 
as when all the plugs are in place, the resistance of 
the outer line (ad) is to -be increased ; if it is de- 
flected as when the connection in ad is broken, the 
resistance is to be diminished. The sensitiveness of 
the galvanometer may be increased if necessary by 
removing the shunt (<S) but the student must not for- 
get to replace the shunt before proceeding to the 
second part of the experiment. The resistance of the 
outer line {ad) causing the deflection of the galvan- 
ometer to disappear is to be recorded. If no such 
resistance can be found, the two nearest resistances 
should be noted, and the deflections (one in one direc- 
tion, the other in the other direction) caused by each 
should be observed. From these results the desired 

1 A " double key " or other mechanical contrivance for closing two 
circuits one after the other will be found useful in this experiment. 



510 ELECTRICAL RESISTANCE. [Exp. 9a 

resistance is to be calculated as in ^ 216» by interpo- 
lation (§ 41). 

So far the resistance in the inner line be has been 
zero. This resistance is now to be increased by re- 
moving the 10-ohm plug. If the keys be closed, the 
galvanometer will be deflected. To reduce the de- 
flection to zero, it will be necessary to increase the 
resistance of tlie outer line (ad). The resistances of 
both parts of the rheostat (be and ad)^ causing equi- 
librium in the galvanometer are to be noted. 

The battery resistance is to be calculated by for- 
mula 6, ^ 228 ; remembering that the values of ad 
correspond to the resistances r^ and r2, common to 
the two circuits, while the values of be correspond to 
the resistances r/ and r,', in the circuit of the stronger 
battery. 



IT 230.] CLASSIFICATION OF METHODS. 511 



ELECTROMOTIVE FORCE. 

^ 230. Different Methods for the Determination of 
Electromotive Forces. 

I. Absolute Methods. Electromotive force 
(see § 137) is defined as the ratio of the power spent 
by any source of electricity to the current which it 
produces. We must distinguish between methods 
(1-4) in which the power thus expended is abso- 
lutely measured and those (5-12) in which compara- 
tive results only are obtained. - 

(1) Method op Heating. The power spent by 
an electric cuiTcnt may be measured in the same way 
as electrical resistance (Exp. 85), by passing a current 
from a battery through a coil of wire surrounded 
with water, and calculating from the rise of tempera- 
ture of the water how much energy has been spent 
by the current in a given length of time.^ If the 
strength of the current be known, the loss of potential 
may be found by the general formula (§ 137) — 

Thus if a current of 2 ampSres is found to heat the 
equivalent of 100 grams of water 15° in 1000 seconds, 
so that it generates 1^ units of heat in one second, 

1 See Glnzebrook and Shaw, Practical Physics, § 74. 



512 ELECTROMOTIVE FORCE. 

since 1 unit of heat per second is equivalent to 4.166 
watts (§ 15), IJ units per second would be equiva- 
lent to 6.249 watts, or 6.249 -5- 2 = 3.124 watts per 
ampdre. We know, therefore, that the difference in 
potential (§ 139) between the two ends of the coil of 
wire must be 3124 volts. It will not do, however, 
to assume that this is equal to the electromotive 
force of the battery ; for we have left out of account 
.the heat generated by the electrical current in the 
connecting wires and in the interior of the battery. 
Unless the electrical resistance of the battery be unu- 
sually small in comparison with that of the coil, a 
considerable portion of the electrical energy will be 
thus wasted. 

At the same time* that the method of heating can 
not in practice be employed to determine directly 
the electromotive force of a battery, it must be 
remembered that all determinations of electromo- 
tive force which involve a measurement of current 
and resistance may depend indirectly upon the 
method of heating, since this is one of the funda- 
mental methods by which resistances are measured 
(Exp. 85). 

(2) Ohm's Method. Having once determined a 
standard of resistance by the Method (»f Heating 
(Exp. 85), we have seen how by various methods 
of comparison (Exp. 86-93) the resistance of any 
part of an electrical circuit may be found. In Ohm*s 
method, we find the current ( (7) in a simple circuit, 
and calculate the resistance (5) of this circuit by 
adding together the resistances of its separate parts. 



1230.] CLASSIFICATION OF METHODS. 618 

Then, by Ohm's Law, we have for the electromotive 
force (^E) the general equation (§ 138) — 

E= CR. 

Substituting in this formula the value of 72, which 
in the absence of any resistance except that of the 
battery, galvanometer, and connecting wires, is given 
by formula 10, ^ 225, namely — 

tan «! — tan a, 

and substituting also the corresponding value of (7, 
namely, I tan a^ we have — 

p JjRa tan g, tan a^ 

tan a^ — tan a, ' 

The student may show that the same formula is ob- 
tained if we multiply the total resistance (jB -|- ^2) 
in the second part of the experiment by the current 
((72 = I tan a^) which flows through it. The agree- 
ment of the two results must not be taken as an 
indication that the electromotive force is the same in 
both parts of the experiment, but as the necessary 
consequence of * the formulae of ^ 225, in framing 
which we have assumed that the electromotive force 
of the battery is constant. 

(3) Poggendorff's Method. It has already 
been shown in Beetz' method (Exp. 93) that the cur- 
rent from a battery may be neutralized by meeting a 
counter current caused by division of a current from 
a more powerful battery into two parts. This is 

33 



514 ELECTROMOTIVE FORCE. 

the principle of Poggendorffs absolute method (see 
Exp. 99), which differs from Beetz' method simply in 
the fact that a tangent galvanometer is introduced 
into the circuit of the more powerful battery (^, 
Fig. 254) as a means of measuring the current (see 
note, ^-228). Given the current, (7, and the resist- 
ance, jR, the electromotive force (^E) is calculated 
by the ordinary foimula (§ 138) — 

E= OR. 

(4.) Electrostatic Methods. The electromo- 
tive force of a powerful battery may be measured by 
the repulsion between two pith-balls charged by the 
battery under certain conditions (see.^ 258). Elec- 
trostatic forces are also measured in absolute elec- 
trometers of various kinds (see ^ 270). It should, 
however, be remembered that results obtained by 
such instruments are strictly in the electrostatic 
system. Since the relation between the electrostatic 
and the ordinary (electromagnetic) systems are not 
known with any great degree of accuracy, the use of 
electrometers, as far as the latter system is concerned, 
is practically confined to the comparison of electro- 
motive forces (see ^ 230, 11, also ^ 270). 

II. Comparison of Electromotive Forces. 
The absolute measurement of electromotive force is, 
like the absolute measurement of resistance upon 
which it depends, a more or less difficult problem. 
The comparison of two electromotive forces may, 
however, be made with a considerable degree of pre- 
cision. 



1 230.] CLASSIFICATION OF METHODS. 615 

(5) The Volt-Metbb. Two electromotive forces 
may be compared by the currents separately produced 
by them through equal resistances. When the re- 
sistance of a battery is unknown, it is evident that 
this method cannot in general be applied; for the 
battery resistance may be a considerable part of the 
resistance of a circuit. In practice, few batteries 
have a resistance of more than 10 ohms ; in fact 1 
ohm would be much nearer the average battery re- 
sistance. Hence if a galvanometer has a resistance 
of several thousand ohms, the battery resistance may 
usually be disregarded. This is the principle on 
which volt-meters are constructed (Exps. 96 and 97). 

(6) Wiedemann's Method. In Wiedemann's 
Method (Exp. 94), two batteries are joined in series 
with a tangent galvanometer of low resistance. 
Whether the batteries act in the same or in opposite 
ways, the total resistance in the circuit is the same 
(see note ^ 197). It follows, therefore, frdm Ohm's 
law (§ 138), that the current is proportional in one 
case to the sum, in the other case to the difference of 
the electromotive forces E and e\ hence the sum 
(j& + ^) is to the difference (J? — e) as the currents 
(7 and c produced, that is — 

E + ei U—e :: : c. 

(7) Method of Opposition. Let us now suppose 
that N cells of the electromotive force U being op- 
posed to N' cells of the electromotive force H' reduce 
the current to zero, then obviously the electromotive 
force NE = N'E'\ or, E' lEiiNi N\ 



616 ELECTROMOTIVE POBCE. 

This is a fundamental method of comparing electro- 
motive forces, the usefulness of which is limited only 
by the difficulty of obtaining enough cells of each 
kind to make an exact balance. We note that, in this 
method, we compare the electromotive forces of two 
batteries when at rest, and not (as in previous meth- 
ods) when in action. The method of opposition is 
essentially a " null method " (§ 42) for the compari- 
son of electromotive forces. 

(8) Beetz* Method. When, as in Experiment 93, 
a battery current is neutralized by part of the current 
from a more powerful battery, we cannot find the 
electromotive force of either battery absolutely, un- 
less, as in (3), the whole current from the stronger 
battery is measured, as well as the resistance which 
it traverses between the poles of the weaker battery. 
We may, however, find the relative electromotive 
forces from formulae 1 and 2, ^ 228. Hence if the 
electromotive force of one battery is known, that of 
the other may be determined. It may be remarked 
that by this method we compare the electromotive 
force of one battery when at rest with that of another 
when in action} 

(9) Clabk's Potentiometer. Again, if a cur- 
rent ((7) flowing through a resistance B neutralizes 
one battery (as in Exp. 93), while the same current 
flowing through a resistance r neutralizes another 

1 By Bubstitutlng one battery, B, for another, R (Fig. 254), as the 
active source in Beetz' Method (Exp. 93) we may compare the two 
successively with a third electromotive force, B"'. This gives us a 
null method by which we may compare the electromotive forces of 
two batteries (B and B') when in action. 



ir230.] CLASSIFICATION OF METHODS. 617 

battery (in the same manner), the electromotive forces 
of these batteries, being CB and Cr respectively, are 
to each other as R is to r. The proportion between 
them may therefore be found, independently of any 
measurement of electrical current. This is the prin- 
ciple of Clark's Potentiometer (Exp. 98), and is 
undoubtedly the best method of comparing the elec- 
tromotive forces of two constant batteries when not in 
action. 

(10) Use of Condensj:es. The relative strength 
of two batteries may be found by charging a con- 
denser (see ^ 257) first by one battery, then by the 
other. The quantity of electricity stored in the con- 
denser is found to be proportional to the electromo- 
tive forces in question. It is estimated by discharging 
the condenser through a ballistic galvanometer, and 
observing, as in Experiments 76 and 77, the throw of 
the needle. 

(11.) Use of Electrometers. The electromo- 
tive force of a battery may be determined by con- 
necting the poles with an electrometer (^ 270) ; but 
in order to interpret the indications of the instru- 
ment, it must first be calibrated by a series of elec- 
tromotive forces of known strength. The chief 
advantage of the use of an electrometer over that of 
a volt-meter is in the case of inconstant electromo- 
tive forces, especially those which disappear as soon 
as a current begins. The use of a condenser has the 
same advantage, and is frequently preferable on ac- 
count of the liability of electrometers to be out of 
order. Neither instrument is suitable for an elemen- 
tary class of students. 



518 ELECTBOMOTIVE FOECE. [Exp. 94. 

(12) UsB OF AN Electbio Spabk. Electromo- 
tive forces may be estimated roughly by the distance 
which an electric spark can be made to jump (see 
Table 86). This method is particularly suited for 
experiments with a Ruhmkorff coil, or other instru- 
ment in which large differences of potential exist for 
an instant only. 



EXPERIMENT XCIV. 

WIEDEMANN'S METHOD. 

^ 281. Determination of ElectrcmctiTe Forces by 
Wiedemann's Method. — (1) Two Daniell cells, A 
and -B, one of which (A) has been used in Ohm's 
method (Exp. 92), are to be connected in series with 
a tangent galvanometer ( (7, Fig. 
256, 1). The connections are 
to be such that the cells act 
together. The deflection of 
the galvanometer is to be ob- 
served. (2) Then the conneo- 
Fio. 266. tions of B are to be reversed 

(Fig. 256, 2), and the deflection again noted. (3) 
The galvanometer connections are then to be inter- 
changed, and the deflection observed (Fig. 256, 8). 
(4) Finally the connections of B are to be inter- 
changed, so that the two cells may act together as at 
first (Fig. 256, 4), and the deflection of the galvan- 
ometer determined. 




1231.] WIEDEMANN'S METHOD. 619 

Let E be the electromotive force of the stronger 
cell, and e that of the weaker cell ; let A be the aver- 
age deflection caused by the joint action of the two 
cells, and O the corresponding current ; let a be the 
average deflection, and c the current produced by the 
two cells when in opposition; then by formula 7, 

11199- 

0=ItanA, (1) 

{? = I tan a. (2) 

Now by Ohm's law (§ 138), as has been explained in 
^ 230, 6, we have — 

E^e-7' ^^> 

or Ec + ec = EC—eCj (4) 

whence eC + ec = EC — Ec^ (5) 

or e(^C+c)=E (^C—c); (6) 

from which we find — 

Substituting the values of C and c from (1) and (2) 
and cancelling the factor J, we have — 

jp tan A — tan a .q. 

« = -fir J , ( 8 ) 

tan A + tan a 

or ^^^ tanA + tana ^ ,9 

• tan A — tan a 

It should be noted that if the reversal of the cell B 
does not affect the direction of the current, — that is. 



520 ELECTROMOTIVE FORCE. [Exp. 96. 

if the deflections in Fig. 256, 2 and 3, are in the same 
direction as in 1 and 4 respectively, — the electromo- 
tive force of the cell -B, being less than that of -4., is 
to be calculated by formula 8 ; but if the reversal of 
B causes a reversal of the current, the electromotive 
force of B is greater than that of A^ and is hence to 
be calculated by formula 9. The electromotive force 
of -4., already computed, may be found from the re- 
sults of Ohm's method by the formulae of ^ 230, 2. 
The electromotive force of the two cells combined is 
now to be calculated by adding E and e together. 

II. The experiment is to be repeated with the bat- 
tery composed of the two cells just employed and a 
Bunsen cell. The cells are first to be set up in series 
with the Bunsen cell and the galvanometer, then 
both of the Daniell cells are to be reversed. * 

The deflections are to be observed and the electro- 
motive force of the Bunsen cell is to be calculated. 



EXPERIMENT XCV. 

THE THERMO-ELECTRIC JUNCTION. 

^ 232. Determination of the Electromotive Force of 
a Thermo-electric Junction — An iron wire (ai, Fig. 
257) and a German-silver wire (ac), insulated by sur- 
rounding them with India-rubber tubes, are soldered 
together at a ; and the junction (a) is enclosed in a 
steam heater. The other ends, h and c, are soldered 
to insulated copper wires, bd and ce. The junctions 



IT 232.] THE THERMO-ELECTRIC JUNCTION. 



521 



b and o are placed in a beaker and covered with melt- 
ing ice. A thermo-element is thus formed with an 
electromotive force of about 3 thousandths of a volt. 
The object of this experiment is to measure the elec- 
tromotive force in question. 

I. The terminals of the thermo-element (rf and e) 
are to be connected with two pole-cups of a differen- 
tial galvanometer (dg) so that the current from the 
thermo-element circulates in one half of the coil of 
the galvanometer. 

The other half of the galvanometer is to be con- 
nected through a rheostat (hi) with the poles (/ and 




Fio. 267. 



K) of a voltaic cell of known electromotive force 
(^ 230, 2). There should be at first, let us say, 
1000 ohms' resistance in the rheostat. The connec- 
tions are to be made so that the current from the 
Daniell cell may produce upon the needle an effect 
opposite to that due to the thermo-element. The re- 
sistance of the rheostat is now to be increased or dimin- 
ished until the two currents exactly neutralize each 
other. The rheostat resistance (22i) is then noted. 

An additional resistance (r) of known amount, 
about equal to that of the galvanometer (see Exp. 
89), is now to be introduced between b and rf, or be- 



522 ELECTEOMOTTVE FORCE. pSxp. 95, 

tween c and e, and the resistance of the rheostat (Az) 
again adjusted so as to produce equilibrium. The 
new value of the resistance (12,) is also to be noted. 

II. If a differential galvanometer cannot be ob- 
tained, the thermo-electric junction is first to be con- 
nected with the galvanometer, and the deflection (D) 
noted ; then the resistance (r) is to be introduced, 
and the deflection (rf) again noted. The Daniell 
cell is then to be connected with the galvanometer 
through a resistance (A)) ^^ch that the deflection of 
the needle is the same as D. Then the rheostat re- 
sistance is increased to a value M^ which produces a 
deflection equal to d. The results of I. and II. are 
to be reduced by formula (10), ^ 233. 

T 233. Calcniatlon of the Electromotive Force of a 
Thermo-electric Junction. — If in the thermo-electric 
circuit (jahdeca^ Fig. 267), e is the electromotive force, 
and b the electrical resistance of the thermo-element, 
g the resistance of the galvanometer, or that part 
of it which is included in the circuit in question, c^ 
the current in the first part of the experiment, c^ the 
current in the second part of the experiment, and r 
the resistance added ; if, furthermore, in the voltaic 
circuit (^fghijkf^ Fig. 267), J^is the electromotive force, 
B the battery resistance, 0- the galvanometer resist- 
ance, Ri and R^ the two rheostat resistances, and C^ 
and (7a the corresponding currents, we have (§ 138), 
since the currents c^ and Q are equal, — 



IT 233.] THE THERMO-ELECTRIC JUNCTION. 523 

and since the currents c^ and (7, are equal — 



^^ = T> X n. X !> ' (2) 



h + 9 + r ' B + a + R, 
From (1) and from (2) we find — 

By either of these formulae (3 or 4) we may calculate 
the value of e from the observed values of r, iZj, and 
R^, if 6, g^ B, G, and JE^ are known (Exps. 87-92). 
The student should bear in mind that the resistance 
of each part of the galvanometer in this experiment 
is about twice that of the two parts in multiple arc 
(§ 140), and half that of the two parts in series. A 
result independent of the battery and galvanometer 
resistances may be obtained by combining the obser- 
vations obtained in the first and second parts of the 
experiment. Dividing (2) by (1) we have — 

b+ff _ B + a + R, ,5. 

b+g + r B^G + R,' ^^ 

whence (i + <7) B+ (b + g) G + ^b+g) R, 

= <ib+g:^B+Cb+g:^G + Cb + g)R, 

^r(B+G+R{). (6) 

that is, — 

(J + i/) iJ, - (6 +i,) A = r (5 + G^ + i2.), (7) 
or (6+5r)(2J,_2J,) = r-(5+(y+iJ,); (8) 



524 ELECTROMOTIVE FORCE. |Exp. 96 

from which we find — 

t + ,- -(J+g+ii,) . (9) 

Substituting this value in (3) and cancelling (-6 + 
G + Ml), we have finally — 



EXPERIMENT XCVI. 

THE VOLT-METER, I. 

^ 234. CaUbration of a Volt-Meter. — The name 
volt-meter is given to any instrument capable of in- 
dicating directly the value of an electromotive force 
in volts. One of the forms ordinarily employed 

(Fig. 258) is similar in ex- 
ternal appearance to the 
ammeter shown in Fig. 231, 
Fio 258. ^ 210. There is, however, 

an essential distinction between these instruments. 
In the ammeter, the coil a is made so as to have the 
smallest possible electrical resistance, in order that this 
resistance may be neglected. In the volt-meter, the 
finest possible wire is employed in this coil, so that 
the current which flows through it may be neglected. 
The simplest way to calibrate a volt-meter is to con- 
nect it with a battery containing different numbers 
of voltaic cells in series (see Fig. 220, ^ 196). Hav- 
ing found the electromotive force of each cell (see 




1 234.1 CALIBRATION OF A VOLT-METER. 525 

^ 230), we may calculate that of the whole battery 
by adding these electromotive forces together. The 
difference between this calculated value arnd the ob- 
served reading of the volt-meter gives the correction 
of the volt-meter for the reading in question. A 
delicate galvanometer ((y, Fig. 259) connected in 
series with a rheostat (jR) is a convenient substitute 
for a volt-meter in the measurements relating to the 
electromotive force of batteries. The resistance in 
the galvanometer circuit should be so great that we 
may entirely neglect the current which flows through 
the instrument in comparison with the other currents 
used in this experiment. To test such a combination, 
it is to be connected with a battery of known electro- 
motive force, as for instance, the Daniell cell em- 
ployed in Experiment 92. If a common astatic 
galvanometer is employed (Fig. 207, ^ 188), the re- 
sistance of the rheostat should be such as to give a 
deflection of about 45°. This resistance should be 
noted, and should remain unchanged through all the 
experiments with the instrument of which it now 
constitutes an essential part. 

An ordinary astatic galvanometer does not obey 
the law of tangents (^ 195) closely enough even for 
rough determinations. It is necessary, accordingly, to 
test the reading of the instrument with a series of 
electromotive forces bearing known ratios to one 
another. 

A simple device by which this object may be at- 
tained consists of a uniform straight wire, traversed 
by a current from a constant battery. The ** bridge- 



526 ELECTROMOTIVE FORCE. [Exp. 96 

wire " of the Wheatstone's apparatus (^\ Fig. 259) may 
be employed. A battery (5) of two Bunsen cells 
in series will probably be required to give the neces- 
sary current. The poles should be connected with 
the ends of the wire by means of screw cups (b and 
/) provided for that purpose. 

Contact is now to be made between this wire and 
the terminals of the volt-meter (^G-K) at points 10 
cm, apart. This may be done by the aid of two 
sliders, similar to the one used in Experiment 87. 
Pressure must be exerted upon the sliders to insure a 
good electrical contact (^ 193, 11). The deflection 




Fio. 259. 

of the galvanometer is to be noted. The experiment 
is to be repeated with contact at two other points the 
same distance apart, but in a different part of the 
wire.^ 

The sliders are now to be interchanged and the 
deflections determined as before. 

The direction of each deflection, whether between 
north and east or between north and west should 
be noted. 

^ A record of the reading of each slider corresponding to a given 
deflection should be preserved, since it maj be useful in comparing 
the resistances of diflerent parts of the wire. 



t 234.] CALIBRATION OF A VOLT-METER. 627 

The experiment is now to be repeated with contacts 
at two points 20 cm, apart, then 30 cm., 40 cm., &c., 
up to 80 or 100 cm. (the length of the wire). The 
observatfons should be repeated in the inverse order 
to eliminate variations in the strength of the battery. 

The average deflections, corresponding respectively 
to 10, 20, . . . 80, or 100 cm., are now to be calculated, 
and the results are to be plotted on co-ordinate paper 
as is Fig. 260. The distance between the sliders is 
here represented by a scale at ^^ o^^-^o r,o .^.> . ^^-^^-^^ 
the top of the figure, and the f 
deflections by a scale at the 
left. The deflection produced 
by the Daniell cell is also to 
be plotted, and the number of 
centimetres corresponding to 
this deflection found (see § 59). 

If the electromotive force of i/ : ^ 

the Daniell cell is E voltg ^WfT^- -^^-^-T--^ '^-^^ 
(^ 230), and if D is the dis- Fio. 260. 

tance between the sliders which produces an equal 
current, the distance d corresponding to 1 volt is — 

This distance is to be indicated on the diagram and 
is to be divided into tenths or smaller parts. The 
division may be extended across the base of the fig- 
ure. The theory and uses to be made of the diagram 
will be explained in the next experiment. 



528 ELECTROMOTIVE FORCE. [Exp. 97. 

EXPERIMENT XCVII. 
THE- VOLT-METER, II. 

^ 235. Determination of XilectromotiTe Forces by 
means of a Volt-meter. — A volt-meter, calibrated as 
in ^ 234, is to be connected with various cells or bat- 
teries, one at a time. The deflection caused by each 
is to be noted. The electromotive force of each is 
then to be found (see § 59) by means of the curve 
already plotted (Fig. 260, ^ 234). A point a is first 
located in the scale of degrees corresponding to the 
deflection in question. Then a point h is found on 
the curve at the right of a, and below h a point c is 
found in the scale of electromotive force into which 
the base of the figure has been divided. 

The student is to determine rapidly in this way the 
electromotive forces of all the cells which he has 
employed. 

The principle upon which this method depends is 
that the difference of potential between two points 
on a wire of uniform resistance is proportional to the 
distance between those points represented by the 
scale at the top of Fig. 260. For if R is the resist- 
ance of 1 cm, of the wire, the resistance of d centi- 
metres will be Bd. Hence from the general formula 
of § 139 — 

g = cr = cRd^ (1) 

e crd* d' ^e^>^ 



1236.] CLARK'S POTENTIOMETER. 529 

If the scale at the bottom of Fig. 260 is con- 
structed so as to give one electromotive force cor- 
rectly, all electromotive forces should be correctly 
represented. 



EXPERIMENT XCVIH. 

CLARK'S POTENTIOMETER. 

^ 236. Comparison of Electromotive Forces by means 
of Clark's Potentiometer. — The positive or carbon 
pole of a battery (J5, Fig. 261), consisting of two 




^•iQW 



Fig. 261. 

Bunsen cells in series, is to be connected with one end, 
rf, of a Wheatstone's Bridge wire. The negative or 
zinc pole is to be connected with the other end (a) of 
the wirer. A key, K^ is to be included in the circuit. 
The negative (or zinc) pole of a Daniell cell (-B') is 
to be connected with a. The positive (or copper) 
pole is to be joined through a key, K\ and a delicate 
galvanometer, 0-\ to a slider (6), by which an electri- 
cal connection may be made at any point of the wire. 
The positive or carbon pole of a Lechanch6 cell is to 
be connected similarly with c?, while the negative 

34, 



630 ELECTROMOTIVE FORCE. [Exp. 98. 

(or zinc) pole is to be connected through a key, K'\ 
and a galvanometer, 0-'\ with a second slider at c. 

The key K' is fii-st pressed for an instant, and the 
direction of the deflection noted. Then K and K' 
are both pressed, the connection being completed 
first in K then in K\ 

If the deflection is in the same direction as before, 
the distance ah is to be increased ; if it is in the oppo- 
site direction the distance is to be diminished. The 
experiment is now repeated until a point 6 is found 
such that in pressing both jBTand K\ no deflection is 
observed. In this case the point 6 has the same po- 
tential as the positive pole of the battery R. 

In the same way a second slider is to be placed at 
a point (?, where the potential is the same as that of 
the negative pole of the Lechanch^ cell. 

The key K being now closed, the keys K' and K'' 
are to be pressed simultaneously. If the adjustments 
have been accurately made, neither galvanometer will 
be deflected. If this is not the case, the adjustments 
must be repeated. 

By the principle explained in ^ 236, if the wire ad 
is of uniform resistance, so that the resistances of ab 
and cd are proportional to their lengths, the differ- 
ence of potential between a and 6 must be to that be- 
tween c and d as ab is to cd^ We have, therefore, — 

JS?" cd ^^ XT// TV cd 

where JP' and -B" represent the electromotive forces, 
respectively, of the batteries B" and jB". By this 



IT 237.] POGGENDORFF'S METHOD. 631 

formula, knowing the electromotive force of the 
Daniell cell (^ 230), we may calculate that of the Le- 
chanch^ cell. In repeating the experiment, the places 
of the Daniell and Lechanch^ elements should be in- 
terchanged. If the two sliders should interfere with 
each other, either 1 or 3 Bunsen cells should be used 
(in -B) instead of 2. The experiment may also be 
repeated with other batteries. Clark's Potentiometer 
is especially adapted to the determination of the 
electromotive forces of inconstant elements. 



EXPERIMENT XCIX. 

POGGENDORFF'S METHOD. 

^ 237. Determination of IilectromotiTe ForceB by 
Poggendor£rs Absolute Method. — The zinc pole d (Fig. 
262) of a Bunsen battery is to be connected with one 




Fig. 202. 

terminal (<?) of the resistance-coil used in the Method 
of Heating (Exp. 85.) The zinc pole (a) of a Daniell 
cell is to be connected with the same terminal 
through a delicate galvanometer, b. The copper pole 
(A) of the Daniell cell is to be connected with the 
terminal (i) of the rheostat, and the carbon pole (k) 
of the Bunsen cell is to be connected through a tan- 
gent galvanometer (glm) with the same terminal (i). 



682 ELECTROMOTIVE FORCE. [Exp. 99. 

A portion (de) of a German-silver wire (de/) having 
in all a resistance about equal to that of the resist- 
ance-coil (ci), let us say 1 ohm, is to be included in 
the circuit of the Bunsen battery. 

The wire def is to be disconnected for a moment, 
and the direction of the galvanometer deflection 
noted. Then the extreme end (/) of the wire {def) 
is to be bound in the clamp e. If the deflection is in 
the same direction as before, a longer wire must be em- 
ployed, and if the two Bunsen cells are still unable to 
reverse the Daniell cell,^ other cells must be added to 
the first, either in series or in multiple arc (§ 140). 

We will suppose that a battery (de) and a wire (def) 
have been found such that when the wire is clamped 
at/, the current in the Daniell cell is reversed ; but 
when clamped at d, the current flows in its natural 
direction. 

The wire (def) is next to be clamped at a point (e), 
found by trial, so that the current in the Daniell 
circuit may be reduced to zero. The galvanometer 
(6) will then show no deflection. 

In practice, we clamp the wire at a point (e) so 
that the Daniell cell is barely reversed, and wait for 
a condition of equilibrium to come about through the 
gradual weakening of the Bunsen cell. At the mo- 
ment when the astatic galvanometer (6) points to 0° 
the reading of the tangent galvanometer (^) is to be 
taken. 

^ The student may be reminded that unless similar poles meet at 
r and at f, it will be impossible in any case to produce a reversal of 
the current 



IT 238.] ELECTRICAL EFFICIENCY. 533 

The experiment is to be repeated with the connec- 
tions of the galvanometers reversed one at a time, 
as in Experiment 79. 

If a is the mean angle of deflection of the tangent 
galvanometer and / its reduction factor, the current 
(7is(seet 199. 7) — 

C =1 tan a amperes. (1) 

If R is the resistance of the coil (ci) in ohms (Exp. 
85) we have a difference of potential («) between its 
terminals c and d (see § 139) equal to — 

e=^ CR = RI tan a volts. (2) 

This is equal to the electromotive force of the Daniell 
cell (see ^ 130, 3). 

For a simplified diagram of PoggendorfTs Method, 
see Fig. 254, 1, ^ 226. The only change to be made 
in this diagram is the introduction of a tangent gal- 
vanometer in the upper circuit (ahe). 



EXPERIMENT C. 

ELECTRICAL EFFICIENCY. 

^ 238. Determination of the Efficiency of an Electric 
Motor. — A small electric motor, such for instance as 
IS represented in Fig. 263, is to be connected through 
an ammeter (Fig. 231, ^ 210) or through a tangent 
galvanometer (^, Fig. 264), with a voltaic battery 
{BB^ containing at least twice as many cells as are 



534 



ELECTRICAL EFFICIENCY. 



[Exp. 100. 



required to keep the motor (-3f) in motion. Thus 
if the motor can be started with 2, but not with 1 
Bunsen cell, a battery of 4 Bunsen cells should be em- 



^■h- 



Fig. 263. 

ployed. The poles of the battery are to be connected 
through a volt-meter or its equivalent (see Exp. 96) 
consisting of an astatic galvanometer (G^) and a 
rheostat (-B). The work done by the motor (if) is 
to be determined as in Experiment 70, by observing 
the readings of a pair of spring balances (^SS") con- 




Fio. 2(54. 



nected by a cord passing round the pulley of the 
motor. Ordinary letter-balances will probably an- 
swer for this experiment. The tension of the cord 
should be such as to reduce the speed of the motor 



^ 



IT 228.] ELECTRICAL EFFICIENCY. 635 

to about one half its maximum ; but different experi- 
ments should be made with different tensions. The 
number of revolutions made by the wheel of the 
motor in a given length of time may be determined 
by an instrument called a " revolution counter " es- 
pecially devised for this purpose. This consists of a 
shaft ab (Fig. 266) which can be easily 
connected with the axle of the motor, 
and a toothed wheel ((?) with teeth fit- 
ting into a thread cut on the shaft at 5, 

° . Fig. 260. 

The revolutions of the shaft are indi- 
cated on a dial (d) by a pointer {e) attached to a 
wheel (<?). The circumference of the pulley is to be 
measured. 

Instead of a revolution counter, we may make a 
band of thread 60 cm. long, passing from the pulley of 
the motor over a second pulley-wheel. Every time 
that the knot in this band passes a given point shows 
that the pulley-wheel has advanced 60 cm. The ve- 
locity of the circumference of the pulley-wheel can 
be found by this method by counting the number of 
times that the knot passes a given point in 1 minute. 
If the band is just 60 cm. long, this number represents 
the velocitj'^ in cm. per sec. without any reduction. 

The power in ergs per sec. utilized by the motor is 
to be calculated from these data as in ^ 174, 1, and 
reduced to watts (§ 15) by dividing by 10,000,000; 
that is, by pointing off 7 places of decimals. The 
power in watts spent upon the motor is found by 
multiplying together the current in amperes indicated 
by the ammeter (or its equivalent) and the electro- 



536 ELECTRICAL EFFICIENCY. [Exp. 100. 

motive force in volts indicated by the volt-meter, or 
its equivalent (see § 137). 

The efficiency of the motor is to be found by divid- 
ing the power utilized by the power spent (see 
t 174, 3). 

II. Instead of an electric motor, we may employ 
a small dynamo-machine, driven by a water-motor. 
The work spent by the water is to be calculated as in 
Experiment 69. The work utilized is to be found as 
above by multiplying together the current in amperes 
and the electromotive force in volts. The former is 
to be measured by an ammeter in the main circuit of 
the dynamo-machine ; the latter by a volt-meter con- 
nected with the poles of the dynamo-machine. The 
experiment should be repeated with greater or less 
resistance interposed in the main circuit. 

The student can hardly fail to notice the similarity 
of the method by which we calculate the work of an 
electrical current to that used in the case of a current 
of water (§ 118). The same general method is em- 
ployed in all measurements of electrical efficiency. 



t239.] THE PEEZOMETER. 637 



EXPERIMENTS FOR ADVANCED 
STUDENTS. 

The principal methods by which physical quantities 
are measured have been considered in the course of 
the 100 experiments which have been described. 
Various modifications of these methods have already 
been alluded to. On account, however, of either the 
practical or the theoretical difficulties involved, and 
the expense of the necessary apparatus, measurements 
of certain physical quantities have been hitherto en- 
tirely omitted. This course would, however, be in- 
complete without an outline, at least, of the methods 
by which some of these quantities may be determined. 
Most of the experiments about to be mentioned are 
suitable only for advanced students. For this reason 
it has been been thought unnecessary to describe 
them in detail, or to include in the text proofs of the 
formulae involved, except when these proofs are neces- 
sary to an understanding of the methods employed. 
The Proofs of other formulae will be considered 
separately in Parts III. and IV. 

^ 239. The Piezometer. — To measure the compressi- 
bility of a liquid, we place it in a glass bulb ((7, Fig. 
266) with a narrow neck or stem (i>) containing a 
small mercury index. The bulb is to be placed in 
a stout glass cylinder filled with water. A consider- 



638 EXPERIMENTS FOR ADVANCED STUDENTS, (t 239. 

able hydrostatic pressure is then generated by means 
of the thumb-screw, -4, and measured by a small air 
manometer, E (see ^77). The contraction of the 
liquid in the stem is observed. Since 
the bulb is at the same pressure in- 
side and out, there is no tendency to 
stretch or to crush it. An allowance 
must, however, be made for the com- 
pression of the sides of the bulb. It 
can be shown geometrically that the 
capacity of a bulb decreases, when 
thus subjected to a uniform pressure, 
in the same proportion as the vol- 
ume of a solid would decrease under 
the same circumstances. The ratio 
of the pressure in dynes per square 
centimetre to the decrease in volume 
of 1 cubic centimetre is called the ''Coefficient of Re- 
silience of Volume." ^ It is usually calculated from 
"Young's Modulus" (F), determined as in Experi- 
ment 65, or as in ^ 248, I., and from the " Simple 
Rigidity" (5) of a solid. The simple rigidity may be 
found from the coefficient of torsion, T^ {i. «., the couple 
necessary to twist a wire 1°, see Exp. 64), and from 
the length, ?, and radius, r, of the wire, by the 

formula — 

g ^ 360 Tl 

It may also be found as in ^ 248, II. Denoting by 
M the "coefficient of resilience of the solid," or 




Fia. 266. 



1 Everett, Units and Physical Constants, Arts. 63-6& 



1 240] THE WEIGHT THERMOMETER. 689 

"modulus of volume elasticity," as it is sometimes 
called, we find — 

sr 



M = 



9aS'— 8 Y' 



A mean value of M for glass may be taken as 400, 
000,000,000 dynes per square centimetre. The quan- 
tities S, Jf, and Y are (in the case of glass and many 
other substances) related to each other in about the 
proportion of the numbers 6, 10, and 15 respectively. 
If C is the capacity in cu. cm. of the bulb (Exp. 
11), and F the pressure to which it is subjected, 
measured in dyneB per sq. cm,^ the contraction of the 
interior volume of the bulb ( V) in cu. cm. is — 

. "^' 

If F' is the apparent contraction in cu. cm. of the 
liquid, its real contraction is F+ V'y and the Coeffi- 
cient of Resilience of volume (^') of the liquid is — 



r + v 



By making the bulb ip two parts, a solid may be in- 
troduced into it and surrounded with liquid. The 
Coefficient of Resilience of the solid may be deduced 
from its effect on the apparent contraction of the 
liquid in question. 

^ 240. Use of a "Weight Thermometer. — If a bulb 
similar to that employed in ^ 239, be filled with mer- 
cury at an observed temperature t^ , then warmed to 
the temperature f^, a certain quantity of mercury will 



640 EXPERIMENTS FOR ADVANCED STUDENTS, [t 240. 

be driven out of it. Let the weight of this mercury 
be w^ and let the whole original weight of the mer- 
cury be TT,, both weights being reduced to vacuo 
(§ 67), then the weight, IF,, remaining in the bulb is 
JTi — tu. If V, and v^ are the specific volumes of 
mercury at the temperatures t^ and t^ (see Table 23, 
A and JS), then the capacities of the bulb (Cj and c^) 
at these temperatures must be — 

Ci = TF, Vi and c^ = W^v^. 

It may be shown by geometry that when a vessel 
is expanded uniformly by heat, its capacity is in- 
creased in the same proportion as the volume of a 
solid would increase under the same circumstances. 
The cubical expansion, e, of glass is accordingly (see 
1163)- 



^i(^a— O 
hence the linear coefficient, e, is (see § 83) — 



€ = 



c, (t, — 1{) 

This is considered to be one of the most accurate 
methods of obtaining the coefficient of expansion of 
various kinds of glass. 

By collecting and weighing the mercury which is 
driven out of a bulb or weight thermometer^ we may 
estimate the relative rise of temperature in differ- 
ent cases. The instrument is useful in determining 
precisely the maximum rise of temperature within an 
enclosure which has to be kept closed at the time 
when the temperature is taken. 



1241.] CONDUCTION OF HEAT. 541 

The weight thermometer has also been employed 
to measure the cubical expansion of solids enclosed in 
the bulb. If <?i is the capacity of the bulb at the 
temperature t^^ and if TFi is the weight of mercury 
required to fill the space between the solid and the 
bulb, the volume of the solid Fi is evidently Ci — 
Wi Vi. If when heated to the temperature ^3, at 
which the capacity of the bulb is c,, w grams of mer- 
cury are driven out, so that W^ (or ^i — w) grams re- 
main, then the volume Vo of the solid is c, — W^ v,; 
hence we may find the cubical coefBcient of expan- 
sion (e) by substituting these values of ¥[ and Vi in 
the ordinary formula (see IT 63) — 

^241. Conduction of Heat. — (I.) The conductiv- 
ity of various insulating materials may be found ap- 
proximately by filling the space between the inner 
and outer cups of a calorimeter (^ 85) with these 
materials, and finding the rate at which heat is lost. 
If A is the mean area of the surfaces between which 
conduction takes place, X the distance between them, 
t the difference of temperature, and T the time in 
which Q units of heat pass from one surface to the 
other, the specific conductivity (e) of the material 

is — 

QL 






tTA 



II. A metallic rod {AD, Fig. 267) is surrounded, 
one end by steam, the other by melting ice. The 




642 EXPERIMENTS FOR ADVANCED STUDENTS, [t 241. 

central portion is covered with insulating material. 
Two thermometers^ B and (7, are inserted in holes in 
the rod, partly filled with mercury. If L is the 
length of the rod between B 
and (7, A the area of its cmss- 
I* section, t the difference of tem- 
perature between the points (^B 
and (7), and w the weight of ice 
Fio. 267 melted in the time T, after a 

steady flow of heat has been established, Um the 
quantity melted in the same time when the rod is re- 
placed hy insulating material^ then since the latent 
heat of liquefaction of water is 79, the specific con- 
ductivity (<?) of the rod is given by the foimula — 

_ 79 wL 

"" tTA ' 

The specific conductivity of a given material repre- 
sents tl e quantity of heat which would flow in one 
second from one side of a unit cube made of that 
material to the opposite side of the cube when 
the difference of temperature between the two 
sides is 1°. 

The results of this experiment will be slightly mod- 
ified by the manner in which heat flows through the 
insulating material which surrounds it. To avoid 
errors from this source, the distance between the 
thermometers should be as small as, or smaller than 
the diameter of the rod. This method should be 
applied only to metals or to substances which are 
good conductors of heat. 



12421 LATITUDE. 643 

^ 242. Latitude. — The latitude of a place is usu- 
ally determined by an observation of the ** altitude " 
of the sun at " apparent noon ; " that is, the time 
when it attains its greatest "altitude," or angular dis- 
tance from the horizon. The true altitude (a) of the 
sun is defined as the angle which a line drawn from 
the centre of the earth to the centre of the sun makes 
with a plane passing through the centre of the earth 
and parallel to the horizon of the place in question. 
The declination (d) of the sun is defined as the angle 
which the same line makes with the earth's equator. 
The sun's declination may be found in nautical alma- 
nacs calculated in advance for every day of the year. 
The difference between local and Greenwich time, 
and the hourly change in declination must generally 
be allowed for. The latitude (I) of a place is by defi- 
nition equal to the complement of the angle between 
the horizontal and equatorial planes. We have, 

accordingly, — 

l=W — a±d. I. 

If the sun is (as in summer) above the equator, the 
sign of d is to be taken as positive ; if the sun is below 
the equator, d is to be called negative. 

I. In nautical observations, the apparent altitude 
of the sun is determined by means of a sextant (see 
Exp. 44). The lower " limb " (or edge) of the sun 
is made to coincide with the sea-horizon. The ob- 
served altitude (A) must be corrected as follows: — 

(1) For Semi-diameter. The apparent semi- 
diameter («) of the sun (not far from IB'), given 
exactly in the nautical almanac for every day in the 



544 EXPERIMENTS FOR ADVANCED STUDENTS. [1242. 

year, is to be added to the observed altitude of the 
lower limb of the sun, since the altitude of the sun*s 
centre is wanted. 

(2) Dip op the Sea-horizon. A line drawn 
from the eye of the observer to the sea-horizon makes 
a certain angle with a true horizontal plane. This is 
called the " dip of the sea horizon." It may be cal- 
culated by the formula — 

A = V m X If (nearly), 

where m is the height in metres of the eye above the 
bea-level. The dip (A) must be subtracted from the 
observed altitude. 

(3) For Refraction. Atmospheric refraction 
tendst to make heavenly bodies appear higher than 
they really are. The correction (r) is accordingly to 
be subtracted from the observed altitude. It is given 
by the equation — 

r =^ cotan -4x1' (nearly). 

(4) For Parallax. The apparent altitude of a 
body as seen from the earth's surface is obviously less 
than if it could be observed at the earth's centre. 
In the case of the stars, on account of their enor- 
mous distance, the difference is imperceptible. The 
correction for parallax (p) is given in general by the 
equation — 

p = Pco8 A (nearly), 

where P is the " horizontal parallax " of the body in 
question; that is, its correction for parallax when 



H 242] LONGITUDE. 545 

seen on the horizon. In observations of the sun with 
an ordinary sextant, since P is less than 9", all cor- 
rections for parallax may usually be neglected. It is 
only in the case of the moon, where P is in the 
neighborhood of 1°, that the correction for parallax 
becomes important. 

The true altitude (a) of a heavenly body is found 
in general from the observed altitude (-4.) by apply- 
ing the corrections for semi-diameter («), dip of the 
horizon (A), refraction (r),and parallax (p) as follows: 

a = A'\'8 — h — r'\-p, 11. 

II. Observations of latitude taken on land are usu- 
ally made with an *' artificial horizon." This may 
consist of a plate^lass mirror (made horizontal by 
two spirit-levels and levelling-screws) or simply a 
dish of mercury (5, Fig. 268) 
The lower limb of the sun is 
made to coincide with its own ^ 
reflection in the horizontal sur- 
face. The observed angle (D) 
between the direct and reflected 
rays reaching the sextant (J., Fig. 268) is measured, 
and halved^ to find the apparent altitude of the sun. 
The result is corrected as above for semi-diameter, 
refraction, and (if sufficiently accurate) for parallax.^ 
We have — 

a = Ji) + 5 — r+j9. m. 

1 The correction for " dip " is obviously to be omitted in the case 
of an " artificial horizon/' since the plane of the reflecting surface 
should be perfectly horizontal. 

85 



•fi^ 



546 EXPERIMENTS FOB ADVANCED STUDENTS, [t 243. 

The latitude is finally calculated by formula I. 
above. 

^ 243. Longitada. — The longitude of a place may 
be determined by a sextant observation of the alti- 
tude of the sun (see ^ 242) an hour or two after 
sunrise or before sunset. For the reduction of the 
results, in general, the student is referred to works 
on navigation. A simple (though not very accurate) 
method of finding the longitude of a place is to 
measure the altitude of the sun at an observed time 
t (about an hour before noon), then to determine 
exactly the time i!' (about an hour after noon) when 
the sun descends to the %ame altitvde. Obviously the 
time of " apparent noon," ^, (neglecting the change 
in the sun's declination), is half-way between t^ and 
r, that is — 

^ = J (e' + r), nearly. I. 

If e is the " equation of time " (given in the nauti- 
cal almanacs for every day of the year), the time (37) 
of "mean noon" is (by definition) given by the 
formula — 

T=t±e. II. 

The sign of the quantity e is positive if the sun is 
fast, but negative if the sun is slow. 

It is assumed that the chronometer employed in 
this experiment has been set so as to indicate cor- 
rectly the time of a given meridian, as for instance 
that of Greenwich, from which it is desired to meas- 
ure longitude. If it does not indicate this time cor- 
rectly, an allowance must be made for the error of 



1244.] INDICES OF REFRACTION. 547 

the chronometer. At sea, several chronometers are 
frequently carried. In certain cases a chronometer 
may have to be set by a lunar observation. For the 
reduction of such results (which is exceedingly com- 
plicated), the student is referred to works on naviga- 
tion. On land, the standard time of a given meri- 
dian is usually obtainable by means of the electric 
telegraph. 

It may be remarked that the longitude of a place 
is given by formula II. in hours, minutes and seconds. 

% 244. Indices of Refraction. — I. If Jl is the angle 
of a prism (Exp. 45), and D the angle of minimum 
deviation (Exp. 46) of a ray of light of a given wave- 
length, the index of refraction (/i) of the material of 
which the prism is composed is (for light of that wave- 
length) — 

8mK^ + i>) 
sin ^ A 

Certain "doubly refracting" substances have two 
indices of refraction instead of one. To determine 
them we employ a prism cut so as to produce the 
maximum separation of the two rays into which a 
single ray of monochromatic light can be decomposed 
by the given prism angle. The minimum deviation 
of each ray is then measured, and the two indices of 
refraction are calculated separately by the ordinary 
formula. 

II. If ^ is a mean radius of curvature of the two 
surfaces of a double convex lens (Exp. 21), and jp 
its principal focal length (Exps. 41-43), the index of 



548 EXPERIMENTS FOR ADVANCED STUDENTS. [IT 245. 

refraction of the material of which the lens is made 
may be found by the formula — 

If the same lens (5, Fig. 269) be enclosed between 
two flat glass plates (Jl and C7), and the space be 
filled with a liquid, with the index of refraction fL\ 
then if F' is the principal focal length of the combi- 
nation, we have — 

If Rx and R^ are the two rddii of curvature of the 
A two sides of the lens, the mean radius of 
^^^ curvature should strictly be calculated hj 
the formula — 

Eio. 269. 

A + ^« 

^ 245. PolarizatioD. — The vibrations which con- 
stitute ordinary light are, according to modern theo- 
ries (§§ 92, 93), at right-angles with the direction in 
which the light is propagated. In a vertical beam of 
light, for instance, the vibrations are supposed to be 
confined to a horizontal plane. The vibrations appear 
in general to be distributed uniformly in every pos- 
sible direction perpendicular to the path of the ray. 
Certain substances and certain optical combinations 
have, however, the property of stopping all the vi- 
brations — or rather all their components (§ 105) — 
except those in a certain direction, as for instance 



ir245.] 



POLARIZATION 



549 



north and south. The light transmitted is then said 
to be polarized. 

In many optical instruments, light passes success- 
ively through two such combinations. The first is 
called the ** polarizer " («, Fig. 270) , the second is 
called the "analyzer" (a). If the polarizer and an- 
alyzer are placed so that the direction of the vibra- 
tions transmitted is the same in both cases, the light 
which has passed through one will also pass freely 




Fio. 270. 

through the other ; but since the polarizer transmits 
only vibrations in a given direction, if the analyzer is 
placed so as to stop all vibrations in this direction, 
a beam of light which has passed through the polar- 
izer will be completely cut off by the analyzer. The 
position of the analyzer when this occurs is indicated 
by a pointer attached to it. The reading of the 
pointer with respect to the graduated circle b deter- 
mines the zero-reading of the instrument. 



650 EXPERIMENTS FOR ADVANCED STUDENTS. [IT 24». 

Certain substances have the property of changing 
the direction of the vibrations in a beam of polarized 
light which they transmit. Thus in passing upward 
through a solution of cane sugar, north and south 
vibrations are gradually changed into a northeast 
and southwest direction.^ 

When a substance producing " rotation of the plane 
of polarization " is placed between the polarizer and 
the analyzer in its zero-position, the analyzer will no 
longer cut off all the light transmitted by the polar- 
izer. To produce perfect darkness, the analyzer must 
obviously be turned through an angle equal to that 
through which the plane of polarization has revolved. 
The instrument shown in Fig. 270 affords, accord- 
ingly, a means of measuring the rotation of the plaue 
of polarization. 

To test the strength of a solution of sugar with 
this instrument, we pour the solution into a tube cd 
with glass ends, and interpose the tube in the path of 
the beam ea of polarized light. The analyzer is then 
turned to the right from its zero-position, until the 
light which it transmits is reduced to a minimum. 

^ When light is polarized by reflection, it is said to be polarized in 
a plane perpendicular to the reflecting surface, and containing both 
the incident and the reflected rays. According to Fresnel's theory 
the vibrations in a beam of polarized light take place at right-angles 
with the ''plane of polarization." The action of a solution of sugar 
upon a beam of polarized light approaching the eye is to rotate the 
plane of polarization (and hence also the direction of the vibration) 
With the hands of a watch. The student should note' that this is called 
a right-handed rotation in optics ; but that it is opposite to the mo- 
tion of an ordinary right-handed screw, which when turned to the 
right moves away from the eye. 



1 246] COLOR. 551 

Let a be the angle in degrees through which it is 
turned when %odium light is employed, and let d be 
the depth of the sugar solution, equal to the distance 
between the glass ends of the ttibe cd ; then experi- 
ments show that the strength of the solution («) in 
grams per cu. cm, is given by the equation (Kohl- 
rausch, § 46), — 

g = 1.5 3 (nearly), 
d 

The rotation varies considerably with lights of 
different colors (see Table 31 2>). For this reason, 
when ordinary white light is employed perfect dark- 
ness can never be attained. 

There are various optical effects (besides the dark- 
ness produced by an analyzer) which depend upon 
the plane in which light is polarized. Many of these 
have been applied to the determination of angles of 
rotation of the plane of polarization. The method 
described above has been chosen because of its 
simplicity. 

^ 246. Color. A piece of colored paper (<?, Fig. 
271) may be mounted in front of a white screen (d) 
and illuminated by a candle (a) through a piece of 

ruby glass (6), all other ^^fs^ ^^ ^^ 

light being cut off. The 

distances ae and ad must ^ 

be adjusted so that c and ^^®- 2^^* 

d appear equally bright when viewed from a point 

near b. The *' relative luminosity " of the surface c 

is then ^equal to (a(?)^-r- (ai)* as far as reflected red 

rays are concerned. 




652 EXPERIMENTS FOR ADVANCED STUDENTS, [f 247. 

A transparent gelatine plate stained with an eme- 
rald green mixture of common green and yellow inks 
is now substituted for the ruby glass (6), and the rel- 
ative luminosity is again determined. Finally, a gela- 
tine plate stained with a violet mixture (Hofmann's 
violet containing a trace of soluble Prussian blue) is 
emploj'^ed. 

The three relative luminosities of the surface <?, 
obtained as above by means of red, green, and violet 
rays, completely determine the color of the surface 
in question (see ^ 115). 

% 247. Velocity of Light. — The velocity of light 
was determined by Fitzeau in 1849.^ A beam of 
light made intermittent by passing between the 
teeth of a revolving wheel, was sent to a distant 
mirror, then reflected back to the eye through the 
same wheel. When the wheel (which had 720 
teeth) made 12.6 revolutions per second, the flashes 
of light, in traversing a total distance of 17,326 
metres, were retarded so as to strike a tooth in- 
stead of the space between two teeth; hence the 
light was cut off. When the speed of the wheel was 
doubled, so that 18,144 teeth passed a given point in 
one second, the light reappeared ; when trebled it dis- 
appeared, &c. It was inferred from this experiment 
that a beam of light required yy^^ of a second to 
traverse 17,326 metres; whence the velocity of light 
would be about 18,144 X 17,326 metres per second, 
or nearly thirty thousand million cm. per sec. 

I See Deschaners Natural Philosophy, §686; Ganofs Physics, 
§507. 



1247] VELOCITY OF LIGHT. 563 

Foucault has measured the time required by light 
to traverse short distances (a few metres only) by the 
use of a revolving mirror.^ A beam of light (-4.5, 
Fig. 272) striking the mirror (5) was reflected td a 
fixed concave mirror ( CO) with its centre of curvature 
in the axis of the revolving mirror (jB), then back on 
its course to the revolving mirror (-B), and thence to 
the eye. The beam strikes the eye only for a very 
short time during each revolution of the mirror, but 
on account of the rapidity of rotation a continuous 
effect is produced. When the speed of rotation 
reaches several hundred revolutions per second, the 
mirror turns through a perceptible angle while the 
light is passing from B to C or to (7' and back again. 




Fio. 272. 

Hence the return path BA' differs slightly from the 
original path AB. 

With a distance 5(7 equal to about 4 metres, and 
with from 600 to 800 revolutions per second, diver- 
gences of about 40'' or 50" were observed. The ve- 
locity of light was found to be 298 (or nearly 80) 
thousand million cm. per sec. 

By passing the beam of light through a tube of 
water {DJS, Fig. 272) it was found that the velocity 
of light in water is about f that in air. 

1 Deschanel's Natural PhiloBophy, § 687; Ganot'e Physics, § 606. 



554 EXPERIMENTS FOR ADVANCED STUDENTS. [H 248. 



^ 248. Velocity of Bound in "Wiraa. — I. If a wire 
stretched between two vises be stroked horizontally 
near one end by a piece of resined cloth, a musical 
note may result from the longitudinal vibrations into 
which the wire is thrown. The pitch of the note is 
to l)P determined by a ** pitch pipe " (Fig. 273) or 
any instrument serving a similar pur- 
I pose. The number of vibrations corres^ 
ponding to the note may be found by 
reference to Table 43. If I is the length 
of the wire between the vises, and n the 
number of vibrations per second, the ve- 
locity of sound (v) is — 

V = 2nl. 

II. If a strip of resined cloth be 
Fig. 273. drawn slowly round the wire (like a belt 
round a pulley) a musical note may result from tor- 
sional vibrations set up in the wire. The velocity of 
these torsional vibations may be found by the same 
formula as above. The note due to longitudinal vi- 
brations is usually about a " sixth " (^ 134) above 
that due to torsional vibrations. Hence the two 
velocities of sound are to each other as 5 to 3, 
nearly. 

If d is the density of the wire, ¥ Young's Modu- 
lus of Elasticity (^ 166) and S the simple rigidity 
of the wire (^ 239) Vi and v^ the velocities of longi- 
tudinal and torsional vibrations, we find — 

Y=vi^d. . I. 




S = v^ d. 



II. 



f 249.J REVERSIBLE PENDULUM. 555 

^ 249. ReverBible Pandolum. — A reversible pendu- 
lum (Fig. 274) may be made of cast iron,^ so that 
although the two knife-edges A and B are at very 
unequal distances from the centre of gravity 
(0) the time of oscillation on both knife- 
edges is nearly the same. The position of 
must be found approximately (Exp. 62), and 
the distances AC and BC measured. The 
distance AB must be accurately determined 
(by measuring BU^ BA^ and BH with a ver- 
nier gauge, and subtracting BA and BJS 
from BB). If if is the time of oscillation 
on the knife-edge A^ and if' that on B (see 
Exp. 58), the time t of oscillation of a simple pendu- 
lum of the length AB is — 

Denoting by I the distance ABy the acceleration of 
gravity (g} may now be calculated by the ordinary 
formula — 

^ For a half-seconds pendulum, the foUowing dimensions are sug- 
gested: extreme length of the shaft {DE), 45 cm., breadth 3| cm., 
thickness 1 cm, ; ends sharpened to an angle of about 70° ; triangular 
knife-edges (steel better than cast iron) 2 cm. long, sides 1 cm. broad; 
distance of each knife-edge from nearest extremity, 10 cm. ; holes 1 
X 2 cm. ; disc 14 cm. in diameter, 2 cm. thick ; centre of disc 24 cm. 
from one knife-edge, 1 cm. from the other. This pendulum should 
weigh about 3 kilograms. The centre of gravity should be about 
5 cm. from one knife-edge, and 20 cm. from the other. In obseryations 
of its time of oscillation, the knife-edges may rest upon the upper 
surface of a short steel rod, 7 mm. square, driven horizontally into 
the wall. 



656 EXPERIMENTS FOR ADVANCED STUDENTS, [t 261. 

^ 250. Coafaciant of Viscosity. — A liquid con- 
tained in a Mariotte's bottle (a, Fig. 275) is fed 
through a rubber tube (6c) into a capillary tube (cd), 
a^ and collected in a small vessel (e). 

^^g^ra^^ The weight (w) which passes through 
Imam^mmm the tube in a given length of time 
I (0 is found, and the height (A) of 

I the inlet (6) above the orifice (d) is 

J determined. The length (l) of the 
^^^^^^^S8g|^ tube (cd) is measured, and its ra- 
SSSSm dius (r) is found (see ^ 170). Then 
Fio. 275. if d is the density of the liquid 
(Exp. 14), and g the acceleration of gravity (Exp. 
58), the coefficient of viscosity of the liquid is given 

by the formula, — 

irgcPhr^t 

This coefficient of viscosity is the force in dynes 
necessary to maintain a difference of velocity equal 
to 1 cm. per see. between two opposite faces of a 
centimetre cube. 

The ordinary coefficient of Lquid friction (see 
^ 172) depends upon the square of the velocity, and 
has no relation to the coefficient of viscosity. 

^ 251. Electro-chemical Equivalents. — If, in Ex- 
periment 81, 1 is the reduction factor of the galvan- 
ometer, determined as in Experiment 83, w the 
weight of copper deposited by the current O in the 
time ^, and a the average angle of deflection, we have 
for the electro-chemical equivalent (j) of copper — 

w w 

^~ Ci~ titan a 






1 252.] CORRECTION OF RHEOSTATS. 567 

By the same formula we may find the electro-chemi- 
cal equivalent of any other substance acted upon by 
the current C> whether that action be to deposit the 
substance in question, or to cause it to go into solu- 
tion. In the case of a gas set free at one of the elec- 
trodes of a voltameter ((7 or i>, Fig. 276), we find 
the weight' indirectly from the volumes 
collected in graduated tubes (A and jB),^ 
originally filled with the liquid QU) 
which is decomposed by the current. A ^^^' ^^^' 
battery of two or three Bunsen cells should be used 
with a gas voltameter. 

If w\ w'\ w"\ &c., are the weights of different sub- 
stances acted upon by a given current traversing a 
series of voltameters for a given time, the electro- 
chemical equivalents q\ q'\ q"\ &c., may be found (if 
any one is known) from the proportion — 

^ 252. Correction of Rheostats. — An arrangement 
of a set of resistances, convenient for the purposes of 
correction, is represented in Fig. 277. The outer 



,..^ ^ jjo r"^jt^"4 ^> sX-s f^ *j ^ ^ + J horse-shoe (ac?) contains 



' L> *t<i ;o4u id^] j<^ f4;^i 



afeaesgaE' 



'^ ^^ . ^ ■ ?t> . T' : X^rx^\X^^ any resistance under 

'} '■■: ^^ ■■ .■ "■'' '» 4g-*-| 4,000 ohms, within one 



18 coils capable of being 
combined so as to give 



Fig. 277. tenth of an ohm. The 

inner horse-shoe (hefc) contains resistances arranged in 
pairs of 1, 10, 100, and 1000 ohms each. Opposite 
a and c are two extra blocks. These are permanently 



558 EXPERIMENTS FOR ADVANCED STUDENTS. [^ 252. 

connected together, underneath, by a thick copper 
rod. One of them is joined to the positive pole of a 
battery. Two blocks opposite b and d are similarly 
joined together, and one of them is connected with 
the negative pole of the battery. 

One terminal of the galvanometer is now carried 
to e (or to/). The other terminal is to be connected 
with one of the blocks in the outer line of resistances 
between two coiUy or sets of coils, which are to be 
compared. A pair of resistances about as great as 
the coils in question is now introduced into the 
inner horse-shoe. When the battery is connected 
with a and (2, the rheostat assumes the form of a 
Wheatstone's Bridge (§ 141). The inner horse-shoe 
furnishes two of the arms be and cf. The connec- 
tions of these arms may be interchanged by breaking 
the battery connections at a and (2, and making them 
at b and c. The arrangement of blocks furnishes in 
fact a commutator within the box of coils. By the 
use of this commutator, errors due to inequality in a 
given pair of resistances may be eliminated (§ 44). 

The 1-ohm coil is first to be tested against the 
smaller coils, together equal to 1 ohm ; then joined in 
series with the smaller coils, and tested against each of 
the 2-ohm coils; then the 5-ohm coil, the 10-ohm coil, 
&c., are to be tested each against its equivalent in 
terms of the coils below it in the line of resistances. 
If differences are observed, the sensitiveness of the 
galvanometer to a change of 1 ohm (or 0.1 ohms) in 
the outer line of resistances must be determined. 
The differences in question may then be estimated by 



1 254-1 ELECTRICAL CAPACITY. 559 

interpolation (see ^ 216). The results are to be re- 
duced as in ^ 217. When the ratios of the different 
coils in the outer series have been found, that of any 
pair of coils in the inner horse-shoe may be deter- 
mined by comparison. 

^ 253. ResiBtanoe of Eleotrolytes. — We may sub- 
stitute in Exp. 87 an alternating current for a com- 
mon battery current ; in this case the galvanometer 
must be replaced by some instrument like the dyna- 
mometer, sensitive to alternating currents. A tele- 
phone is sometimes found to give satisfactory results 
with a rapidly alternating current. Usually a loud 
note is heard in the telephone ; but when the Wheat- 
stone's bridge is in adjustment, the sound either com- 
pletely ceases or reaches a minimum. 

The advantage of using alternating currents is 
that, in the short time during which they last, the 
effects of polarization are so small as to be almost 
inappreciable. The method is especially valuable in 
the determination of the resistances of batteries and 
electrolytes. It is not, however, always successful, 
on account of various causes tending to destroy the 
minima of sound. To obtain satisfactory results, the 
resistance to be measured should be not less than 10 
or 15 ohms. The electrodes should consist of plat- 
inum strips, at least 10 sq. cm. in area, and freshly 
coated with platinum through electrolytic action 
(Kohlrausch, 6th ed. 72 II.), 

^ 254. Measurement of Electrical Capacity. — A 
*' condenser" consists of two sets of thin metallic 
plates, arranged alternately, as in Fig. 278, so that 




560 EXPERIMENTS FOR ADVANCED STUDENTS, [t 254. 

although the plates are very close together, there is 
no metallic coDDection between the two sets. The 
plates are generally separated by thin layers of glass, 
mica, or paper dipped in paraf- 
fine. The plates of one set are 
all connected with one binding- 
post (J.) ; those of the other 
Fig. 278. set with another binding-post 

(-B). A condenser is charged by connecting A and 
B each with one pole of a battery. It may then 
be disconnected from the battery, , and discharged 
through a galvanometer by carrying the terminals to 
A and B. Care must be taken not to touch both 
terminals at the same time. 

The capacity of a condenser is defined as the quan- 
tity of electricity which can thus be stored in it by a 
battery having an electromotive force equal to 1 unit 
in absolute measure. The "farad" is a thousand 
millionth part of the electro-magnetic unit of capacity. 
The distance between the plates of a condenser is 
usually very small in comparison with the area of the 
separate plates. To calculate the electrical capacity 
of such a condenser, we measure the thickness (<) 
and total area (^) of the insulating layers, then if s 
is the " specific inductive capacity " of the insulating 
material (^ 256), the capacity (C) of the condenser 
is given in electrostatic units by the equation — 

C=^ I. 

or, since it has been found by experiment that 1 mi- 
crofarad is equivalent to about 900,000 electrostatic 



1254.] ELECTRICAL CAPACITT. 561 

units,^ the capacity (<;) in microparads may be calcu- 
lated by the formula — 

As 
c = ^^ ^^^ ^^^ — microfarads (rfearly). II. 

36,000,000 TT « ^ ^-^ 

The specific inductive capacity («) of the insulat- 
ing material must in general be found as in IT 256 ; 
but when the plates of a condenser are separated by 
air spaces, since the specific inductive capacity of air 
is taken as 1, the capacity of a condenser may be 
calculated from direct measurements of the area and 
thickness of the insulating material. 

The capacity of any condenser may be determined 
by measuring the quantity of electricity stored in it 
by a battery of known electromotive force. With 
the aid of clockwork, a condenser is to be charged 
by a battery and discharged through a galvanometer 
n times a second ; the deflection of the galvanometer 
being noted. Then if M is the resistance in ohms 
through which the same battery produces the same 
deflection (see Exp. 95, II.) we have — 

e= — — ^ — microfarads. III. 

nJt 

In practice we must employ a very sensitive gal- 
vanometer capable of measuring currents at least 
in millionths of an ampdre. The time of oscillation 
of the needle should be 10 seconds or more, in order 
that the intermittent discharge through the instru- 
ment may produce a sensibly constant effect. An 
ordinary condenser, of 1 microfarad capacity cannot 

1 Everett, Units and Physical Constants, Arts. 177, 185. 
36 



662 EXPERIMENTS FOR ADVANCED STUDENTS. [1 255. 

be charged and discharged satisfactorily more than 
10 or 100 times per second.^ To avoid large errors 
due to this cause, the speed of the mechanism should 
be reduced until an approximate agreement is ob- 
tained between two or more results. 

The experiment may be performed with an ordi- 
nary astatic galvanometer, but only by the use of a 
condenser of great capacity and a battery of high 
electromotive force. 

^ 255. Comparison of Condensers. — The capacities 
of two condensers may be compared by charging 
them, successively, by a given battery, then discharg- 
ing them successively through a ballistic galvan- 
ometer (see ^ 187). The capacities will then be 
approximately as the chords of the throws (§ 109). 

The capacities of two condensers may be compared 
with great precision by including the condensers in 
two adjacent arms of a Wheatstone's bridge (see Exp. 
87). One pole of the battery must be applied be- 
tween the two condensers. The resistances in the 
other two arms of the bridge should be great, and 
adjusted so that a sudden reversal of the battery 
current causes no sudden deflection of the galvan- 
ometer.2 If Ci and 0^ are the capacities of the two 

1 Owing to effects of " electrical absorption " and " residual 
charge," the quantity of electricity stored in or obtained from a con- 
denser depends somewhat upon the time during which connections 
are made. See Ganot's Pliysics, § 773. When a condenser is rapidly 
charged and discharged, these phenomena almost entirely disappear; 
but the resistance of the various conductors may reduce the quantity 
of electricity which can flow in and out of the condenser to an indefi- 
nitely small amount. 

2 See Glazebrook and Shaw, Practical Physics, §§ 81, 82. 



IT 256.] SPECIFIC INDUCTIVE CAPACITY. 668 

condensers, Bi and i2s the resistances adjacent to 
them, respectively, we have — 

d : C; :: B^ : B^. 

We have seen (^ 264) that the capacity of a con- 
denser with air spaces between its plates may be 
measured. The capacity of such condensers is gen- 
erally so small that comparisons cannot be made by 
ordinary methods. By substituting an alternating 
current for the battery and a telephone for the gal- 
vanometer (see ^ 263) in the combination described 
above, comparisons of these and even smaller capaci- 
ties should be possible. 

^ 266. Specific Inductiva Capacity. — When two 
condensers are similar in every respect except the 
nature of the insulating materials used in their con- 
struction, their capacities (<? and f') are to each other 
as the " specific inductive capacities " (« and s') of 
these materials. Since the specific inductive capacity 
of air may be taken as 1, we have in general, from 

^254,1.,- 

Aired 

The specific inductive capacity of a given insulating 
material may accordingly be found by constructing a 
condenser with that material between its plates, 
measuring the area of and distance between these 
plates, and determining as in ^ 264 or as in ^ 266 
the capacity of the condenser. 

Winkelmann's method for testing specific inductive 
capacities consists in the use of three parallel plates. 



564 EXPERIMENTS FOB ADVANCED STUDENTS. [7 256. 

-4, 5, and C (Figs. 279 and 280), equal in area, and 16 
or 20 cm, in diameter. A and B are separated by 
an air space of the thickness a, while B and C are 
separated by an air space of the thickness (, and by 
a thickness c of the material whose specijGc inductive 
capacity is to be determined. The outer plates A 
and C are connected either through a telephone (7, 
Fig. 279) with each other, or through a diflPerential 
telephone (DT, Fig. 270), and through a metallic 
conductor {Q) with the ground. The central plate 





Fig. 279. 



{B) is joined to one pole of an induction coil, the 
other pole of which is connected through Q- with 
the ground. The distances a and h are then adjusted 
so that the sound heard in the telephone is reduced 
to a minimum. The specific inductive capacity («) is 
then given by the formula — ' 



% = 



a — 6' 



1258.1 ELECTROSTATIC SYSTEM. 665 

In Winkelmann's method we may consider that the 
plates A and B form one condenser, while the plates 
B and C form another condenser. When the capaci- 
ties of these two condensers are equal, a given charge 
of electricity on B must raise A and C to the same 
potential ; hence if the effect be simultaneoiLS no cur- 
rent will flow through the telephone. In practice, 
most dielectrics cause a slight retardation in the 
chai-ging of a condenser, so that although the tele- 
phone gives a minimum of sound, it never becomes 
perfectly silent. 

^ 257. CompariBon of Electromotive Forces by 
means of a Condenser. — The pole Cups of a condenser 
{A and jB, Fig. 278) are to be connected as in ^ 254 
with the poles of a battery, then disconnected from 
the battery, and connected with the terminals of a 
ballistic galvanometer, the throw of which is to be 
observed. The experiment is to be repeated with a 
second battery. If a' and a" are the throws, H' and 
-E?" the electromotive forces, we have (see § 109), if 
the angles are small, — 

H' chord d d y 

In this experiment it is important that the duration 
of charging, discharging, and changing connections 
should be exactly the same in the two cases. 

^ 258. Electrostatic System. — Two gilt pith-balls 
(h and c, Fig. 281), of equal weight (i^) and diame- 
ter (d) are both to be suspended from an insulated 
point a, by fine cotton threads of equal length (Z). 



566 EXPERIMENTS FOE ADVANCED STUDENTS. [T 258. 

The threads may be blackened with a lead-pencil to 
make sure that they will conduct electricity. One 
pole of a battery (de), of several hundred volts, is 
to be connected with the point (a) of suspension ; the 
other pole with the ground. 

The balls b and c^ being similarly charged, will 
now repel each other. A considerable divergence 
should be observed. The distance («) between the 
centres of the two balls is to be found by a sextant 
placed at a fixed distance (see ^ 124). The electro- 




Fio. 281. 



motive force (e) of the battery in electrostatic units 
is then (roughly) — 



,= /: 



l(P 



The pith-balls should be about 1 cm. in diameter, and 
not over .05 g. in weight. The cords ab and ac 
should be at least 100 cm. long, but not over 0,01 g. 
in weight. All electrical conductors should be re- 
moved as far as possible from the neighborhood of 
the balls b and c, 

A water battery (de^ Fig. 281) will be found con- 
venient for this experiment. It may be constructed 



1258] ELECTROSTATIC SYSTEM. 567 

of alternate strips of zinc and copper soldered to- 
gether in pairs and attached with pitch to the under 
side of a board so that drops of water or dilute 
sulphuric acid may be taken up between adjacent 
pairs (as A and B). 

It has been found by experiment that one unit of 
electromotive force in the electrostatic system is 
equal to about 300 volts, or 80 thousand million ab- 
solute units in the electromagnetic system. It is an 
interesting fact that the ratio between the absolute 
units of the two systems is equal, within the limits of 
errors of observation, to the velocity of light (see 
§ 93). 



568 



INSTRUMENTS OF PRECISION. 



INSTRUMENTS OF PRECISION. 



The apparatus employed in the course of experi- 
ments which has been described is of the simplest 
possible form. The most accurate results can be ob- 
tained only by the use of instruments especially de- 
signed for a given purpose. The following sections 







Fig 282. 

contain a brief description of the construction and 
adjustments of certain instruments of precision, 
which though unsuitable for an elementary class of 
students, might be advantageously employed by ad- 
vanced students in place of the ordinary apparatus. 



ir259.] ANALTTICAL BALANCES. 669 

^259. Analytical Balances. — The adjastments of 
an analytical balance (Fig. 282) and the precautions 
in using it are essentially the same as those described 
in Experiment 6. In addition to the mechanism, oper- 
ated from outside the case, by which in a fine balance 
all weight may be removed from the knife-edges, there 
is often a pan-arrester, which has to be moved before 
two weights can be exactly balanced. A preliminary 
adjustment of the weights should be carried as far as 
centigrams on an ordinary balance. The weights may 
then be transferred to the analytical balance, and a 
finer adjustment made by means of a rider (e. Fig. 
283) made of platinum wire. The rider can be placed 




Fig. 283. 

at any point (e) of a graduated scale on the balance- 
beam by means of a hook (d) attached to a rod (ac) 
passing through a tube (6) in the side of the balance- 
case. The necessary motion is given to the hook by 
pushing, pulling, or twisting the rod (ac). 

The indication of the pointer is always found while 
it is in oscillation (^ 20) ; but since the weights may 
be adjusted by means of the rider with any degree of 
precision, the method of interpolation (^ 20), though 
generally quicker, need not be employed. 

In finding the position of the rider necessary for 
an exact balance, the same method of approximation 
should be employed, at first, as in the adjustment of 
weights ; that is, the rider should be placed midway 



670 INSTRUMENTS OF PRECISION. [t 259. 

between two distances on the scale, one too great 
the other too small, until the deflection of the pointer 
and the sensitiveness of the balance indicate directly 
where it should be placed. When finally observations 
of the swings of the pointer show that it would come 
to rest at its zero-position, the position of the rider is 
noted. 

The accuracy of the rider is tested by weighing a 
small weight with it. To obtain results accurate to 
a tenth of a milligram, the set of weights employed 
(even the best) should be most carefully tested (^ 25). 

The advantage of weighing with a rider is that the 
final adjustment of two weights may be made with 
the balance-case closed. The air within the case 
should always be kept perfectly dry with chloride of 
calcium (or with concentrated sulphuric acid), which 
must be renewed from time to time. Neither arm of 
the balance should be exposed to the heat of a fire or 
lamp, or to the cold glass of a window. The method 
of double weighings should if possible be employed. 
If it is not employed, care must be taken that the 
pans are equal in weight, and that in the zero- 
position, the balance-beam is horizontal and the 
pointer vertical.^ 

1 When the greatest accuracy is desired, arrangements must be 
made to carrj on the ordinary processes of weighing from a dis- 
tance. Thus at the International Bureau of Weights and Measures 
at St. Cloud, not only the suspension of weights from the balance- 
beam, but also the interchange of the contents of the scale-pans is 
accomplished by a series of shafts leading from each instrument 
nearly to the centre of a large room in which the finest balances are 
contained. Mechanical contrivances are also employed for the final 
adjustment of weights In vacuo. 



ir26l.] THE DIVIDING-ENGINE. 671 

^ 260. Comparators. — A simple form of comp'dra' 
tor is represented in Fig. 284. It consists of two 
reading microscopes {A and B) mounted on supports 
{E and F) which slide along a rail ^GH). The slid- 
ing supports may be clamped at any point of the rail 
by thumb-screws ((7 and i>). A small scale of tenths 
of millimetres (6 and b\ Fig. 284) is placed in the 
tube of each microscope at a distance from the object 
glass (<?) equal to twice its focal length. The eye- 




FiG. 284. 

piece (a) is first focussed upon this scale, then raised 
or lowered until a given object is in focus. Let us 
suppose that the two microscopes are thus set, one 
upon each end of a scale. It is obvious that if a 
standard scale be now substituted any di£Perence be- 
tween the two will be not only readily detected, but 
easily measured in tenths of a millimetre and such 
fractions of a tenth as may be estimated bj'^ the eye 
(§ 26). 

Care must be taken to have the upper surfaces of 
the two scales on the same level, so that both scales 
may be in focus, and to have the microscopes firmly 
clamped, and not subjected to any strain between ob- 
servations. 

^ 261. The Dividiag-Engine. — A dividing-engine 
(Fig. 285) consists essentially of a micrometer (js) with 



672 INSTEUMENTS OF PRECISION. [IT 262, 

a long screw {DQ) fixed in position, so that when 
the micrometer is turned, a nut {EF) gives a slow 
motion to a slide (J?) to which a reading microscope 
{A) is usually attached. The length of an object 
parallel to the screw is determined by the number of 
turns of the micrometer necessary to make the micro- 
scope travel from one end of the object to the other. 
The microscope is of course provided with cross- 
hairs; so that it may be set exactly on a given point. 
The screw is always to be turned in a given direction 
in measuring a given distance ; otherwise an error 
due to looseness of the screw (" backlash ") may be 
made. The pitch of the screw in diiBFerent parts is 




Fig. 286. 

found by measuring with it a standard scale of 
known length (see % 52). If the nut is long and fits 
equally well in all parts of the screw, no great varia- 
tions of pitch can occur. 

The dividing-engine is especially useful in measur- 
ing distances between the lines of a scale, or lengths 
of columns of mercury in the calibration of a tube 
(see % 71). The results may be more precise than 
those obtained with any other instrument for the 
measurement of length. 

^ 262. The Cathetometer. — {icaThy down, Ti0f}fLh to 



IT 263.] 



THE CATHETOMETER. 



673 



place, and pirpov^ measure) is an instrument for meas- 
uring vertical distances (Fig. 286). It consists of a 
horizontal telescope or reading 
microscope (i) sliding on a verti- 
cal shaft (aA), which is capable 
of rotating about its own axis. 
Sometimes the shaft is gradua- 
ted, the carriage to which the 
telescope is attached being pro- 
vided with a vernier, so that the 
height of the telescope may be 
read. Slow motion may also be 
given by a micrometer screw 
(«/). The cathetometer may 
then be used for measuring small 
vertical distances, just as the div- 
iding-engine (^ 261) is used for 
horizontal distances. The mi- 
crometer is useful in measuring 
precisely, for instance, the dis- 
tance through which a wire is 
stretched (Exp. 65). For ordi- 
nary purposes, neither the micrometer nor the ver- 
nier is required. The shaft is first adjusted by the eye 
so as to be as nearly perpendicular as possible, by 
means of the le veiling-screws (A, t, and Z) at the base 
of the instrument, then the telescope is made hori- 
zontal according to a spirit-level (c) with which it 
is provided. Then the shaft is rotated about its axis. 
If the axis is not vertical, the bubble in the spirit- 
level will tend to move in a given direction. The 



Fig. 286. 



574 INSTRUMENTS OF PRECISION. [IT 263. 

top of the shaft is to be inclined slightly in this di- 
rection. After a series of trials the axis may in this 
way be made perfectly vertical. 

The object to be measured is to be set up with the 
aid of a plumb-line, beside a vertical scale, so as to 
be at the same distance from the cathetometer as the 
scale is, both at the top and at the bottom. The tele- 
scope of the cathetometer, accurately levelled, is to 
be focussed by means of the cross-hairs upon one end 
of the object (T[ 116, 3), then rotated so as to bear 
upon the scale, and the reading of the scale noted. 
If the spirit-level is disturbed, the cathetometer must 
be readjusted and the reading redetermined. The 
reading of the lower end of the object is to be found 
in the same way. By putting a graduated scale in 
place of the cross-hairs, the divisions of a scale may 
be divided into very small parts. This method is not 
so precise as that depending upon the use of a ver- 
nier or micrometer attached to the cathetometer, 
but may, in unskilled hands, give fully as accurate 
results. 

^263. Bilorometer Eye-Pieoes. — Instead of mov- 
ing a telescope or a reading microscope bodily, as in 
^T[ 261 and 262, it is sometimes convenient to mount 
the cross-hairs upon a small slide within the eye- 
piece of an instrument, and to give a slow motion to 
the slide by means of a micrometer screw. The value 
of the micrometer divisions must be found for each 
instrument. A micrometer eye-piece gives indica- 
tions much more precise than a fixed scale ; but care 
must be taken not to alter the setting of an instru- 



T 264.] KEGULATORS. 575 

ment by pressure upon the eye-piece in adjusting the 
micrometer, and, as in the dividing-engine (^ 261), to 
turn the instrument always in a given direction up 
to a setting. If the micrometer is turned too far, it 
must be turned backward a considerable way, then 
forward to the desired point.* 

In the best optical circles two microscopes with 
micrometer eye-pieces are usually provided. These 
are placed on opposite sides of the circle, in order 
that errors due to excentricity may be avoided. 

^ 264. Regulators. — For experiments involving 
the accurate measurement of time, a clock with a 
compensating pendulum, or a chronometer with a 
compensating balance is indispensable. The clock 
or chronometer should be provided with an electric 
break-circuit, and must be rated by observations with 
either a sextant (^ 243) or a transit (see Pickering's 
Physical Manipulation, § 178), or by comparison with 
time signals from some observatory. 

In the Physical Laboratory of Harvard College, the 
regulator employed is a common seconds-clock with 
a wooden pendulum-rod controlled by an electrical 
time circuit. The control consists simply of a fine 
spiral spring connecting the pendulum with the arma- 
ture of a telegraph instrument in the circuit. Elec- 
trical signals,' sent from the Astronomical Observatory 
at intervals of two seconds, are thus made to act 
iriechanically upon the pendulum. When the latter 

^ The *' backlash " should be taken up, in so far as possible, by the 
action of a spring. Errors due to " backlash " may be thus greatlj- 
diminished, but not completely eliminated. 



5T6 INSTRUMENTS OF PRECISION. (T265. 

has been carefully rated without the control, Tery 
small impulses are suflScient to prevent it from gain- 
ing or losing. 

^ 265. Kater's Pendulum (Fig. 287). — In Kater's 
form of reversible pendulum (see 249) the rod (de) is 
usually made of brass, a little over a metre long, 2 
or 3 em. wide and about 5 mm. thick. Two 
steel knife-edges, be and fg^ are attached 
firmly to this rod with a distance of about 
1 metre between them. They are supported 
when the pendulum is in use, by agate 
planes, b and c. The bob (A) is a brass cylin- 
der, weighing 1 or 2 kilograms. Movable 
counterpoises, d and «, serve to adjust the 
centre of oscillation. Two light and firm 
metallic pointers (a and i) may be used to 
magnify the oscillations. 

In addition to these adjustments, clampa 
with tangent-screws may be employed to ob- 
tain a slow motion of the counterpoises. 
The knife-edges be and fg are sometimes 
made movable (one or both of them). In 
this case, verniers are usually attached, so 
Fig. 287. that the distance between the knife-edges 
may be read by a scale on the shaft de. The zero- 
reading of the vernier is found by bringing the knife- 
edges together against a pressure equal to the whole 
weight of the pendulum. The accuracy of the main 
scale is tested by a comparator (^ 260) at the ordinary 
temperature of the experiments, and under a strain 
equal to the average weight which the shaft sustains. 




ir267.] THE SIREN. 677 

^ 266. Chronographs. — A chronograph consists 
generally of a cylindrical drum (il, Fig. 288) rotated 
uniformly by clock-work. The surface of the drum 
is coated with lampblack, 
so that a style (B), at- 
tached to the armature (c) 
of a telegraph instrument 
may make a mark upon it. 
The line AB represents the 
trace caused by an ordinary 
seconds break-circuit. At 
the point D there is an 
extra break due to a signal Fig. 288. 

given by hand. If the drum revolves uniformly, 
the exact time of such a break can evidently be de- 
termined by measuring the distance from it to the 
nearest second-mark, and comparing this with the 
distance between two second-marks. 

The pitch of a tuning-fork may be determined very 
exactly by the trace made on the surface of a chron- 
ograph (see ^ 139). 

It may be said in general that the chronograph is 
valuable as a means of determining precisely the 
interval of time between any two phenomena which, 
with or without the agency of electricity, are capable 
of affecting the motion of a style. 

% 267. The Siren. — The siren (Fig. 289) is an in- 
strument for producing a musical note of any pitch, 
and at the same time registering the number of vibra- 
tions constituting that note. It is operated by a 
constant air pressure from a bellows, specially con- 

37 



678 



INSTRUMENTS OP PRECISION. 



1267. 



structed for this purpose. The air entets the wind- 
chest of the instrument at (^), issues obliquely from 
a series of holes (of which ^ is one) in the top of the 
wind-chest, and strikes obliquely against the sides of 
a series of holes (of which D is one) in a disc (C'), 
which is thereby set in motion. When the two series 
of holes come opposite, the air escapes freely from 
the wind-chest; when they are not opposite, the cur- 
rent of air is nearly cutoff. The irregular flow of 
the air sets the atmosphere in vibration. The num- 



mm 





Fio. 289. 



ber of vibrations in a given length of time is indicated 
by the dials A and B. 

In practice the speed of the siren is regulated by 
pressure on the top of the bellows used to drive it. 
The note is slowly raised until it agrees with one 
whose pitch is to be determined. When the two 
notes are nearly in unison beats will be heard (^ 140). 
By a slight change of air pressure, perfect unison 
may generally be obtained. This will be shown by a 



1 268.] 



MIRROR GALVANOMETERS. 



579 



cessation of beats. The unison is maintained for a 
given length of time during wliich the number of 
vibrations made by the siren is registered. In some 
instruments the dials may be thrown in and out of 
gear at a given moment. This facilitates the obser- 
vations of the dials, but care must be taken that the 
speed of the siren is not afiEected. 

It must be remembered that beats occur not only 
when two notes are in unison, but also when they are 
nearly an octave apart, and to a somewhat less extent, 
when they are separated by any other musical inter- 
val (^ 134). A musical ear is therefore almost a 
necessity in the adjustment of a siren. The chief ad- 
vantage of the siren is that it enables us to find the 
pitch of notes not easily determined (as is Exps. 52, 
54, and 55), by either optical or graphical methods. 

^ 268. Mirror Galvanometers. — A very sensitive 
galvanometer is made by suspending a small mirror 
(F, Fig. 290) in the middle of a coil H of insulated 
wire, by means of a single fibre of 
cocoon silk (DE). Small bits of 
"hair-spring" (used in watches) 
highly magnetized, all in the same 
manner, are fastened with the 
smallest possible quantity of wax 
to the back of the mirror. A 
large curved magnet (BC) capable 
of sliding up and down the tube ~^^l 
(A) or turning round it, is ad- Fio. 290. 

justed so as to nearly neutralize the eiffect of the 
earth's magnetism on the magnets attached to the 




580 INSTRUMENTS OF PRECISION. [t 269. 

mirror. The sensitiveness of this instrument when 
accurately adjusted, though less permanent than that 
of an astatic combination, is for the time being fully 
as great. 

In some galvanometers a converging mirror is used, 
so that a spot of light may be projected on a trans- 
parent screen. The existence of a current is indi- 
cated by the motion of the spot of light with respect 
to a scale graduated on the screen. 

In other instruments a plane mirror is employed, 
with a long-focus lens mounted permanently in front 




-i % 

;y£:";v-T.v^^x:^^ ..;m 



Fig. 291. 

of it. The deflection of the mirror is frequently ob- 
served by means of the reflection (£, Fig. 291) of a 
scale (B) in the mirror ((7), seen from a point (-4), 
where either the eye or a telescope may be placed.^ 

^ 269. Electrical Standarda. — Copies of " standard 
ohms" may be obtained from most dealers in electri- 
cal apparatus. The terminals should be thick copper 

^ Prof. B. O. Feirce has shown that excellent resalts may be ob- 
tained without any telescope (-4), by placing beneath the mirror C 
H fixed mirror D, so that the two reflections (E and F) of the scale 
(/i) very nearly coincide. When the two mirrors are parallel, the 
zeros of the two scales are opposite, no matter where the eye may be 
placed. The slightest deflection of the mirror causes an apparent 
motion of the scale reflected in it. 



1269.] ELECTRICAL STANDARDS. 681 

rods, capable of being amalgamated with mercury 
and connected by mercury cups with a Wheatstone's 
Bridge Apparatus. Unless special care be taken 
in making these connections, the most accurate 
standards of resistance may lead to very erroneous 
results. 

Standard cells of Latimer Clark's pattern may 
easily be obtained. Their electromotive force is 
about 1-435 volts at IS"". The decrease is about 
•00077 volts for a rise of temperature of 1° Centigrade. 
The uses of a constant cell have been alluded to in 
^^ 228, 230. 

*' Standard amperes " are now being made by some 
dealers. When the attraction of a coil of wire for 




Fig. 292. 

a piece of soft iron is balanced by gravity (Fig. 292), 
an allowance must be made for variations in gravity 
when the instrument is transported from one latitude 
to another. A standard ampere depending upon the 
action of a spring, though subject to many theoretical 
objections, would be practically useful as a check 
upon results obtained by other methods. Let us sup- 
pose that such an instrument is connected in series 
with a rheostat and a tangent galvanometer, that a 
current, sent through both, is increased until the in- 
strument indicates 1 ampere, and that the galvan- 
ometer is then read. The reciprocal of the tangent 



682 mSTEUMENTS OF PRECISION. [If 270. 

of the angle of deflection should agree closely with 
the redaction factor already found (Exp. 83). 

^ 270. Electxoiiieters. — Various forms of quadrant 
electrometer may now be obtained from manufactu- 
rers. The theory of these instruments is exceedingly 
complicated, and the results are more or less uncer- 
tain. The principal use of the instrument is in the 
case of inconstant cells, to confirm results obtained 
by the use of a condenser. Such instruments in gen- 





Fig 203. 



eral have to be calibrated by means of cells of known 
electromotive force. 

Thomson's absolute electrometer (Figs. 293 and 
294) depends upon the attraction between two plates 
j and £, when charged oppositely with electricity. 
The plate j is suspended from one end (ci) of a bal- 
ance-beam (ac). The force exerted upon it is coun- 
terpoised by weights in a pan (/) suspended from the 
other end of the beam (a.) The deflection of the beam 
is observed by means of a sight (d) and a lens (e). 
The plate i is very much larger than /, which is sur- 
rounded by a ring (A) charged to the same potential 



1 270.] ELECTROMETERS. 683 

as the movable disk (j)^ to equalize the distribution of 
electricity upon the latter. 

If w is the weight required to balance the attrac- 
tion of the two plates, d the distance between them, 
and a the area of the suspended plate 0), then the 
difference of potential (e) between the plates is given 
in electrostatic measure by the formula — 



= d|/Z^Z5. 



It is said that an absolute electrometer may be made 
sensitive to the difference in potential between the 
two poles of a Daniell cell. It is especially valuable 
for the calibration of other forms of electrometer 
better suited for actual use, and for determinations 
of the fundamental relations between the electrostatic 
and electro-magnetic systems. 



END OP PART II. 



CABOT SCIENCE LIBRARY 



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