LIBRARY 
 
 THE UNIVERSITY OF CALIFORNIA 
 
 MISS ROSE WHITING. 
 
 Deceived September, 1896. 
 Accession No.btoO . Chns No. 
 
UNITS AND 
 PHYSICAL CONSTANTS. 
 
UNITS AND 
 PHYSICAL CONSTANTS. 
 
 BY 
 
 J. D. EVERETT, M.A., D.C.L., F.R.S., F.R.S.E, 
 
 PROFESSOR OF NATURAL PHILOSOPHY IN QUEEN'S COLLEGE, BELFAST. 
 
 UHIVBBSIT7 
 
 IP oribon : 
 
 MACMILLAN AND CO. 
 1879. 
 
 [The right of translation and reproduction is reserved. ~\ 
 
GLASGOW : PRINTED AT THE UNIVERSITY PRESS 
 BY ROBERT MACLEUOSE. 
 
PREFACE TO FIRST EDITION. 
 
 THE quantitative study of physics, and especially of 
 the relations between different branches of physics, is 
 every day receiving more attention. 
 
 To facilitate this study, by exemplifying the use of 
 a system of units fitted for placing such relations in 
 the clearest light, is the main object of the present 
 treatise. 
 
 A complete account is given of the theory of units 
 ab initio. The Centimetre-Gramme-Second (or C.G.S.) 
 system is then explained; and the remainder of the 
 work is occupied with illustrations of its application 
 to various branches of physics. As a means to this 
 end, the most important experimental data relating to 
 each subject are concisely presented on one uniform 
 scale a luxury hitherto unknown to the scientific 
 calculator. 
 
 I am indebted to several friends for assistance in 
 special departments but especially to Professor Clerk 
 Maxwell and Professor G. C. Foster, who revised the 
 entire manuscript of the work in its original form. 
 
 Great pains have been taken to make the work 
 correct as a book of reference. Readers who may 
 discover any errors will greatly oblige me by pointing 
 them out. 
 
trilVBHSITY 
 
 PREFACE TO SECOND EDITION. 
 
 THIS Book is substantially a new edition of my 
 " Illustrations of the C.G.S. System of Units" published 
 in 1875 by the Physical Society of London, supplemented 
 by an extensive collection of physical data. The title 
 has been changed with the view of rendering it more 
 generally intelligible. 
 
 Additional explanations have been given upon some 
 points of theory, especially in connection with Stress 
 and Strain, and with Coefficients of Diffusion. Under 
 the former head, I have ventured to introduce the terms 
 "resilience" and "coefficient of resilience," in order to 
 avoid the multiplicity of meanings which have become 
 attached to the word " elasticity." 
 
 A still greater innovation has been introduced in an 
 extended use of the symbols and processes of multiplica- 
 tion and division, in connection with equations which 
 express not numerical but physical equality. The ad- 
 vantages of this mode of procedure are illustrated by its 
 application to the solution of the most difficult problems 
 on units that I have been able to collect from standard 
 text-books (chiefly from Wormell's ' Dynamics '). 
 
 A correction has been made in the definition of the 
 Weber (p. 139). 
 
viii PREFACE TO SECOND EDITION. 
 
 I am indebted to several friends for contributions of 
 experimental data. 
 
 A Dutch translation of the first edition of this work 
 has been made by DR. C. J. MATTHES, Secretary of the 
 Royal Academy of Sciences of Amsterdam, and was 
 published in that city in 1877. 
 
 Though the publication is no longer officially con- 
 nected with the Physical Society, the present enlarged 
 edition is issued with the Society's full consent and 
 approval. 
 
 J. D. EVERETT. 
 BELFAST, June, 1879. 
 
nfXVEBSITY 
 
 CONTENTS. 
 
 PAGES 
 
 Tables for reducing other measures to C. G. S. 
 
 measures, xiii xvi 
 
 CHAPTER I. 
 
 GENERAL THEORY OF UNITS, . . . . 115 
 
 Units and derived Units, 1-5. Dimensions, 6-9. 
 Meaning of "per," 10. Extended sense of "Multipli- 
 cation" and "Division," 11-12. Examples, 13. 
 Dimensions of mechanical and geometrical quantities, 
 14. 
 
 CHAPTER II. 
 CHOICE OF THREE FUNDAMENTAL UNITS, . 1620 
 
 Three independent units, 15. Their selection, 16. 
 Standards of mass, 17. Standards of length, 18. 
 Standard of time, 19. -Necessity for one common scale, 
 20. The C.G.S. system, 21. Powers of 10 as factors, 
 
 CHAPTER III. 
 MECHANICAL UNITS, 21 -29 
 
 Acceleration ; value of g; seconds' pendulum, 23. 
 Force ; the dyne ; gravitation measure of force, 24-26. 
 The poundal, 27. Work and energy ; the erg, 28. 
 Kinetic energy, 29. Gravitation-measure of work, 30. 
 Rate of working ; horse-power, 31. Examples in dyna- 
 mics, p. 25. Centrifugal force, 32. Examples on centri- 
 fugal force, p. 27. 
 
x CONTENTS. 
 
 CHAPTER IV. 
 
 PAGES 
 
 HYDROSTATICS, 3043 
 
 Relative density of water at various temperatures, 
 33. Absolute density of water, 34. Formulas for ex- 
 pansion of water, 35. Table of densities of solids and 
 liquids, 36. Volume by weighing in water, 37. 
 Examples in hydrostatics, p. 33. Barometric pressure ; 
 C.G.S. unit of pressure ; standard atmosphere adopted in 
 this volume, 38. Examples on barometric pressure, p. 
 35. Density of dry air, with example, 39. Absolute 
 densities of gases, 40. Pressure-height at a point in the 
 air '(height of homogeneous atmosphere) 41, 42. Exam- 
 ples on pressure-height, p. 39. Variation of density with 
 height in the atmosphere, 43. Examples on height at 
 which density is halved, p. 41. Pressure of aqueous 
 vapour, 44. Pressures of various vapours, 45. Super- 
 ficial tensions of liquids, 46. Correction of barometer 
 for capillarity, 46 A. 
 
 CHAPTER V. 
 STRAIN, STRESS, AND RESILIENCE, . . . 4456 
 
 Strain, 47-49. Stress, 50, 51. Coefficients of 
 resilience, 52. Resilience of volume ; Young's modulus ; 
 simple rigidity, 53. Shear, 54-58. Shearing stress, 
 59-6o. Resistance to shearing, 61. Resilience of 
 volume of liquids, 62, 63. Resilience of solids, 
 64-69. 
 
 CHAPTER VI. 
 ASTRONOMY, 57 61 
 
 Size and figure of the earth ; lengths of day and year, 
 70. Attraction ; sun, earth, and moon, 71. Attraction 
 of a given number of grammes at a given distance, 72. 
 The three fundamental units reducible to two, 73-75. 
 System of units based on a given spectrum-line, 76. 
 
CONTENTS. xi 
 
 CHAPTER VII. 
 
 PAGES 
 
 VELOCITY OF SOUND, 6266 
 
 General formula, 77. Gases, 78. Liquids, 79. 
 Solids, 80-82. Musical strings, with example, 83. 
 Faintest audible sound, 84. 
 
 CHAPTER VIII. 
 LIGHT, 6777 
 
 Velocity, 85. Wave-lengths, 86. Index of refrac- 
 tion of air, 87. Vibrations per second, 88. Indices of 
 refraction; glass, 89. Liquids, 90, 91. Indices of 
 double refraction, 92. Indices of refraction of miscel- 
 laneous substances, 93. Gases, 94. Dispersion in 
 gases, 95, 96. 
 
 CHAPTER IX. 
 HEAT, 78119 
 
 Unit of heat, 97-99. Capacity and specific heat, 
 98-102. Table of specific heats and atomic weights of 
 elements, 103. Variation of specific heat with tempera- 
 ture, 104. Specific heats in solid and liquid form, 105. 
 Specific heats of gases, * 106, 107. Of vapours, 108. 
 Melting points, 109. Change of volume from liquid to 
 vapour, no. Latent heats of fluidity, and melting points, 
 IIT. Latent heats of vaporization, 112. Latent and 
 total heat of steam, 113. Effect of temperature on 
 gases, 114. Boiling points of water, 115. Departures 
 from Boyle's law, 116. Specific heats of the same 
 substances in different states, 117. Boiling points of 
 various liquids, 118. Change of volume in melting, 119. 
 Dry-bulb, wet-bulb, and dew-point, 120. 
 
 Conductivity; definition, 121. Dimensions, 122. 
 
 Thermometric conductivity, , 123. Analogous to co- 
 efficient of diffusion, 124, 125. Coefficients of diffusion 
 
CONTENTS. 
 
 of certain gases, 126. Conductivity of air, 127. Results 
 of experiments on conductivity of solids, 128-135. 
 
 J. D. Forbes, 128. Neumann, 129. Results from 
 underground thermometers, 130. Angstrom, 131. G. 
 Forbes, 132. A. S. Herschel and Lebour, 133. Pe"clet 
 134, 135- 
 
 Emission and surface-conduction, M'Farlane, 136. 
 Tait, 137. 
 
 Mechanical equivalent of heat, 138, 139. 
 
 Heat and energy of combination, 140. 
 
 Two specific heats of a gas, 141. Change of freezing- 
 point with pressure, 142. Change of temperature pro- 
 duced by adiabatic compression, 143. Resilience as 
 affected by heat of compression, with examples, 144. 
 Tables of expansions, 145. 
 
 CHAPTER X. 
 MAGNETISM, 120 127 
 
 Magnetic units and their dimensions, 146-150. 
 Examples, pp. 122-125. Gauss's pound magnet, p. 122. 
 Maximum of permanent magnetism, p. 123. Maximum 
 magnetization of iron, nickel, and cobalt, p. 123. Moment 
 and magnetization of earth, p. 124. Different units em- 
 ployed by Gauss, p. 125. Distribution of magnetic potential 
 and force over surface of earth, 151. Magnetic ele- 
 ments at Greenwich, 152, 152*. Magneto-optic rota- 
 tions, 153. 
 
 CHAPTER XL 
 ELECTRICITY, 128161 
 
 Electrostatic units and their dimensions, 154-159. 
 Electromagnetic units and their dimensions, 160-164. 
 Table of dimensions in both systems, 165. Heat gene- 
 rated by current, 166. Ratios of the two sets of electric 
 units expressed in terms of a velocity, 167, 168. Deter- 
 minations of this velocity, and comparison with velocity of 
 
CONTENTS. xiii 
 
 PAGES 
 
 light, 169. Dimensions in terms of density, length, and 
 time, 170. 
 
 Specific inductive capacity, 171. Of gases, 172. 
 Ohm, theoretical and practical, 173. Volt, Weber, and 
 Farad, 174. Earth-quadrant per second, 175. 
 
 Length of spark for various differences of potential, 
 176, 177. 
 
 Resistance, and specific resistance, 178. Tables of 
 specific resistance, 179, 180. Resistance of insulators, 
 1 8 1. Resistance of water and ice at various tempera- 
 tures, 182. 
 
 Electro-motive forces of various cells, 183. Electro- 
 motive forces of contact, 184. Thermoelectric force, 
 with examples, 185, 186. Electro-chemical equivalents, 
 187. Computation of electromotive force from heat 
 of combination, 188. Examples on electricity, 189, 
 pp. 155-158. Capacity of earth, p. 156. Electromotive 
 force due to revolving coil, p. 158. Problem on selection of 
 fundamental units, 190. 
 
 Electrodynamics. Ampere's formula, with example, 
 
 APPENDIX, ..... .... 163169 
 
 Reports of Units Committee of British Association. 
 
 INDEX, ........ 171175 
 
TABLES FOR REDUCING OTHER MEASURES 
 TO C.G.S. MEASURES. 
 
 The abbreviation cm. is used for centimetre or centimetres, 
 
 m ' gramme or grammes, 
 
 scc - second m seconds, 
 
 S 2- square, 
 
 b. cubic. 
 
 Length. 
 
 J ;. nch > - - = 2 -5400 cm. 
 I foe*, 
 
 j? 
 
 yard, - . - 91-4392 
 innle, - . = 160933 
 i nautical mile, - = 185230 
 
 More exactly, according to Captain Clarke's com- 
 parisons of standards of length (printed in 1866), the 
 metre is equal to 1-09362311 yard, or 3-2808693 feet, or 
 39370432 inches, the standard metre being taken as 
 correct at o C, and the standard yard as correct at i6f 
 ience the inch is 2-5399772 centimetres. 
 
 Area. 
 
 I square inch, - . . 6-4516 sq. cm. 
 
 I square foot, - - = o 2 o - O i 
 
 I square yard, - . =8361-13 
 
 i square mile, - . =2'59xio 10 
 
 I cubic inch, 
 I cubic foot, 
 I cubic yard, 
 I pint, 
 i gallon, - 
 
 Volume. 
 
 = 16-387 cub. cm. 
 
 - - 28316 
 
 4541 
 
TABLES. 
 
 Mass. 
 
 i grain, - - = '0647990 gm. 
 
 i ounce avoirdupois, - = 28 '3495 , , 
 
 i pound ,, - = 453'59 
 
 i ton, - - - =i'oi6o5xio 6 ,, 
 
 More exactly, according to the comparison made by 
 Professor W. H. Miller in 1844 of the "kilogramme des 
 Archives," the standard of French weights, with two 
 English pounds of platinum, and additional weights, also 
 of platinum, the kilogramme is 15432*34874 grains, of 
 which the new standard pound contains 7000. Hence 
 the kilogramme is 2*2046212 pounds, and the pound is 
 453'59 26 5 grammes. 
 
 Velocity. 
 
 i foot per second, - - =30*4797 cm. per sec. 
 
 I statute mile per hour, - =44704 ,, 
 
 I nautical mile per hour, - =51 '453 
 
 I kilometre per hour, - - =27777 
 
 Density. 
 
 Pure water at temperature of / , 
 
 maximum density, - .- j = ' OOOI 3 m - P er cub * cm ' 
 I pound per Cubic foot, ' - = '016019 
 
 Force (assuming -=981). 
 
 Weight of i 
 
 grain, 
 
 = 63-57 dynes, 
 
 i 
 
 ounce avoirdu- 
 
 =278x10* 
 
 
 pois, 
 
 
 ] 
 
 pound avoirdu- 
 pois, 
 
 =4-45x105 
 
 
 cwt. , 
 
 =4-98x107 
 
 
 ton, - 
 
 =9'97xio 8 
 
 
 gramme, - 
 
 = 981 
 
 
 kilogramme, 
 
 =9-81 x io 5 
 
 
 tonne, 
 
 =9-81 xio 8 
 
TABLES. 
 
 foot-pound, 
 foot-grain, 
 foot-ton, - 
 milligram-millimetre, 
 gramme-centimetre, - 
 kilogrammetre, - 
 tonne-metre, 
 
 Work (assuming -=981). 
 
 - = i '356 xio 7 ergs, nearly. 
 
 - = i -937 xio 3 
 . =3-04 x io 10 
 
 =9-81 x io- 2 
 
 =9-81 xio 2 
 
 =9-81 xio 7 
 
 =9-81 xio 10 
 
 Work in a second by one ) , .- 
 
 f,Vol T,.c =746 
 
 theoretical "horse 
 
 Pressure (assuming "=981). 
 
 i pound per square foot, - = 479 dynes per sq. cm., nearly, 
 i pound per square inch, - =6 '9 xio 4 ,, ., 
 
 i kilogramme per square ( _ o 
 
 metre, - - - 
 l kilogramme per square) g . 
 
 decimetre, - - ) 
 I kilogramme per square g . 
 
 centimetre, - - ) 
 I kilogramme per square g 7 
 
 millimetre, - - ~ y 
 Pressure of 760 millimetres _ T . m . x Tn c 
 
 of mercury at o C, - ~ J 14 X IO 
 Pressure of 30 inches of ) 6 
 
 mercury at o C., - \ 
 ) 
 
 mercury 
 Pressure of I inch of mercury ) _ .,.~gg x Io t 
 
 Heat. 
 
 i gramme-degree Centigrade, =4 '2 x io 7 ergs = 42 million ergs. 
 i pound-degree, ,, = 1*91 x io 10 ergs. 
 
 i ,, Fahr., - - = i -06 x io 10 ergs. 
 

F JhH A * ** 
 
 &sjw83fiJ* 
 
 CHAPTER I. 
 
 GENERAL THEORY OF UNITS. 
 Units and Derived Units. 
 
 i. THE numerical value of a concrete quantity is its 
 ratio to a selected magnitude of the same kind, called 
 the unit. 
 
 Thus, if L denote a definite length, and / the unit 
 
 length, is a ratio in the strict Euclidian sense, and 
 
 is called the numerical value of L. 
 
 The numerical value of a concrete quantity varies 
 directly as the concrete quantity itself, and inversely as 
 the unit in terms of which it is expressed. 
 
 2. A unit of one kind of quantity is sometimes defined 
 by reference to a unit of another kind of quantity. For 
 example, the unit of area is commonly defined to be the 
 area of the square described upon the unit of length; 
 and the unit of volume is commonly defined as the volume 
 of the cube constructed on the unit of length. The units 
 of area and volume thus defined are called derived units, 
 and are more convenient for calculation than indepen- 
 dent units would be. For example, when the above 
 definition of the unit of area is employed, we can assert 
 
2 UNITS AND PHYSICAL CONSTANTS. [CHAP. 
 
 that [the numerical value of] the area of* any rectangle is 
 equal to the product of [the numerical values of] its 
 length and breadth ; whereas, if any other unit of area 
 were employed, we should have to introduce a third 
 factor which would be constant for all rectangles. 
 
 3. Still more frequently, a unit of one kind of quantity 
 is defined by reference to two or more units of other 
 kinds. For example, the unit of velocity is commonly 
 defined to be that velocity with which the unit length 
 would be described in the unit time. When we specify 
 a velocity as so many miles per hour, or so many feet per 
 second, we in effect employ as the unit of velocity a mile 
 per hour in the former case, and a foot per second in 
 the latter. These are derived units of velocity. 
 
 Again, the unit acceleration is commonly defined to 
 be that acceleration with which a unit of velocity 
 would be gained in a unit of time. The unit of acceler- 
 ation is thus derived directly from the units of velocity 
 and time, and therefore indirectly from the units of 
 length and time. 
 
 4. In these and all other cases, the practical advantage 
 of employing derived units is, that we thus avoid the intro- 
 duction of additional factors, which would involve needless 
 labour in calculating and difficulty in remembering.' 1 ' 
 
 * An example of such needless factors may be found in the rules 
 commonly given in English books for finding the mass of a body 
 when its volume and material are given. " Multiply the volume in 
 cubic feet by the specific gravity and by 62 '4, and the product will be 
 the mass in pounds ; " or " multiply the volume in cubic inches by 
 the specific gravity and by 253, and the product will be the mass in 
 grains." The factors 62 '4 and 253 here employed would be avoided 
 that is, would be replaced by unity, if the unit volume of water 
 were made the unit of mass. 
 
i.] GENERAL THEORY OF UNITS. 3 
 
 5. The correlative term to derived is fundamental. 
 Thus, when the units of area, volume, velocity, and 
 acceleration are denned as above, the units of length 
 and time are called the fundamental units. 
 
 Dimensions. 
 
 6. Let us now examine the laws according to which 
 derived units vary when the fundamental units are 
 changed. 
 
 Let V denote a concrete velocity such that a concrete 
 length L is described in a concrete time T ; and let v, /, t 
 denote respectively the unit velocity, the unit length, and 
 the unit time. 
 
 The numerical value of V is to be equal to the 
 numerical value of L divided by the numerical value of 
 
 V L T 
 T. But these numerical values are -, -> ', 
 
 hence we must have 
 
 V L / 
 
 This equation shows that, when the units are changed 
 (a change which does not ^affect V, L, and T), v must 
 vary directly as / and inversely as t ; that is to say, the 
 unit of velocity varies directly as the unit of length, and 
 
 inversely as the unit of time. 
 
 y 
 Equation (i) also shows that the numerical value - of a 
 
 given velocity varies inversely as the unit of length, and 
 directly as the unit of time. 
 
 7. Again, let A denote a concrete acceleration such 
 that the velocity V is gained in the time T', and let a 
 denote the unit of acceleration. Then, since the 
 
4 UNITS AND PHYSICAL CONSTANTS. [CHAP. 
 
 numerical value of the acceleration A is the numerical 
 value of the velocity V divided by the numerical value of 
 the time T, we have 
 
 A=Y L 
 
 a vT' 
 But by equation (i) we may write for . We 
 
 thus obtain 
 
 A = L / t_ 
 
 a I T T' (2) 
 
 This equation shows that when the units a, /, / are 
 changed (a change which will not aifect A, L, T or T'), a 
 must vary directly as /, and inversely in the duplicate 
 
 A 
 
 ratio of // and the numerical value will vary inversely 
 
 a 
 
 as /, and directly in the duplicate ratio of /. In other 
 words, the unit of acceleration varies directly as the unit of 
 length, and inversely as the square of the unit of time ; and 
 the numerical value of a given acceleration varies inversely 
 as the unit of length, and directly as the square of the unit 
 of time. 
 
 It will be observed that these have been deduced as 
 direct consequences from the fact that [the numerical 
 value of] an acceleration is equal to [the numerical value 
 of] a length, divided by [the numerical value of] a time, 
 and then again by [the numerical value of] a time. 
 
 The relations here pointed out are usually expressed by 
 
 saying that the dimensions of acceleration* are - en ^ .^ or 
 
 (time) 2 
 
 that the dimensions of the unit of acceleration* are 
 
 * Professor James Thomson ('Brit. Assoc. Report,' 1878, p. 452) 
 objects to these expressions, and proposes to substitute the follow- 
 
I.] GENERAL THEORY OF UNITS. 5 
 
 unit of length 
 (unit of time) 2 ' 
 
 8. We have treated these two cases very fully, by way 
 of laying a firm foundation for much that is to follow. 
 We shall hereafter use an abridged form of reasoning, 
 such as the following : 
 
 length 
 
 velocity = T 5 - ; 
 time 
 
 velocity length 
 
 acceleration = . = . . * w 
 
 time (time)" 1 
 
 Such equations as these may be called dimensional 
 equations. Their full interpretation is obvious from what 
 precedes. In all such equations, constant numerical 
 factors can be discarded, as not affecting dimensions. 
 
 9. As an example of the application of equation (2) we 
 shall compare the unit acceleration based on the foot and 
 second with the unit acceleration based on the yard and 
 minute. 
 
 Let / denote a foot, L a yard, / a second, T a minute, 
 T a minute. Then a will denote the unit acceleration 
 based on the foot and second, and A will denote the unit 
 acceleration based on the yard and minute. Equation 
 (2) becomes 
 
 6 = M-) 1 = ~; . . (i) 
 
 a i 60' T2oo **' 
 
 that is to say, .an acceleration in which a yard per minute 
 
 Change-ratio of unit of acceleration = 
 
 This is very clear and satisfactory as a full statement of the meaning 
 intended ; but it is necessary to tolerate some abridgment of it for 
 practical -working. 
 
6 UNITS AND PHYSICAL CONSTANTS. [CHAP. 
 
 of velocity is gained per minute, is i of an acceleration 
 
 1200 
 
 in which a foot per second is gained per second. 
 
 Meaning of "per" 
 
 10. The word per, which we have several times 
 employed in the present chapter, denotes division of the 
 quantity named before it by the quantity named after it. 
 Thus, to compute velocity in feet per second, we must 
 divide a number of feet by a number of seconds.* 
 
 If velocity is continuously varying, let x be the number 
 of feet described since a given epoch, and t the number 
 
 of seconds elapsed, then is what is meant by the 
 
 dt 
 
 number of feet per second. The word should never be 
 employed in the specification of quantities, except when 
 the quantity named before it varies directly as the quantity 
 named after it, at least for small variations as, in the 
 above instance, the distance described is ultimately 
 proportional to the time of describing it. 
 
 Extended Sense of the terms "Multiplication " and 
 11 Division" 
 
 11. In ordinary multiplication the multiplier is always 
 a mere numerical quantity, and the product is of the same 
 nature as the multiplicand. Hence in ordinary division 
 either the divisor is a mere numerical quantity and the 
 quotient a quantity of the same nature as the dividend; 
 
 * It is not correct to speak of interest at the rate of Five Pounds 
 per cent. It should be simply Five per cent. A rate of five pounds 
 in every hundred pounds is not different from a rate of five shillings 
 in every hundred shillings. 
 
i.] GENERAL THEORY OF UNITS. 7 
 
 or else the divisor is of the same nature as the dividend, 
 and the quotient a mere numerical quantity. 
 
 But in discussing problems relating to units, it is con- 
 venient to extend the meanings of the terms " multiplica- 
 tion " and " division." A distance divided by a time 
 will denote a velocity the velocity with which the given 
 distance would be described in the given time. The dis- 
 tance can be expressed as a unit distance multiplied by 
 a numerical quantity, and varies jointly as these two 
 factors ; the time can be expressed as a unit time multi- 
 plied by a numerical quantity, and is jointly proportional 
 to these two factors. Also, the velocity remains unchanged 
 when the time and distance are both changed in the 
 same ratio. 
 
 1 2. The three quotients 
 
 i mile 5280 ft. 22 ft. 
 i "hour* 3600 sec. ' 15 sec 
 
 all denote the same velocity, and are therefore to be 
 regarded as equal. In passing from the first to the 
 second, we have changed the units in the inverse ratio 
 to their numerical multipliers, and have thus left both 
 the distance and the tirrfe unchanged. In passing from 
 the second to the third, we have divided the two numeri- 
 cal factors by a common measure, and have thus changed 
 the distance and the time in the same ratio. A change 
 in either factor of the numerator will be compensated 
 by a proportional change in either factor of the denom- 
 inator. 
 
 Further, since the velocity ------ - is of the velo- 
 
 15 sec. 15 
 
 city J-*r_ we are entitled to write 22 tj 2 
 
 i sec. 15 sec 15 sec. 
 
8 UNITS AND PHYSICAL CONSTANTS. [CHAP. 
 
 thus separating the numerical part of the expression from 
 the units part. 
 
 In like manner we may express the result of Art. 9 by 
 writing 
 
 yard i foot 
 
 (minute) 2 1200 (second) 2 ' 
 
 Such equations as these may be called " physical 
 equations," inasmuch as they express the equality of 
 physical quantities, whereas ordinary equations express 
 the equality of mere numerical values. The use of 
 physical equations in problems relating to units is to be 
 strongly recommended, as affording a natural and easy 
 clue to the necessary calculations, and especially as 
 obviating the doubt by which the student is often 
 embarrassed as to whether he ought to multiply or 
 divide. 
 
 13. In the following examples, which illustrate the 
 use of physical equations, we shall employ / to denote 
 the unit length, m the unit mass, and t the unit time. 
 
 Ex. i. If the yard be the unit of length, and the 
 acceleration of gravity (in which a velocity of 32-2 ft. per 
 sec. is gained per sec.) be represented by 2415, find the 
 unit of time. 
 
 We have / = yard, and 
 
 32>2 = 24r5 * = ' 
 
 sec/ .> t = 
 
 32-2 
 
 'Ex. 2. If the unit time be the second, the unit density 
 162 Ibs. per cub. ft, and the unit force* the weight of an 
 
 * For the dimensions of density and force, see Art. 14. 
 
[.] GENERAL THEORY OF UNITS. 
 
 We have / m sec., ~ 4 = 162 _ 5> 
 
 ounce at a place where g (in foot-second units) is 32, 
 what is the unit length ? 
 
 m , Ib. 
 
 7' = l62 (ft) 3 
 
 and jf!L = 32 . oz ^, or ml = 32 oz. ft. - 2 Ib. ft. 
 sec. 2 sec. 2 
 
 Hence by division 
 
 /* = Q r - (ft.) 4 , / - 1 ft. = 4 in. 
 
 ol 
 
 Ex. 3. If the area of a field of 10 acres be represented 
 by 100, and the acceleration of gravity (taken as 32 
 foot-second units) be 58?, find the unit of time. 
 
 We have 48400 (yd.) 2 = 100 / 2 , whence /= 22 yd. ; 
 
 whence / 2 m I7 22 sec. 2 - 121 sec. 2 , /- n sec. 
 
 32 
 
 Ex. 4. If 8 ft. per sec. be the unit velocity, and the 
 acceleration of gravity (32 foot-second units) the unit 
 acceleration, find the units of length and time. 
 We have the two equations 
 
 / ft / ft. 
 
 *- 8 sb: / 2 : 32 ^. 2 ' 
 
 whence by division / | sec., and substituting this value 
 of/ in the first equation, we have 4 / = 8 ft, / = 2 ft. 
 
 Ex. 5. If the unit force be 100 Ibs. weight, the unit 
 length 2 ft, and the unit time J sec., find the unit mass, 
 the acceleration of gravity being taken as 32 foot-second 
 units. 
 
 We have / - 2 ft, / = J sec., 
 
 ,, ft. ;;// m 2 ft. 
 
 100 Ib. 32 = - = , 
 
 sec. 2 / 2 T V sec. 2 
 
 that is 100 x 32 Ib. = 32 ;;/, m 100 Ib. 
 
io UNITS AND PHYSICAL CONSTANTS. [CHAP. 
 
 Ex. 6. The number of seconds in the unit of time is 
 equal to the number of feet in the unit of length, the unit 
 of force is 750 Ibs weight [g being 32], and a cubic foot 
 of the standard substance [substance of unit density] con- 
 tains 13500 oz. Find the unit of time. 
 
 Let t = x sec., then /= x ft. ; also let m =y Ib. Then 
 we have 
 
 ml _ y Ib. x ft. . y_ Ib. ft. ^ Ib. ft. 
 
 /* ' x* sec. 2 3 x sec. 2 " 75 * 32 sec. 2 
 
 or ? - 750 x 32. 
 
 oc 
 
 .1 771 V ID. 
 
 AIso p = ^ ?T = I35 ?T' 
 
 * c-v 1L. 1 L. 
 
 m v Ib. oz. 
 
 73 = 77 Z7 3 = J 35 
 
 * c-v AL 
 
 whence = 13500 x . 
 
 x* 1 6 
 
 Hence by division 
 
 7so x 72 x 16 i6 2 16 16 
 
 #2 = 15 6 = x = , /= sec. 
 
 13500 3 2 3 3 
 
 Ex. 7. When an inch is the unit of length and / the 
 
 unit of time, the measure of a certain acceleration is a ; 
 
 when 5 ft. and i min. are the units of length and time 
 
 respectively, the measure of the same acceleration is 
 
 io a. Find /. 
 Equating the two expressions for the acceleration, we 
 
 have a = io a 5 
 
 f 2 (mm.) 
 
 whence f> = (min.) 2 = ' = 6 (sec). 2 
 
 50 ft. 600 
 
 / P J6 sec. 
 
 Ex. 8. The numerical value of a certain force is 56 
 when the pound is the unit of mass, the foot the unit of 
 length, and the second the unit of time ; what will be the 
 
I.] GENERAL THEORY OF UNITS. 11 
 
 numerical value of the same force when the hundredweight 
 is the unit of mass, the yard the unit of length, and the 
 minute the unit of time ? 
 
 Denoting the required value by x we have 
 , Ib. ft. cwt. yard . 
 
 , Ib. ft. /min.\ 2 
 5 cwt. ydT \sec7 ) 
 
 = 56 x L- x x 6o 2 = 600. 
 
 112 
 
 O 
 
 Dimensions of Mechanical and Geometrical Quantities. 
 
 14. In the following list of dimensions, we employ the 
 letters L, M, T as abbreviations for the words Length, 
 Mass, Time. The symbol of equality is used to denote 
 sameness of dimensions. 
 
 Area = L 2 , Volume = L 3 , Velocity = ^ , 
 
 Acceleration = , Momentum = - . 
 
 M 
 Density = , density' being defined as mass per unit 
 
 J / 
 
 volume. 
 
 ML 
 
 Force = -<p since a force is measured by the momen- 
 tum which it generates per unit of time, and is therefore 
 the quotient of momentum by time or since a force is 
 measured by the product of a mass by the acceleration 
 generated in this mass. 
 
 Work = - , being the product of force and distance. 
 
12 UNITS AND PHYSICAL CONSTANTS. [CHAP. 
 
 TV/TT 2 
 
 Kinetic energy = =^-, being half the product of mass 
 
 by square of velocity. The constant factor % can be 
 omitted, as not affecting dimensions. 
 
 Moment of couple =--, being the product of a force 
 
 by a length. 
 
 T 
 
 arc 
 
 The dimensions of angle*, when measured by -. , 
 
 J radius 
 
 are zero. The same angle will be denoted by the same 
 number, whatever be the unit of length employed. In 
 
 fact. we have arc = - = L. 
 radius L 
 
 The work done by a couple in turning a body through 
 any angle, is the product of the couple by the angle. 
 The identity of dimensions between work and couple is 
 thus verified : 
 
 Angular velocity . 
 Angular acceleration = -L . 
 
 Moment of inertia = ML 2 . 
 
 ML 2 
 
 Angular momentum = moment of momentum = ' , 
 
 being the product of moment of inertia by angular 
 velocity, or the product of momentum by length. 
 
 Intensity of pressure, or intensity of stress generally, 
 
 * It is proposed to give the name radian to the angle whose 
 arc is equal to radius. Thus, instead of "an angle whose value 
 in circular measure is P," we shall be able to say "an angle of t) 
 radicns." 
 
i.J GENERAL THEORY OF UNITS. 13 
 
 being force per unit of area, is of dimensions -; that 
 
 area 
 
 . M 
 
 Intensity of force of attraction at a point, often called 
 simply force at a point, being force per unit of attracted 
 
 mass, is of dimensions - - or - . It is numerically 
 mass T 2 
 
 equal to the acceleration which it generates, and has 
 accordingly the dimensions of acceleration. 
 
 The absolute force of a centre of attraction, better called 
 the strength of a centre, may be defined as the intensity of 
 force at unit distance. If the law of attraction be that of 
 inverse squares, the strength will be the product of the 
 intensity of force at any distance by the square of this 
 
 L 3 
 
 distance, and its dimensions will be . 
 
 Curvature (of a curve) = ]- , being the angle turned by 
 Li 
 
 the tangent per unit distance travelled along the curve. 
 
 Tortuosity = , being the angle turned by the osculat- 
 L 
 
 ing plane per unit distance travelled along the curve. 
 
 The solid angle or aperture of a conical surface of any 
 form is measured by the area cut off by the cone from a 
 sphere whose centre is at the vertex of the cone, divided 
 by the square of the radius of the sphere. Its dimensions 
 are therefore zero ; or a solid angle is a numerical quan- 
 tity independent of the fundamental units. 
 
 The specific curvature of a surface at a given point 
 (Gauss's measure of curvature) is the solid angle de- 
 scribed by a line drawn from a fixed point parallel to the 
 
14 UNITS AND PHYSICAL CONSTANTS. 
 
 normal at a point which travels on the surface round the 
 given point, and close to it, divided by the very small 
 
 area thus enclosed. Its dimensions are therefore . 
 
 The mean curvature of a surface at a given point, in 
 the theory of Capillarity, is the arithmetical mean of the 
 curvatures of any two normal sections normal to each 
 
 other. Its dimensions are therefore - . 
 
CHAPTER II. 
 
 CHOICE OF THREE FUNDAMENTAL UNITS. 
 
 15. NEARLY all the quantities with which physical 
 science deals can be expressed in terms of three funda- 
 mental units ; and the quantities commonly selected to 
 serve as the fundamental units are 
 
 a definite length, 
 
 a definite mass, 
 
 a definite interval of time. 
 
 This particular selection is a matter of convenience 
 rather than of necessity ; for any three independent units 
 are theoretically sufficient. For example, we might em- 
 ploy as the fundamental units 
 
 a definite mass, 
 
 a definite amount of energy, 
 
 a definite density. 
 
 1 6. The following are the most important consider- 
 ations which ought to guide the selection of fundamental 
 units : 
 
 (i) They should be quantities admitting of very 
 accurate comparison with other quantities of the same 
 kind. 
 
16 UNITS AND PHYSICAL CONSTANTS. [CHAP. 
 
 (2) Such comparison should be possible at all times. 
 Hence the standards must be permanent that is, not 
 liable to alter their magnitude with lapse of time. 
 
 (3) Such comparison should be possible at all places. 
 Hence the standards must not be of such a nature 
 as to change their magnitude when carried from place to 
 place. 
 
 (4) The comparison should be easy and direct. 
 
 Besides these experimental requirements, it is also 
 desirable that the fundamental units be so chosen that 
 the definition of the various derived units shall be easy, 
 and their dimensions simple. 
 
 17. There is probably no kind of magnitude which so 
 completely fulfils the four conditions above stated as a 
 standard of mass, consisting of a piece of gold, platinum, 
 or some other substance not liable to be affected by 
 atmospheric influences. The comparison of such a 
 standard with other bodies of approximately equal 
 mass is effected by weighing, which is, of all the 
 operations of the laboratory, the most exact. Very ac- 
 curate copies of the standard can thus be secured ; and 
 these can be carried from place to place with little risk 
 of injury. 
 
 The third of the requirements above specified forbids 
 the selection of a force as one of the fundamental units. 
 Forces at the same place can be very accurately mea- 
 sured by comparison with weights ; but as gravity varies 
 from place to place, the force of gravity upon a piece 
 of metal, or other standard weight, changes its mag- 
 nitude in travelling from one place to another. A 
 spring-balance, it is true, gives a direct measure of 
 
n.j CHOICE OF THREE FUNDAMENTAL UNITS. 17 
 
 force; but its indications are too rough for purposes of 
 accuracy. 
 
 1 8. Length is an element which can be very accurately 
 measured and copied. But every measuring instrument 
 is liable to change its length with temperature. It is 
 therefore necessary, in denning a length by reference to a 
 concrete material standard, such as a bar of metal, to 
 state the temperature at which the standard is correct. 
 The temperature now usually selected for this purpose is 
 that of a mixture of ice and water (o C), observation 
 having shown that the temperature of such a mixture is 
 constant. 
 
 The length of the standard should not exceed the 
 length of a convenient measuring-instrument ; for, in 
 comparing the standard with a copy, the shifting of the 
 measuring-instrument used in the comparison introduces 
 additional risk of error. 
 
 In end-standards, the standard length is that of the bar 
 as a whole, and the ends are touched by the instrument 
 every time that a comparison is made. This process is 
 liable to wear away the ends and make the standard 
 false. In line-standards, the^ standard length is the dis- 
 tance between two scratches, and the comparison is 
 made by optical means. The scratches are usually at 
 the bottom of holes sunk halfway through the bar. 
 
 19. Time is also an element which can be measured 
 with extreme precision. The direct instruments of mea- 
 surement are clocks and chronometers ; but these are 
 checked by astronomical observations, and especially by 
 transits of stars. The time between two successive tran- 
 sits of a star is (very approximately) the time of the 
 earth's rotation on its axis ; and it is upon the uniformity 
 
i8 UNITS AND PHYSICAL CONSTANTS. [CHAP. 
 
 of this rotation that the preservation of bur standards of 
 time depends. 
 
 Necessity for a Common Scale. 
 
 20. The existence of quantitative correlations between 
 the various forms of energy, imposes upon men of science 
 the duty of bringing all kinds of physical quantity to one 
 common scale of comparison. Several such measures 
 (called absolute measures) have been published in recent 
 years; and a comparison of them brings very promi- 
 nently into notice the great diversity at present existing 
 in the selection of particular units of length, mass, and 
 time. 
 
 Sometimes the units employed have been the foot, the 
 grain, and the second; sometimes the millimetre, milli- 
 gramme, and second ; sometimes the centimetre, gramme, 
 and second; sometimes the centimetre, gramme, and 
 minute ; sometimes the metre, tonne, and second ; 
 sometimes the metre, gramme, and second ; while some- 
 times a mixture of units has been employed ; the area of 
 a plate, for example, being expressed in square metres, 
 and its thickness in millimetres. 
 
 A diversity of scales may be tolerable, though undesir- 
 able, in the specification of such simple matters as length, 
 area, volume, and mass when occurring singly; for the 
 reduction of these from one scale to another is generally 
 understood. But when the quantities specified involve 
 a reference to more than one of the fundamental units, 
 and especially when their dimensions in terms of these 
 units are not obvious, but require careful working out, 
 there is great increase of difficulty and of liability to 
 mistake. 
 
ii.] CHOICE OF THREE FUNDAMENTAL UNITS. 19 
 
 A general agreement as to the particular units of 
 length, mass, and time which shall be employed if not 
 in all scientific work, at least in all work involving com- 
 plicated references to units is urgently needed; and 
 almost any one of the selections above instanced would 
 be better than the present option. 
 
 21. We shall adopt the recommendation of the Units 
 Committee of the British Association (see Appendix), 
 that all specifications be referred to the Centimetre, the 
 Gramme, and the Second. The system of units derived 
 from these as the fundamental units is called the C.G.S. 
 system; and the units of the system are called the C.G.S. 
 units. 
 
 The reason for selecting the centimetre and gramme, 
 rather than the metre and gramme, is that, since a 
 gramme of water has a volume of approximately i cubic 
 centimetre, the former selection makes the density of 
 water unity; whereas the latter selection would make 
 it a million, and the density of a substance would 
 be a million times its specific gravity, instead of being 
 identical with its specific gravity as in the C.G.S. 
 system. 
 
 Even those who may have a preference for some other 
 units will nevertheless admit the advantage of having a 
 variety of results, from various branches of physics, re- 
 duced from their original multiplicity and presented in 
 one common scale. 
 
 22. The adoption of one common scale for all quan- 
 tities involves the frequent use of very large and very 
 small numbers. Such numbers are most conveniently 
 written by expressing them as the product of two factors, 
 
20 UNITS AND PHYSICAL CONSTANTS. 
 
 one of which is a power of 10; and it is usually advan- 
 tageous to effect the resolution in such a way that the 
 exponent of the power of 10 shall be the characteristic of 
 the logarithm of the number. Thus 3240000000 will 
 be written 3*24 x ic, and '00000324 will be written 
 3-24 x lo" 6 . 
 
CHAPTER III. 
 
 MECHANICAL UNITS. 
 Value of g. 
 
 23. ACCELERATION is defined as the rate of increase of 
 velocity per unit of time. The C.G.S. unit of accelera- 
 tion is the acceleration of a body whose velocity increases 
 in every second by the C.G.S. unit of velocity namely, 
 by a centimetre per second. The apparent acceleration 
 of a body falling freely under the action of gravity in 
 vacuo is denoted by g. The value of g in C.G.S. units 
 at any part of the earth's surface is approximately given 
 by the following formula, 
 
 g= 980*6056 - 2-5028 cos 2 A '000003^, 
 
 A denoting the latitude, and h the height of the station 
 (in centimetres) above sea-level. 
 
 The constants in this formula have been deduced from 
 numerous pendulum experiments in different localities, 
 the length / of the seconds' pendulum being connected 
 with the value of g by the formula g ir 2 /. 
 
 Dividing the above equation by ?r 2 we have, for the 
 length of the seconds' pendulum, in centimetres, 
 
 1= 99*3562 - '2536 COS 2A - '0000003//. 
 
22 UNITS AND PHYSICAL CONSTANTS. [CHAP. 
 
 At sea-level these formulae give the following values for 
 the places specified : 
 
 
 Latitude. 
 
 Value of g. 
 
 Value of /. 
 
 Equator . 
 
 o / 
 O O 
 
 Q?8'IO 
 
 QQ'IO^ 
 
 Latitude 45, . . . 
 
 45 o 
 
 4-8 Q 
 
 980-61 
 980-88 
 
 99'356 
 99 "384 
 
 Paris, . . . 
 
 4.8 so 
 
 080*04. 
 
 QQ"?QO 
 
 Greenwich, .... 
 Gottingen, .... 
 Berlin, . . . 
 
 5 1 29 
 
 5 1 32 
 
 C2 3O 
 
 98I-I7 
 98I-I7 
 08 1 '2 <J 
 
 99'4I3 
 99-4I4 
 QQ*422 
 
 Dublin, 
 
 S3 21 
 
 981 "12 
 
 QQ'42Q 
 
 Manchester, .... 
 Belfast, 
 
 53 29 
 
 C4. -26 
 
 981-34 
 08 1 "4"? 
 
 99-430 
 
 90'44-O 
 
 Edinburgh, .... 
 Aberdeen, .... 
 Pole, 
 
 55 57 
 57 9 
 
 QO O 
 
 98I-54 
 981-64 
 
 o8vn 
 
 99-45I 
 99-461 
 99'6lO 
 
 
 
 
 
 The difference between the greatest and least values 
 
 (in the case of both g and /) is about -i- of the mean 
 value. '9 6 
 
 Force. 
 
 24. The C.G.S. unit of force is called the dyne. It is 
 the force which, acting upon a gramme for a second, 
 generates a velocity of a centimetre per second. 
 
 It may otherwise be defined as the force which, acting 
 upon a gramme, produces the C.G.S. unit of acceleration, 
 or as the force which, acting upon any mass for i 
 second, produces the C.G.S. unit of momentum. 
 
 To show the equivalence of these three definitions, let 
 m denote mass in grammes, v velocity in centimetres per 
 second, t time in seconds, F force in dynes. 
 
 Then, by the second law of motion, we have 
 
in.] MECHANICAL UNITS. 23 
 
 force 
 
 acceleration = 
 
 mass 
 
 that is, if a denote acceleration in C.G.S. units. a = - 
 
 m } 
 
 hence, when a and m are each unity, F will be unity. 
 
 Again, by the nature of uniform acceleration, we have 
 v - at, v denoting the velocity due to the acceleration a, 
 continuing for time t. 
 
 Hence we have F = ma = . Therefore, if mv = i 
 
 and t = i, we have F = i. 
 
 As a particular case, if m i, v = i, /= i, we have 
 F= i. 
 
 25. The force represented by the weight of a gramme 
 varies from place to place. It is the force required to 
 sustain a gramme in vacuo, and would be nil at the 
 earth's centre, where gravity is nil. To compute its 
 amount in dynes at any place where g is known, observe 
 that a mass of i gramme falls in vacuo with acceleration 
 g. The force producing this acceleration (namely, the 
 weight of the gramme) must be equal to the product of 
 the mass and acceleration, jthat is, to g. 
 
 The weight (when weight means force) of i gramme is 
 therefore g dynes ; and the weight of m grammes is mg 
 dynes. 
 
 26. Force is said to be expressed in gravitation-measure 
 when it is expressed as equal to the weight of a given 
 mass. Such specification is inexact unless the value of 
 g is also given. For purposes of accuracy it must always 
 be remembered that the pound, the gramme, &c., are, 
 strictly speaking, units of mass. Such an expression as 
 " a force of 100 tons " must be understood as an abbrevia- 
 
24 UNITS AND PHYSICAL CONSTANTS. [CHAP. 
 
 tion for " a force equal to the weight [at the locality in 
 question] of 100 tons." 
 
 27. The name poundal has recently been given to the 
 unit force based on the pound, foot, and second ; that is, 
 the force which, acting on a pound for a second, gene- 
 rates a velocity of a foot per second. It is - of the 
 
 weight of a pound, g denoting the acceleration due 
 to gravity expressed in foot-second units, which is about 
 32-2 in Great Britain. 
 
 To compare the poundal with the dyne, let x denote 
 the number of dynes in a poundal ; then we have 
 
 gm. cm. Ib. ft. 
 sec. 2 sec.''* 
 
 Ib. ft. 
 * = - * - : =453'59 x 3'4797 = i3 82 5- 
 
 Work and Energy. 
 
 28. The C.G.S. unit of work is called the erg. It is 
 the amount of work done by a dyne working through a 
 distance of a centimetre. 
 
 The C.G.S. unit of energy is also the erg, energy being 
 measured by the amount of work which it represents. 
 
 29. To establish a rule for computing the kinetic energy 
 (or energy due to the motion) of a given mass moving with 
 a given velocity, it is sufficient to consider the case of 
 a body falling in vacuo. 
 
 When a body of m grammes falls through a height of h 
 centimetres, the working force is the weight of the body 
 that is, gm dynes, which, multiplied by the distance 
 worked through, gives gmh ergs as the work done. But 
 
in.] MECHANICAL UNITS. 25 
 
 the velocity acquired is such that v 1 2gh. Hence we 
 have gmh = \miP-. 
 
 The kinetic energy of a mass of m grammes moving 
 with a velocity of v centimetres per second is therefore 
 \mv^ ergs ; that is to say, this is the amount of work 
 which would be required to generate the motion of 
 the body, or is the amount of work which the body 
 would do against opposing forces before it would come 
 to rest. 
 
 30. Work, like force, is often expressed in gravitation- 
 measure. Gravitation units of work, such as the foot- 
 pound and kilogramme-metre, vary with locality, being 
 proportional to the value of g. 
 
 One gramme-centimetre is equal to g ergs. 
 One kilogramme-metre is equal to 100,000 g ergs. 
 One foot-poundal is 453'59 x (3'4797) 2 ^4 2I 39 er g s - 
 One foot-pound is 13,825 g ergs, which, if^be taken 
 as 981, is 1*356 x io 7 ergs. 
 
 31. The unit rate of working is i erg per second. 
 Watt's "horse-power" is defined as 550 foot-pounds per 
 second. This is 7-46 x io 9 ergs per second. The "force 
 de cheval" is denned as 75 kilogrammetres per second. 
 This is 7-36 x i o 9 ergs per second. We here assume 
 =981. 
 
 Examples. 
 
 i. If a spring balance is graduated so as to show 
 the masses of bodies in pounds or grammes when used at 
 the equator, what will be its error when used at the poles, 
 neglecting effects of temperature ? 
 
26 UNITS AND PHYSICAL CONSTANTS. [CHAP. 
 
 Ans. Its indications will be too high by about L- of 
 the total weight. 
 
 2. A cannon-ball, of 10,000 grammes, is discharged 
 with a velocity of 45,000 centims. per sec. Find its 
 kinetic energy. 
 
 Ans. J x 10000 x (45ooo) 2 = 1-0125 x io 13 ergs. 
 
 3. In last question find the mean force exerted upon 
 the ball by the powder, the length of the barrel being 
 200 centims. 
 
 Ans. 5*0625 x io 10 dynes. 
 
 4. Given that 42 million ergs are equivalent to i 
 gramme-degree of heat, and that a gramme of lead at 
 10 C. requires 15^6 gramme-degrees of heat to melt it ; 
 find the velocity with which a leaden bullet must strike a 
 target that it may just be melted by the collision, sup- 
 posing all the mechanical energy of the motion to be 
 converted into heat and to be taken up by the bullet. 
 
 We have i^ 2 =i5'6xj, where J = 42xio 6 . Hence 
 ^=1310 millions; # = 36-2 thousand centims. per 
 second. 
 
 5. With what velocity must a stone be thrown vertically 
 upwards at a place where g is 981 that it may rise to 
 a height of 3000 centims. ? and to what height would 
 it ascend if projected vertically with this velocity at the 
 surface of the moon, where g is 150 ? 
 
 Ans. 2426 centims. per second ; 19620 centims. 
 
 Centrifugal Force. 
 
 32. A body moving in a curve must be regarded as 
 continually falling away from a tangent. The accelera- 
 
 z/ 2 
 tion with which it falls away is , v denoting its velocity 
 
in.] MECHANICAL UNITS. 27 
 
 and r the radius of curvature. The acceleration of a 
 body in any direction is always due to force urging it in 
 that direction, this force being equal to the product of 
 mass and acceleration. Hence the normal force on a 
 body of ;;/ grammes moving in a curve of radius r 
 centimetres, with velocity v centimetres per second, is 
 
 
 dynes. This force is directed towards the centre of 
 
 curvature. The equal and opposite force with which the 
 body reacts is called centrifugal force. 
 
 If the body moves uniformly in a circle, the time 
 
 of revolution being T seconds, we have v - -7=- ; 
 
 hence = 1-7=- 1 r -> an d the force acting on the bodv is 
 r \T / 
 
 /27T\ 2 , 
 
 mr(~\ dynes. 
 
 If n revolutions are made per minute, the value of T is 
 
 , and the force is mr( \ dynes. 
 \3/ 
 
 Examples. 
 
 i. A body of m grammes moves uniformly in a circle 
 of radius 80 centims., the time of revolution being \ of a 
 second. Find the centrifugal force, and compare it with 
 the weight of the body. 
 
 Ans. The centrifugal force is ;;/ x / 2 ^ J x 80 = m x 647^ 
 
 x 80 = 50532 m dynes. 
 
 The weight of the body (at a place where g is 981) 
 is 981 m dynes. Hence the centrifugal force is about 
 52 J times the weight of the body. 
 
28 UNITS AND PHYSICAL CONSTANTS. [CHAP. 
 
 2. At a bend in a river, the velocity in a certain part 
 of the surface is 170 centims. per second, and the radius 
 of curvature of the lines of flow is 9100 centims. Find 
 the slope of the surface in a section transverse to the 
 lines of flow. 
 
 Ans. Here the centrifugal force for a gramme of the 
 
 water is iZ2Z =3*176 dynes. If^be 981 the slope will 
 
 be - ' = - ; that is. the surface will slope upwards 
 981 309' 
 
 from the concave side at a gradient of i in 309. The 
 general rule applicable to questions of this kind is that 
 the resultant of centrifugal force and gravity must be 
 normal to the surface. 
 
 3. An open vessel of liquid is made to rotate rapidly 
 round a vertical axis. Find the number of revolutions 
 that must be made per- minute in order to obtain a slope 
 of 30 at a part of the surface distant 10 centims. from 
 the axis, the value of g being 981. 
 
 Ans. We must have tan 30 - , where f denotes the 
 
 & 
 
 intensity of centrifugal force that is, the centrifugal force 
 per unit mass. We have therefore 
 
 lnir\ 2 . n denoting the number of 
 981 tan 30 = lol ) . . 
 
 \3o/ revolutions per minute, 
 
 90 
 Hence n- 71-9. 
 
 4. For the intensity of centrifugal force at the equator 
 due to the earth's rotation, we have r = earth's radius 
 = 6-38 x io 8 , T = 86164, being the number of seconds in 
 a sidereal day. 
 
in.] 
 
 MECHANICAL UNITS. 
 
 29 
 
 /2Y 
 
 (x) = 3 ' 39 ' 
 
 This is about of the value of r. 
 289 
 
 If the earth were at rest, the value of g at the equator 
 would be greater than at present by this amount. If the 
 earth were revolving about 17 times as fast as at present, 
 the value of g at the equator would be nil. 
 
 UBI7BRSITY 
 
CHAPTER IV. 
 
 HYDROSTATICS. 
 
 33. THE following table of the relative density of water 
 at various temperatures (under atmospheric pressure), the 
 density at 4 C. being taken as unity, is from Rossetti's 
 results deduced from all the best experiments (Ann. Ch. 
 Phys. x. 461 ; xvii. 370, 1869) : 
 
 Temp. 
 Cent. 
 
 Relative 
 Density. 
 
 Temp. 
 Cent. 
 
 Relative 
 Density. 
 
 Temp. 
 Cent. 
 
 Relative 
 Density. 
 
 o 
 
 
 
 
 
 
 o 
 
 999871 
 
 i3 
 
 999430 
 
 35 
 
 99418 
 
 I 
 
 999928 
 
 H 
 
 999299 
 
 40 
 
 99235 
 
 2 
 
 999969 
 
 15 
 
 999160 
 
 45 
 
 99037 
 
 3 
 
 999991 
 
 16 
 
 999002 
 
 50 
 
 98820 
 
 4 
 
 I '000000 
 
 17 
 
 998841 
 
 55 
 
 98582 
 
 5 
 
 999990 
 
 18 
 
 998654 
 
 60 
 
 98338 
 
 6 
 
 999970 
 
 19 
 
 998460 
 
 65 
 
 98074 
 
 8 
 
 999933 
 999886 
 
 20 
 22 
 
 998259 
 997826 
 
 70 
 75 
 
 97794 
 97498 
 
 9 
 
 10 
 
 999824 
 999747 
 
 11 
 
 997367 
 
 996866 ; 
 
 80 
 
 85 
 
 97194 
 96879 
 
 ii 
 
 999655 
 
 28 
 
 996331 j 
 
 90 
 
 96556 
 
 12 
 
 999549 
 
 30 
 
 995765 1 
 
 100 
 
 95865 
 
 34. According to Kupffer's observations, as reduced 
 by Professor W. H. Miller, the absolute density (in 
 grammes per cubic centimetre) at 4 is not i, but 
 
HYDROSTATICS. 
 
 1*000013. Multiplying the above numbers by this 
 factor, we obtain the following table of absolute den- 
 sities : 
 
 Temp. 
 
 Density. 
 
 Temp. Density. 
 
 Temp. 
 
 Density. 
 
 
 
 999884 
 
 13 
 
 "999443 
 
 35 
 
 99469 
 
 I 
 
 999941 
 
 14 
 
 999312 
 
 40 
 
 99236 
 
 2 
 
 999982 
 
 15 
 
 999173 
 
 45 
 
 99038 
 
 3 
 
 I '000004 
 
 16 
 
 999015 
 
 5 
 
 98821 
 
 4 
 
 I -OOOOI3 
 
 17 
 
 998854 
 
 55 
 
 98583 
 
 5 
 
 I -000003 
 
 18 
 
 998667 
 
 60 
 
 98339 
 
 6 
 
 999983 
 
 19 
 
 998473 
 
 65 
 
 9 8075 
 
 7 
 
 999946 
 
 20 
 
 998272 
 
 70 
 
 '97795 
 
 8 
 
 999899 
 
 22 
 
 997839 
 
 75 
 
 '97499 
 
 9 
 
 999837 
 
 24 
 
 997380 
 
 80 
 
 97195 
 
 10 
 
 999760 
 
 26 
 
 996879 
 
 85 
 
 96880 
 
 ii 
 
 999668 
 
 28 ^96344 
 
 90 
 
 96557 
 
 12 
 
 999562 
 
 30 '99S77 8 
 
 100 
 
 95866 
 
 35. The volume, at temperature t, of the water which 
 occupies unit volume at 4, is approximately 
 
 ! + A(/- 4 ) 2 -B(/- 4 ) 2 - 6 + C(/- 4 ) 3 , 
 where 
 
 A = 8-38 x io- , 
 
 B - 379'' x i~ 7 > 
 C 2*24 x io~ 8 ; 
 
 and the relative density at temperature f is given by the 
 same formula with the signs of A, B, and C reversed. 
 The rate of expansion at temperature f is 
 
 2 A (/- 4 )-2-6B .(/- 4)1-6 + 3 C (/- 4 ) 2 . 
 
 In determining the signs of the terms with the frac- 
 tional exponents 2-6 and r6, these exponents are to be 
 regarded as odd. 
 
UNITS AND PHYSICAL CONSTANTS. [CHAP. 
 
 36. Table of Densities (chiefly taken from Rankine's 
 " Rules and Tables," pp. 149 and 150) : 
 
 Solids. 
 
 , , wire 
 
 8-1:4. 
 
 Brick 
 
 O " 
 
 2 to 2*17 
 
 Bronze 
 
 
 Brickwork 
 
 1-8 
 
 Copper, cast 
 
 8-6 
 
 Chalk 
 
 I '8 tO 2'i 
 
 ,, sheet.. . . 
 
 8-8 
 
 Clay 
 
 I '92 
 
 ,, hammered.. 
 Gold 
 
 8-9 
 10 to I0'6 
 
 Glass, crown 
 ,, flint . 
 
 
 Iron, cast 
 , , wrought 
 
 6 '95 to 7-3 
 7 '6 to 7 '8 
 
 Quartz (rock-cry- 
 stal) 
 
 2'6- 
 
 Lead 
 
 1 1 -4 
 
 Sand (dry) 
 
 I '42 
 
 Platinum 
 
 21 tO 22 
 
 Fir, spruce 
 
 48 to 7 
 
 Silver .... 
 
 
 Oak, European . . 
 
 69 to "99 
 
 Steel 
 
 7 '8 to 7 '9 
 
 Lignum vitse 
 
 65 to i 'V 
 
 Tin . 
 
 7 '3 to 7 '5 
 
 Sulphur, octahedral 
 
 
 Zinc 
 
 6 "8 to 7 "2. 
 
 ,, prismatic. 
 
 1-98 
 
 Ice .. 
 
 02 
 
 
 
 Liquids at o C. 
 
 Sea- water, ordinary i '026 
 
 Alcohol, pure 791 
 
 ,, proof spirit ... '916 
 
 Ether 716 
 
 Mercury 13 -596 
 
 Naphtha '848 
 
 Oil, linseed '940 
 
 olive -915 
 
 whale -923 
 
 ,, of turpentine '870 
 
 37. If a body weighs m grammes in vacuo and m' 
 grammes in water of density unity, the volume of the body 
 is m m cubic centims.; for the mass of the water dis- 
 placed is m - m grammes, and each gramme of this 
 water occupies a cubic centimetre. 
 
 Examples. 
 
 i. A glass cylinder, / centims. long, weighs m grammes 
 in vacuo and m grammes in water of unit density. Find 
 its radius. 
 
iv.] HYDROSTATICS. 33 
 
 Solution. Its section is 7r;- 2 , and is also m ; hence 
 2 m m' 
 
 ~^T 
 
 2. Find the capacity at o C. of a bulb which holds m 
 grammes of mercury at that temperature. 
 
 Solution. The specific gravity of mercury at o being 
 13-596 as compared with water at the temperature of 
 maximum density, it follows that the mass of i cubic 
 centim. of mercury is 13*596 x 1-000013 = 13-59618, say 
 
 13-596. Hence the required capacity is - - cubic 
 
 centims. 
 
 3. Find the total pressure on a surface whose area is A 
 square centims. when its centre of gravity is immersed to 
 a depth of h centims. in water of unit density, atmos- 
 pheric pressure being neglected. 
 
 Ans. A/i grammes weight ; that is gKh dynes. 
 
 4. If mercury of specific gravity 13*596 is substituted 
 for water in the preceding question, find the pressure. 
 
 Ans. 13-596 Kh grammes weight- that is, 13-596 ghh 
 dynes. 
 
 5. If h be 76, and A be unity in example 4, the 
 answer becomes 1033-3 grammes weight, or 1033-3^- 
 dynes. 
 
 For Paris, where g is 980-94, this is 1-0136 x io r> 
 dynes. 
 
 Barometric Pressure. 
 
 38. The C.G.S. unit of pressure intensity (that is, of 
 pressure per unit area) is the pressure of a dyne per 
 square centim. 
 
34 UNITS AND PHYSICAL CONSTANTS. [CHAP. 
 
 At the depth of h centims. in a uniform liquid whose 
 density is d [grammes per cubic centim.], the pressure 
 due to the weight of the liquid is ghd dynes per square 
 centim. 
 
 The pressure-intensity due to the weight of a column of 
 mercury at o C, 76 centims. high, is found by putting 
 ^ = 76, d= 13*596, and is 1033-3^-. It is therefore 
 different at different localities. At Paris, where 'g is 
 980-94, it is i -0136 x io 6 ; that is, rather more than a 
 megadyne * per square centim. To exert a pressure of 
 exactly one megadyne per square centim., the height of 
 the column at Paris must be 74*98 centims. 
 
 At Greenwich, where gis 981*17, the pressure due to 76 
 centims. of mercury at o C. is 1*0138 x io 6 ; and the 
 height which would give a pressure of io 6 is 74*964 
 centims., or 29*514 inches. 
 
 Convenience of calculation would be promoted by 
 adopting the pressure of a megadyne per square centim., 
 or io 6 C.G.S. units of pressure-intensity, as the standard 
 atmosphere. 
 
 The standard now commonly adopted (whether 76 
 centims. or 30 inches) denotes different pressures at 
 different places, the pressure denoted by it being pro- 
 proportional to the value of g. 
 
 We shall adopt the megadyne per square centim. as our 
 standard atmosphere in the present work. 
 
 Examples. 
 i. What must be the height of a column of water of 
 
 * The prefix mega denotes multiplication by a million. A 
 megadyne is a force of a million dynes. 
 
iv.] HYDROSTATICS. 35 
 
 unit density to exert a pressure of a megadyne per square 
 centim. at a place where g is 981 ? 
 
 Ans. I00 000 = 1019-4 centims. This is 33*445 f eet - 
 981 
 
 2. What is the pressure due to an inch of mercury at 
 o C. at a place where g is 981. (An inch is 2-54 
 centims.) 
 
 Ans. 981 x 2-54 x 13*596 =33878 dynes per square 
 centim. 
 
 3. What is the pressure due to a centim. of mercury at 
 o C. at the same locality ? 
 
 Ans. 981 x 13-596 = 13338. 
 
 4. What is the pressure due to a kilometre of sea-water 
 of density 1*027, g being 981 ? 
 
 Ans. 981 x io 5 x 1-027 IP 75 x io 8 dynes per square 
 centim., or 1*0075 x r 2 niegadynes per square centim. ; 
 that is, about 100 atmospheres. 
 
 5. What is the pressure due to a mile of the same 
 water ? 
 
 Ans. i '62 14 x io 8 C.G.S. units, or 162 '14 atmospheres 
 [of a megadyne per square xTentim.]. 
 
 Density of Air. 
 
 39. Regnault found that at Paris, under the pressure of 
 a column of mercury at o, of the height of 76 centims., 
 the density of perfectly dry air was -0012932 gramme per 
 cubic centim. The pressure corresponding to this height 
 of the barometer at Paris is 1-0136 x io 6 dynes per square 
 centim. Hence, by Boyle's law, we can compute the 
 density of dry air at o C. at any given pressure. 
 
36 UNITS AND PHYSICAL CONSTANTS. [CHAP. 
 
 At a pressure of a megadyne (io 6 dynes) per square 
 
 centim. the density will be OOI2 93 2 _ -ooi^^o. 
 
 1-0136 
 
 The density of dry air at o C. at any pressure / (dynes 
 per square centim.) is 
 
 /x 1-2759 x io- 9 . . . - - (4) 
 
 Example. 
 
 Find the density of dry air at o C., at Edinburgh, 
 under the pressure of a column of mercury at o 9 C., of 
 the height of 76 centims. 
 
 Here we have p = 981-54 x 76 x 13-596= 1-0142 x io 6 . 
 
 Ans. Required density = 1*2940 x io~ 3 =-0012940 
 gramme per cubic centim. 
 
 40. Absolute Densities of Gases, in grammes per cubic 
 centim., at o C., and a pressure of io 6 dynes per 
 square centim. 
 
 Mass of a cubic Volume of a gramm 
 
 centim. in grammes. in ubic centims. 
 
 Air, dry, -0012759 783*8 
 
 Oxygen, -0014107 708*9 
 
 Nitrogen, '0012393 806*9 
 
 Hydrogen, '00008837 11316-0 
 
 Carbonic acid, -0019509 512*6 
 
 ,, oxide -0012179 821-1 
 
 Marsh-gas, '0007173 I394'i 
 
 Chlorine, '0030909 323*5 
 
 Protoxide of nitrogen,... '0019433 5 r 4*6 
 
 Binoxide ,. ... '0013254 754*5 
 
 Sulphurous acid, '0026990 37o*5 
 
 Cyanogen, '0022990 435' 
 
 Olefiant gas '0012529 798*1 
 
 Ammonia, '0007594 1316*8 
 
iv.] HYDROSTATICS. 37 
 
 The numbers in the second column are the reciprocals 
 of those in the first. 
 
 The numbers in the first column are identical with the 
 specific gravities referred to water as unity. 
 
 Assuming that the densities of gases at constant pres- 
 sure and temperature are directly as their atomic weights, 
 we have for any gas at zero 
 
 pvp 1-1316 x io w m ; 
 
 v denoting its volume in cubic centims., m its mass in 
 grammes,/ its pressure in dynes per square centim., and 
 /* its atomic weight referred to that of hydrogen as unity. 
 
 Height of Homogeneous Atmosphere. 
 
 41. We have seen that the intensity of pressure at 
 depth /i, in a fluid of uniform density d, is ghd when the 
 pressure at the upper surface of the fluid is zero. 
 
 The atmosphere is not a fluid of uniform density ; but 
 it is often convenient to have a name to denote a height 
 H such that/=-HD, where/ denotes the pressure and 
 D the density of the air at a given point. 
 
 It may be defined as the height of a column of uniform 
 fluid having the same density as the air at the point 
 which would exert a pressure equal to that existing at the 
 point. 
 
 If the pressure be equal to that exerted by a column of 
 mercury of density 13 -5 96 and height /*, we have 
 
 / = g#x 13-596; 
 /.HD = Ax 13-596, H = ^LIp6. 
 
 If it were possible for the whole body of air above the 
 point to be reduced by vertical compression to the 
 
38 UNITS AND PHYSICAL CONSTANTS. [CHAP, 
 
 density which the air has at the point, the height from the 
 point up to the summit of this^compressed atmosphere 
 would be equal to H, subject to a small correction 
 for the variation of gravity with height. 
 
 H is called the height of the homogeneous atmosphere at 
 the point considered. Pressure-height would be a better 
 name. 
 
 The general formula for it is 
 
 and this formula will be applicable to any other gas 
 as well as dry air, if we make D denote the density 
 of the gas (in grammes per cubic centim.) at pressure/. 
 
 If, instead of/ being given directly in dynes per square 
 centim. , we have given the height h of a column of liquid 
 of density d which would exert an equal pressure, the 
 formula reduces to 
 
 (6) 
 
 42. The value of - in formula (5) depends only on the 
 
 nature of the gas and on the temperature; hence, for 
 a given gas at a given temperature, H varies inversely 
 as.g-. 
 
 For dry air at zero we have, by formula (4), 
 
 g 
 
 At Paris, where g is 980-94, we- find 
 H = 7-990 x io 5 . 
 
iv.] HYDROSTATICS. 39 
 
 At Greenwich, where g is 981-17, 
 H = 7-988 xio. 
 
 Examples. 
 
 1. Find the height of the homogeneous atmosphere at 
 Paris for dry air at 10 C., and also at 100 C. 
 
 Ans. For given density, p varies as i + "00366 /, / 
 denoting the temperature on the Centigrade scale. 
 Hence we have, at 10 C., 
 
 H = 1-0366 x 7-99 x io 5 = 8-2825 x io 5 ; 
 and at 100 C.. 
 
 H= 1-366 x 7-99 x io 5 1*0914 x io 6 . 
 
 2. Find the height of the homogeneous atmosphere for 
 hydrogen at o, at a place where g is 981. 
 
 Here we have 
 
 Diminution of Density with increase of Height in the 
 Atmosphere. 
 
 43. Neglecting the variation of gravity with height, the 
 variation of H as we ascend in the atmosphere would 
 depend only on variation of temperature. In an atmos- 
 phere of uniform temperature H will be the same at 
 all heights. In such an atmosphere, an ascent of i 
 centim. will involve a diminution of the pressure (and 
 
 therefore of the density) by -^ of itself, since the layer of 
 
 H 
 
 air which has been traversed is -^ of the whole mass of 
 
 H 
 
 superincumbent air. The density therefore diminishes 
 
40 UNITS AND PHYSICAL CONSTANTS. [CHAP. 
 
 by the same fraction of itself for every centim. that we 
 ascend; in other words, the density and pressure dim- 
 inish in geometrical progression as the height increases 
 in arithmetical progression. 
 
 Denote height above a fixed level by x, and pressure 
 by/. Then, in the notation of the differential calculus, 
 
 , dx dp 
 
 we have = - -^ , 
 
 H / 
 
 and if p v / 2 are the pressures at the heights x v x z , we 
 deduce 
 
 * 2 - x l = H log e - - H x 2-3026 Iog 10 i ... (7) 
 fi Pi 
 
 In the barometric determination of heights it is usual 
 to compute H by assuming a temperature which is the 
 arithmetical mean of the temperatures at the two heights. 
 
 For the latitude of Greenwich formula (7) becomes 
 
 x% Xi = (i + '00366 /) 7-988 x io 5 x 2*3026 log^ 1 
 
 A 
 
 = (i + '00366 /) 1,839,300 log A . . . (8) 
 
 A 
 
 t denoting the mean temperature, and the logarithms 
 being common logarithms. 
 
 To find the height at which the density would be 
 halved, variations of temperature being neglected, we 
 
 must put 2 for ^ in these formulae. The required height 
 
 P\ 
 
 will be H \og e 2, or, in the latitude of Greenwich, for 
 temperature o C., will be 
 
 1-8393 x io 6 x '30103 = 553700. 
 
 The value of log e 2, or 2-3026 Iog 10 2, is 
 2-3026 x -30103 = '69315. 
 
iv.j HYDROSTATICS. 41 
 
 Hence for an atmosphere of any gas at uniform tempera- 
 ture, the height at which the density would be halved is 
 the height of the homogeneous atmosphere for that gas, 
 multiplied by '69315. The gas is assumed to obey 
 Boyle's law. 
 
 Examples. 
 
 1. Show that if the pressure of the gas at the lower 
 station and the value of g be given, the height at which 
 the density will be halved varies inversely as the density. 
 
 2. At what height, in an atmosphere of hydrogen at 
 o C., would the density be halved, g being 981 ? 
 
 Ans. 7 '9954 x io 6 . 
 
 44. Pressure of Aqueous Vapour at various temperatures, 
 in dynes per square centim. 
 
 -20 1236 
 
 -15 1866 
 
 -io 2790 
 
 - 5 415 
 
 o 6133 
 
 5 8710 
 
 10 12220 
 
 50 i '226 x io 5 
 
 60 1-985 
 
 80 4729 
 
 loo 1-014 x Ic6 
 
 120 1-988 ,, 
 
 140 3-626 
 
 6-210 
 
 15 1693 ! J 8o 1-006 x io 7 
 
 20 23190 I 200 i '560 
 
 25 3HOO 
 
 30 42050 
 
 40 73200 
 
 The density of saturated steam, at any temperature /, 
 is approximately 
 
 622 x -0012759 / 
 i -f '00366 / 10 ' 
 
 p denoting the pressure as given in the above table. 
 
UNITS AND PHYSICAL CONSTANTS. [CHAP. 
 
 45. Pressure of Vapour of various Liquids, in dynes per 
 square centim. 
 
 
 Alcohol. 
 
 Ether. S cS D on. f Chloroform. 
 
 \ 
 
 o 
 
 20 
 
 4455 
 
 9-19 x io 4 6*31 x io 4 
 
 10 
 
 8630 
 
 i -53 x io 5 , i -058 x io 5 
 
 
 
 16940 
 
 2-46 
 
 
 i -706 
 
 
 10 
 
 32310 
 
 3-826 
 
 2-648 
 
 
 20 
 
 593io 
 
 5-772 
 
 i 3 '975 
 
 j 2*141 x io 5 
 
 30 
 
 1-048 x lo 5 ! 8-468 
 
 I 5799 
 
 ! 3-3oi 
 
 
 40 
 
 1783 
 
 
 I-2IO X 
 
 io 6 j 8-240 
 
 ! 4-927 
 
 
 50 
 
 2-932 
 
 1-687 
 
 
 1-144 x 
 
 io 6 7-14 
 
 
 60 
 
 4-671 
 
 
 2-30I 
 
 
 i '554 
 
 1-007 x 
 
 IO 6 
 
 So 
 
 I '084 x 
 
 io 6 ; 4-031 
 
 
 2-711 
 
 1-878 
 
 
 100 
 
 2-265 
 
 
 6-608 
 
 
 4-435 
 
 3-24 
 
 
 120 
 
 4'3i 
 
 
 1-029 x J 7 
 
 6-87 
 
 5-24 
 
 
 46. The phenomena of capillarity, soap-bubbles, &c., 
 can be reduced to quantitative expression by assuming a 
 tendency in the surface of every liquid to contract. The 
 following table exhibits the intensity of this contractile 
 force for various liquids at the temperature of 20 C. The 
 contractile force diminishes as the temperature increases. 
 
 Superficial Tensions at 20 C., in dynes per linear centim,, 
 deduced from Qiiincke's results. 
 
 
 Density. 
 
 Tension of surface separating 
 the liquid from 
 
 Air. 
 
 Water. 
 
 Mercury. 
 
 Water, 
 
 0-9982 
 13-5432 
 I -2687 
 1-4878 
 7906 
 9136 
 8867 
 
 7977 
 i'i 
 
 1-1248 
 
 81 
 
 540 
 32-1 
 
 30-6 
 
 25-5 
 36-9 
 297 
 
 3 1 7 
 70-1 
 
 77-5 
 
 O 
 418 
 4^75 
 29'5 
 
 20-56 
 
 H'55 
 27-8 
 
 418 
 
 
 372-5 
 
 399 
 399 
 335 
 250-5 
 284 
 377 
 
 442-5 
 
 Mercury, ... 
 
 Bisulphide of carbon, .. . 
 
 Chloroform, 
 
 Alcohol, 
 
 Olive-oil, 
 
 Turpentine, 
 
 Petroleum, 
 
 Hydrochloric acid, 
 
 Solution of hyposulphite ) 
 of soda, } 
 
 
[IV. 
 
 HYDROSTATICS. 
 
 43 
 
 46 A. Depression of the barometrical column due to 
 capillarity, according to Pouillet : 
 
 Internal 
 
 
 Internal 
 
 1 Internal 
 
 diameter 
 of tube. 
 
 Depression. 
 
 Diameter 
 of tube. 
 
 Depression. 
 
 Diameter 
 of tube. 
 
 m.m. 
 
 m.m. 
 
 m.m. 
 
 m.m. 
 
 m.m. 
 
 2 
 
 4'579 
 
 8-5 
 
 604 
 
 15 
 
 2'5 
 
 3-595 
 
 9 
 
 '534 
 
 15-5 
 
 3 
 
 2 '9O2 
 
 9-5 
 
 '473 
 
 16 
 
 3'5 
 
 2'4I5 
 
 10 
 
 419 
 
 16-5 
 
 4 
 
 2-053 
 
 10-5 
 
 372 
 
 17 
 
 4'5 
 
 1752 
 
 ii 
 
 330 
 
 I7'5 
 
 5 
 
 I-507 
 
 *5 
 
 293 
 
 18 
 
 5'5 
 
 I-306 
 
 12 
 
 260 
 
 18-5 
 
 
 I-I36 
 
 12-5 
 
 230 
 
 19 
 
 6'5 
 
 '995 
 
 13 
 
 204 
 
 I9'5 
 
 7 
 
 877 
 
 I3-5 
 
 181 
 
 20 
 
 7'5 
 
 775 
 
 14 
 
 161 
 
 20'5 
 
 8 
 
 684 
 
 I4-5 
 
 143 
 
 21 
 
 Depression. 
 
 m.m. 
 127 
 "112 
 
 099 
 087 
 077 
 068 
 060- 
 53- 
 047 
 041 
 036 
 032- 
 028 
 
44 
 
 CHAPTER V. 
 
 STRESS, STRAIN, AND RESILIENCE. 
 
 47. IN the nomenclature introduced by Rankine and 
 adopted by Thomson and Tait, any change in the shape 
 or size of a body is called a strain, and an action of force 
 tending to produce a strain is called a stress. We shall 
 always suppose strains to be small; that is, we shall 
 suppose the ratio of the initial to the final length of every 
 line in the strained body to be nearly a ratio of equality. 
 
 48. A strain changes every small spherical portion of 
 the body into an ellipsoid ; and the strain is said to be 
 homogeneous when equal spherical portions in all parts of 
 the body are changed into equal ellipsoids with their 
 corresponding axes parallel. When the strain consists in 
 change of volume, unaccompanied by change of shape, 
 the ellipsoids are spheres. 
 
 When strain is not homogeneous, but varies con- 
 tinuously from point to point, the strain at any point 
 is defined by attending to the change which takes place 
 in a very small sphere or cube having the point at its 
 centre, so small that the strain throughout it may be 
 regarded as homogeneous. In what follows we shall 
 suppose strain to be homogeneous, unless the contrary is 
 expressed. 
 
STRESS, STRAIN, AND RESILIENCE. 45 
 
 49. The axes of a strain are the three directions at 
 right angles to each other, which coincide with the 
 directions of the axes of the ellipsoids. Lines drawn in 
 the body in these three directions will remain at right 
 angles to each other when the body is restored to its 
 unstrained condition. 
 
 A cube with its edges parallel to the axes will be 
 altered by the strain into a rectangular parallelepiped. 
 Any other cube will be changed into a parallelepiped not 
 in general rectangular. 
 
 When the axes have the same directions in space after 
 as before the strain, the strain is said to be unaccompanied 
 by rotation. When such parallelism does not exist, the 
 strain is accompanied by rotation, namely, by the 
 rotation which is necessary for bringing the axes from 
 their initial to their final position. 
 
 The numbers which specify a strain are mere ratios, 
 and are therefore independent of units. 
 
 50. When a body is under the action of forces which 
 strain it, or tend to strain it ; if we consider any plane 
 section of the body, the portions of the body which it 
 separates are pushing each other, pulling each other, or 
 exerting some kind of force^-upon each other, across the 
 section, and the mutual forces so exerted are equal and 
 opposite. The specification of a stress must include a 
 specification of these forces for all sections, and a body is 
 said to be homogeneously stressed when these forces are 
 the same in direction and intensity, for all parallel 
 sections. We shall suppose stress to be homogeneous, in 
 what follows, unless the contrary is expressed. 
 
 51. When the force-action across a section consists of 
 a simple pull or push normal to the section, the direction 
 
46 UNITS AND PHYSICAL CONSTANTS. [CHAP. 
 
 of this simple pull or push (in other words, the normal to 
 the section) is called an axis of the stress. A stress (like 
 a strain) has always three axes, which are at right angles 
 to one another. The mutual forces across a section not 
 perpendicular to one of the three axes are in general 
 partly normal and partly tangential one side of the sec- 
 tion is tending to slide past the other. 
 
 The force per unit area which acts across any section 
 is called the intensity of the stress on this section, or 
 simply the stress on this section. The dimensions of 
 
 "force per unit area," or ~ are - 1 , , which we shall 
 area LT 2 
 
 therefore call the dimensions of stress. 
 
 52. The relation between the stress acting upon a 
 body and the strain produced depends upon the resilience 
 of the body, which requires in general 21 numbers for its 
 complete specification. When the body has exactly the 
 same properties in all directions, 2 numbers are sufficient. 
 These specifying numbers are usually called coefficients of 
 .elasticity ; but the word elasticity is used in so many 
 senses that we prefer to call them coefficients of resilience. 
 A coefficient of resilience expresses the quotient of a 
 stress (of a given kind) by the strain (of a given kind) 
 which it produces. A highly resilient body is a body 
 which has large coefficients of resilience. Steel is an 
 example of a body with large, and cork of a body with 
 small, coefficients of resilience. 
 
 In all cases (for solid bodies) equal and opposite strains 
 (supposed small) require for their production equal and 
 opposite stresses. 
 
 53. The coefficients of resilience most frequently 
 referred to are the three following : 
 
v.] STRESS, STRAIN, AND RESILIENCE. 47 
 
 (1) Resilience of volume, or resistance to hydrostatic 
 compression. If V be the original and V - v the strained 
 
 volume, ^- is called the compression, and when the body 
 
 is subjected to uniform normal pressure P per unit area 
 over its whole surface, the quotient of P by the compres- 
 sion is the resilience of volume. This is the only kind of 
 resilience possessed by liquids and gases. 
 
 (2) Young's modulus, or the longitudinal resilience of 
 a body which is perfectly free to expand or contract 
 laterally. In general, longitudinal extension produces 
 lateral contraction, and longitudinal compression produces 
 lateral extension. Let the unstrained length be L and 
 
 the strained length L /, then is taken as the measure 
 
 J_< 
 
 of the longitudinal extension or compression. The stress 
 on a cross section (that is, on a section to which the stress 
 is normal) is called the longitudinal stress, and Young's 
 modulus is the quotient of the longitudinal stress by the 
 longitudinal extension or compression. If a wire of cross 
 section A sq. cm. is stretched with a force of F dynes, 
 and its length is thus altered from L to L + /, the value 
 
 " F T 
 
 of. Young's modulus for the wire is - . ~. 
 
 A / 
 
 (3) " Simple rigidity " or resistance to shearing. This 
 requires a more detailed explanation. 
 
 54. A shear maybe defined as a strain by which a sphere 
 of radius unity is converted into an ellipsoid of semiaxes 
 i, i+e, ie; in other words, it consists of an extension 
 in one direction combined with an equal compression in 
 a perpendicular direction. 
 
 55. A unit square (Fig. i) whose diagonals coincide 
 
4 8 
 
 UNITS AND PHYSICAL CONSTANTS. [CHAP. 
 
 with these directions is altered by the strain into a 
 rhombus whose diagonals are (1+^)^/2 and (1^)^/2, 
 and whose area, being half the product of the diagonals, 
 is i -e 2 , or, to the first order of small quantities, is i, 
 the same as the area of the original square. The length 
 of a side of the rhombus, being the square root of the 
 
 Fig.1. 
 
 sum of the squares of the semi-diagonals, is found to be 
 J i + e 1 or _+i, and is therefore, to the first order of 
 small quantities, equal to a side of the original square. 
 
 56. To find the magnitude of the small angle which a 
 side of the rhombus makes with the corresponding side of 
 the square, we may proceed as follows : Let acb (Fig. 2) 
 be an enlarged representation of one of the small tri- 
 angles in Fig. i. Then we have ab = J, cb = \e ^2 = ~- 
 
 angle cba - - . 
 4 
 
 Hence the length of the perpendicular cd 
 
 is cb sin = - = ; and since ad is ultimately 
 
 equal to ab, we have, to the first order of small quan- 
 tities, 
 
 angle tf = -^-= 4~ = e - 
 aa * 
 
v.] 
 
 STRESS, STRAIN, AND RESILIENCE. 
 
 The semi-angles of the rhombus are therefore - e, 
 
 4 
 
 and the angles of the rhombus are - ze ; in other 
 
 words, each angle of the square has been altered by the 
 amount 2e. This quantity 2e is adopted as the measure 
 of the shear. 
 
 57. To find the perpendicular distance between oppo- 
 site sides of the rhombus, we have to multiply a side by 
 the cosine of 2*, which, to the first order of small quan- 
 tities, is i. Hence the perpendicular distance between 
 opposite sides of the square is not altered by the shear, 
 and the relative movement of these sides is represented 
 
 Fig 3. 
 
 by supposing one of them to remain fixed, while the 
 other slides in the direction of its own length through a 
 distance of 2^, as shown in pig. 3 or Fig. 4. Fig. 3, in 
 fact, represents a shear combined with right-handed rota- 
 tion, and Fig. 4 a shear combined with left-handed rota- 
 tion, as appears by comparing these figures with Fig. i, 
 which represents shear without rotation. 
 
 58. The square and rhombus in these three figures may 
 be regarded as sections of a prism whose edges are per- 
 pendicular to the plane of the paper, and figures 3 and 4 
 show that (neglecting rotation) a shear consists in the 
 
 D 
 
50 UNITS AND PHYSICAL CONSTANTS. [CHAP. 
 
 relative sliding of parallel planes without change of dis- 
 tance, the amount of this sliding being proportional to the 
 distance, and being in fact equal to the product of the 
 distance by the numerical measure of the shear. A good 
 illustration of a shear is obtained by taking a book, and 
 making its leaves slide one upon another. 
 
 It may be well to remark, by way of caution, that the 
 selection of the planes is not arbitrary as far as direction 
 is concerned. The only planes which are affected in 
 the manner here described are the two sets of planes 
 which make angles of 45 with the axes of the shear 
 (these axes being identical with the diagonals in Fig. i). 
 
 .59. Having thus denned and explained the term 
 " shear," which it will be observed denotes a particular 
 species of strain, we now proceed to define a shearing 
 stress. 
 
 A shearing stress may be defined as the combination of 
 two longitudinal stresses at right angles to each other, 
 these stresses being opposite in sign and equal in magni- 
 tude ; in other words, it consists of a pull in one direction 
 combined with an equal thrust in a 
 D r perpendicular direction. 
 
 60. Let P denote the intensity 
 of each of these longitudinal 
 stresses; we shall proceed to cal- 
 culate the stress upon a plane in- 
 I B clined at 45 to the planes of these 
 . stresses. Consider -a unit cube so 
 
 taken that the pull is perpendicular 
 
 to two of its faces, AB and D C (Fig. 5), and the thrust 
 is perpendicular to two other faces, AD, B C. The 
 forces which hold the half-cube ABC in equilibrium are 
 
v.] STRESS, STRAIN, AND RESILIENCE. 5* 
 
 (1) An outward force P, uniformly distributed over the 
 face A B, and having for its resultant a single force P 
 acting outward applied at the middle point of AB. 
 
 (2) An inward force P, having for its resultant a single 
 force P acting inwards at the middle point of B C. 
 
 (3) A force applied to the face A C. 
 
 To determine this third force, observe that the other 
 two forces meet in a point, namely the middle point ot 
 A C, that their components perpendicular to A C destroy 
 one another, and that their components along A C, or 
 
 P ' 
 
 rather along C A, have each the magnitude ; hence 
 
 V 2 
 
 their resultant is a force P ^/2, tending from C towards A. 
 The force (3) must be equal and opposite to this. Hence 
 each of the two half-cubes ABC, ADC exerts upon the 
 other a force P ^2, which is tangential to their plane of 
 separation. The stress upon the diagonal plane A C is 
 therefore a purely tangential stress. To compute its 
 intensity, we must divide its amount P J2 by the area of 
 the plane, which is ^2, and we obtain the quotient P. 
 Similar reasoning applies to the other diagonal plane B D. 
 P is taken as the measure of the shearing stress. The 
 above discussion shows that it may be defined as the 
 intensity of the stress either on the planes of purely normal 
 stress, or on the planes of purely tangential stress. 
 
 6 1. A shearing stress, if applied to a body which has 
 the same properties in all directions (an isotropic body), 
 produces a simple shear with the same axes as the stress; 
 for the extension in the direction of the pull will be equal 
 to the compression in the direction of the thrust ; and in 
 the third direction, which is perpendicular to both of 
 these, there is neither extension nor contraction, since 
 
52 UNITS AND PHYSICAL CONSTANTS. [CHAP. 
 
 the transverse contraction due to the pull is equal to 
 the transverse extension due to the thrust. 
 
 A shearing stress applied to a body which has not the 
 same properties in all directions, produces in general a 
 shear with the same axes as the stress, combined with 
 some other distortion. 
 
 In both cases, the quotient of the shearing stress by 
 the shear produced is called the resistance to shearing. 
 In the case of an isotropic body, it is also called the 
 simple rigidity. 
 
 62. The following values of the resilience of liquids 
 under compression are reduced from those given in 
 Jamin, ' Cours de Physique,' 2nd edition, torn. i. pp. 168 
 and 169 : 
 
 
 Temp. 
 Cent. 
 
 1 
 
 Coefficient of 
 Resilience. 
 
 Compression for 
 one Atmosphere 
 (megadyne per 
 square centim.) 
 
 Me 
 Wa 
 
 rcury, 
 
 o 
 O'O 
 
 O'O 
 
 I -5 
 
 4'i 
 
 10-8 
 
 I3'4 
 18-0 
 25-0 
 34'5 
 43'o 
 53'o 
 o-o) 
 
 i 'C 
 f 14-0 \ 
 
 7'3 
 (I3'i i 
 i7'5 
 
 3'436 x 
 
 2'02 X 
 
 i '97 
 2-03 
 
 2'II 
 2'13 
 2'2O 
 
 2'22 
 2-24 
 2'29 
 2-30 
 
 9-2 x 
 
 7-8 
 
 7-2 
 
 I '22 X 
 
 I '12 
 
 2'33 
 
 [O 11 
 I0 1 * 
 
 O 9 
 
 O io 
 
 2-91 x 
 
 4-96 x 
 
 5-08 
 
 4.92 
 
 473 
 470 
 
 4'55 
 4'5o 
 4 '47 
 4-36 
 4'35 
 i '09 x 
 1-29 
 1-38 
 8-17 
 8-91 x i 
 
 4-30 , 
 
 [o- 6 
 to- 5 
 
 ter, 
 
 
 - 
 
 ';. 
 
 o * 
 
 o- 6 ! 
 i 
 
 
 
 
 
 
 Eth 
 
 Ale 
 Sea 
 
 
 
 
 er, 
 
 ohol, 
 
 water, . 
 
 
STRESS, STRAIN, AND RESILIENCE. 
 
 53 
 
 63. The following are reduced from the results ob- 
 tained by Amaury and Descamps, ' Comptes Rendus,' 
 torn. Ixviii. p. 1564 (1869), and are probably more 
 accurate than the foregoing, especially in the case of 
 mercury : 
 
 
 
 Coefficient of 
 Resilience. 
 
 Compression for 
 one megadyne per 
 square centim. 
 
 Distilled water,. 
 
 
 
 1C 
 
 2 '22 X IO 10 
 
 s 
 4-51 x io~ 
 
 Alcohol 
 
 o 
 
 I '21 ,, 
 
 S'24. 
 
 
 1C 
 
 I'll ,, 
 
 8'99 ,, 
 
 Ether, 
 
 o 
 
 9*30 x io 9 
 
 I "08 x io 
 
 
 
 7 '02 
 
 1*26 
 
 Bisulphide of carbon,.. 
 Mercury, 
 
 14 
 
 1C 
 
 I -60 X IO 10 
 C/42 X IO 11 
 
 6-26 x lo-jj 
 
 I '84 X IO~ 
 
 
 
 
 
 64. The following values of the coefficients of resilience 
 for solids are reduced from those given in my own papers 
 to the Royal Society (see ' Phil. Trans.,' Dec. 5th, 1867, p. 
 369), by employing the value of g at the place of obser- 
 vation, namely 981*4. 
 
 
 Young's 
 
 Modulus. 
 
 Simple 
 Rigidity. 
 
 * 
 
 Resilience of 
 Volume. 
 
 Density 
 
 Glass flint, 
 Another specimen 
 Brass, drawn, 
 Steel 
 
 6-03 xio 11 
 
 574 ,, 
 i -075 x io 12 
 
 2'IIQ 
 
 2-40 x io 11 
 
 2'35 
 3-66 
 8'IQ 
 
 4-15 xio 11 
 
 3'47 
 I '841 X IO 12 
 
 2-942 
 
 2*935 
 8'47I 
 7-840 
 
 Iron, wrought, ... 
 ,, cast, 
 Copper, . 
 
 I'963 
 1-349 
 I '2^4 
 
 7-6 9 
 5'32 
 4- '4-7 
 
 1-456 
 
 9 '64 x io 11 
 
 I '684 X IO 12 
 
 7-677 
 
 7-235 
 8-841 
 
 
 
 
 
 
 65. The resilience of volume was not directly ob- 
 served, but was calculated from the values of " Young's 
 
54 UNITS AND PHYSICAL CONSTANTS. [CHAP. 
 
 modulus "' and " simple rigidity," by a formula which 
 is strictly true for bodies which have the same properties 
 in all directions. The contraction of diameter in lateral 
 directions for a body which is stretched by purely longi- 
 tudinal stress was also calculated by a formula to which 
 the same remark applies. The ratio of this lateral con- 
 traction to the longitudinal extension is called " Poisson's 
 ratio," and the following were its values as thus calculated 
 for the six bodies experimented on : 
 
 Glass, flint, '258 Iron, wrought, ... "275 
 
 Another specimen, '229 ,, cast, '267 
 
 Brass, drawn, '469 (?) Copper, '378 
 
 Steel, "310 
 
 Kirchhoff has found for steel the value '294, and Clerk 
 Maxwell has found for iron '267. Cornu (' Comptes 
 Rendus,' Aug. 2, 1869) has found for different 
 specimens of glass the values '225, '226, "224, '257, 
 236, -243, -250, giving a mean of '237, and maintains 
 (with many other continental savants) that for all isotropic 
 solids (that is, solids having the same properties in all 
 directions) the true value is J. 
 
 66. The following are reduced from Sir W. Thomson's 
 results (Troc. Roy. Soc.,' May, 1865), the value of g being 
 981-4: 
 
 Simple Rigidity. 
 Brass, three specimens, ... 4 '03 3^48 3*44 \ 
 
 Copper, two specimens, ... 4*40 4 '40 
 
 Other specimens of copper in abnormal states gave 
 results ranging from 3*86 x io n to 4*64 x lo 11 . 
 
 67. The following are reduced from Wertheim's re- 
 sults ('Ann. de Chim.,' ser. 3. torn, xxiii), g being taken as 
 981 : 
 
v.j STRESS, STRAIN, AND RESILIENCE. 55 
 
 Different Specimens of Glass (crystal}. 
 
 Young's modulus, 3-41 to 4-34, mean 3*96 ) 
 
 Simple rigidity, 1*26 to 1*66, ,, 1*48 V X io 11 
 
 Volume resilience, ... 3*50 to 4*39, ,, 3'^9 ) 
 
 Different Specimens of Brass. 
 
 Young's modulus, 9*48 to 10*44, mean 9*86 \ 
 
 Simple rigidity, 3 -53 to 3-90, ,, 3-67 V x io' 1 
 
 Volume resilience, 10*02 to 10*85, ,, 10*43 J 
 
 68. Savart's experiments on the torsion of brass wire 
 ('Ann. de Chim.' 1829) lead to the value 3-61 x io 11 for 
 simple rigidity. 
 
 Kupffer's values of Young's modulus for nine different 
 specimens of brass, range from 7-96 x io 11 to 11-4 x 
 io 11 , the value generally increasing with the density. 
 
 For a specimen, of density 8-4465, the value was 10*58 
 
 X 10 11 ' 
 
 For a specimen, of density 8*4930, the value was 11-2 
 x ro 11 . 
 
 The values of Young's modulus found by the same 
 experimenter for steel, range from 20*2 x io 11 to 21*4 
 x io 11 . 
 
 69. The following are reduced from Rankine's c Rules 
 and Tables,' pp. 195 and ^196, the mean value being 
 adopted where different values are given : 
 
 Tenacity. Young's Modulus. 
 
 Steel bars, 7*93 x io 9 2*45 x io 12 
 
 Iron wire, $'86 ,, I *745 > 
 
 Copper wire, 4*14 ,, 1*172 ,, 
 
 Brass wire, 3*38 ,, 9'8i x io 11 
 
 Lead, sheet, 2*28 x io 8 5-0 x io 10 
 
 Tin, cast, 3*17 ,, 
 
 Zinc, 5*17 
 
56 UNITS AND PHYSICAL CONSTANTS. 
 
 Tenacity. Young's Modulus. 
 
 Ash, I-I72XI0 9 i'io xio 11 
 
 Spruce, 8*55 x io 8 i'io ,, 
 
 Oak, i*o26xio 9 i'O2 ,, 
 
 Glass, 6-48 xio 8 5*52 xio 11 
 
 Brick and cement, ... 2*0 xio 7 
 
 The tenacity of a substance may be denned as the 
 greatest longitudinal stress that it can bear without tear- 
 ing asunder. The quotient of the tenacity by Young's 
 modulus will therefore be the greatest longitudinal exten- 
 sion that the substance can bear. 
 
57 
 
 CHAPTER VI. 
 
 ASTRONOMY. 
 70. Size and Figure of the Ea 
 
 ACCORDING to the latest determination, as published by 
 Capt. Clarke in the ' Philosophical Magazine ' for August, 
 1878, the semiaxes of the ellipsoid which most nearly 
 agrees with the actual earth are, in feet, 
 
 a 20926629, b 20925105, c = 20854477, 
 which, reduced to centimetres, are 
 a = 6*37839 x io 8 , 3 = 6-37792 x io 8 , c = 6*35639 x io 8 , 
 
 giving a mean radius of 6*3709 x io 8 , and a volume of 
 1*0832 x io 27 cubic centims. 
 The ellipticities of the two principal meridians are 
 
 T and _ T _ 
 289-5 295-8 
 
 The longitude of the greatest axis is 8 15' W. The mean 
 length of a quadrant of the meridian is 1*00074 x io 9 . 
 
 The length of a minute of latitude is approximately 
 185200-940 cos. 2 lat. of middle of arc. 
 
 The mass of the earth, assuming Baily's value 5*67 for 
 the mean density, is 6*14 x io 27 grammes. 
 
58 UNITS AND PHYSICAL CONSTANTS. [CHAP. 
 
 Day and Year. 
 
 Sidereal day, 86164 mean solar seconds. 
 
 Sidereal year, 31,558,150 ,, 
 
 Tropical year, 3 I >55 6 >9 2 9 >, 
 
 Angular velocity of earth's rotation, - 71 " -- = 
 
 80164 
 
 Velocity of points on the equator ) , 
 
 due to earth's rotation,. . 1 46510 centims per second. 
 
 Velocity of earth in orbit, about... 2960600 ,, 
 
 Centrifugal force at equator due ) , 
 
 to earth's rotation,.. ( 3'39o8 dynes per gramme. 
 
 Attraction in Astronomy. 
 
 7 1 . The mass of the moon is the product of the earth's 
 mass by '01 1364, and is therefore to be taken as 6-98 x io 25 
 grammes, the doubtful element being the earth's mean 
 density, which we take as 5*67. 
 
 The mean distance of the centres of gravity of the 
 earth and moon is 60*2734 equatorial radii of the earth 
 that is, 3-8439 x io 10 centims. 
 
 The mean distance of the sun from the earth is about 
 1-487 x io 13 centims., or 92-39 million miles, correspond- 
 ing to a parallax of 8*848 * 
 
 The intensity of centrifugal force due to the earth's 
 motion in its orbit (regarded as circular) is ( ?? j *i r de- 
 noting the mean distance, and T the length of the sidereal 
 year, expressed in seconds. This is equal to the accele- 
 ration due to the sun's attraction at this distance. Putting 
 for r and T their values, 1-487 x io 13 and 3*1558 x to 7 , 
 
 (27T\ 2 
 ~T~) r = *5*94- : ] 
 
 * This is the value of the mean solar parallax determined by Pro- 
 fessor Newcomb, and is adopted in the ' Nautical Almanac ' for 
 1882. 
 
vi.] ASTRONOMY. 59 
 
 This is about ~ of the value of F at the earth's sur- 
 1660 
 
 face. 
 
 The intensity of the earth's attraction at the mean dis- 
 tance of the moon is about 
 
 081 
 
 -z or -2701. 
 (6o-27) 2 
 
 This is less than the intensity of the sun's attraction upon 
 the earth and moon, which is '5894 as just found. Hence 
 the moon's path is always concave towards the sun. 
 
 72. The mutual attractive force F between two masses 
 m and m', at distance /, is 
 
 F = C 
 
 where C is a constant. To determine its value, consider 
 the case of a gramme at the earth's surface, attracted by 
 the earth. Then we have 
 
 F = 981, m = i, m 6*14 x io 27 , / = 6*37 x io 8 ; 
 whence we find 
 
 c = 6-48 ^ _j_ 
 
 io s 1-543 x io 7 ' 
 
 To find the mass m which, at the distance of i centim. 
 from an equal mass, would- 'attract it with a force of i 
 dyne, we have i = Cm 2 ; 
 
 whence m = /I = 3928 grammes. 
 
 N, C 
 
 73. To find the acceleration a produced at the distance 
 of /centims. by the attraction of a mass of m grammes, 
 
 i F r^m 
 
 we have a = - C -, 
 
 m P 
 
 where C has the value 6*48 x io~ 8 as above. 
 
60 UNITS AND PHYSICAL CONSTANTS. [CHAP. 
 
 72 
 
 To find the dimensions of C we have C = a , where the 
 
 m 
 dimensions of a are LT~ 2 . 
 
 The dimensions of C are therefore 
 
 L'M-'LT- 2 ; that is, L'M-'T-*. 
 
 74. The equation a = C^- shows that when a = i 
 and / = i, m must equal -*, that is to say, the mass 
 
 T^S 
 
 which produces unit acceleration at the distance of i 
 centimetre is 1*543 x io 7 grammes. If this were taken 
 as the unit of mass, the centimetre and second being 
 retained as the units of length and time, the acceleration 
 produced by the attraction of any mass at any distance 
 would be simply the quotient of the mass by the square 
 of the distance. 
 
 It is thus theoretically possible to base a general 
 system of units upon two fundamental units alone ; one 
 of the three fundamental units which we have hitherto 
 employed being eliminated by means of the equation 
 
 mass = acceleration x (distance) 2 , 
 
 which gives for the dimensions of M the expression 
 L 3 T~ 2 . 
 
 Such a system would be eminently convenient in 
 astronomy, but could not be applied with accuracy to 
 ordinary terrestrial purposes, because we can only roughly 
 compare the earth's mass with the masses which we weigh 
 in our balances. 
 
 75. The mass of the earth on this system is the 
 product of the acceleration due to gravity at the earth's 
 surface, and the square of the earth's radius. This 
 product is 
 
vi.] ASTRONOMY. 61 
 
 981 x (6-37 x io 8 ) 2 = 3-98 x I0 20 , 
 and is independent of determinations of the earth's 
 density. 
 
 The new unit of force will be the force which, acting 
 upon the new unit of mass, produces unit acceleration. 
 It will therefore be equal to 1*543 x io r dynes; and its 
 dimensions will be 
 
 mass x acceleration = (acceleration) 2 x (distance) 2 
 
 = L 4 T~ 4 . 
 
 76. If we adopt a new unit of length equal to / 
 centims., and a new unit of time equal to / seconds, while 
 we define the unit mass as that which produces unit 
 acceleration at unit distance, the unit mass will be 
 
 /V~ 2 x i '543 x i o 7 grammes. 
 
 If we make / the wave-length of the line F in vacuo r 
 say, 
 
 4-86 x io~ 5 , 
 
 and / the. period of vibration of the same ray, so that - 
 
 is the velocity of light in vacuo, say, 
 3 x io 10 , 
 
 the value of / 3 /~ 2 or /(-) is 
 
 4*374 -* i 16 > 
 
 and the unit mass will be the product of this quantity 
 into i '543 x io 7 grammes. This product is 675 x io 23 
 grammes. 
 
 The mass of the earth in terms of this unit is 
 3-98 x io 20 + (4'374 x i 16 ) = 9 IOO > 
 and is independent of determinations of the earth's 
 density. 
 
62 
 
 CHAPTER VII. 
 
 VELOCITY OF SOUND. 
 
 77. THE propagation of sound through any medium is 
 due to the elasticity or resilience of the medium ; and 
 the general formula for the velocity of propagation s is 
 
 where D denotes the density of the medium, and E the 
 coefficient of resilience. 
 
 78. For air, or any gas, we are to understand by E 
 the quotient 
 
 increment of pressure 
 corresponding compression ' 
 
 that is to say, if P, P + p be the initial and final pres- 
 sures, and V, V - v the initial and final volumes, / and 
 ^ being small in comparison with P and V, we have 
 
 v v 
 V 
 
 If the compression took place at constant temperature, 
 -we should have 
 
 
VELOCITY OF SOUND. 63 
 
 But in the propagation of sound, the compression is 
 effected so rapidly that there is not time for any sensible 
 part of the heat of compression to escape, and we have 
 
 where y= i 41 for dry air, oxygen, nitrogen, or hydrogen. 
 
 p 
 The value of for dry air at / Cent, (see p. 38) is 
 
 (i + -00366*) x 7-838+ io s . 
 Hence the velocity of sound through dry air is 
 
 s= io 4 x/i'41 x (i + -00366/) x 7-838 
 
 = 33240 n/i + 00366 /; 
 
 or approximately, for atmospheric temperatures, 
 * 33240 + 60*. 
 
 79. In the case of any liquid, E denotes the resilience 
 of volume.* 
 
 For water at 8-i C. (the temperature of the Lake of 
 Geneva in Colladori's experiment) we have 
 
 E - 2-08 x i o 10 , D - i sensibly ; 
 " \'L> = N/E^ 1 44ooo, 
 
 the velocity as determined by Colladon was 143500. 
 
 80. For the propagation of sound along a solid, in the 
 form of a thin rod, wire, or pipe, which is free to expand 
 or contract laterally, E must be taken as denoting Young's 
 modulus of elasticity.* The values of E and D will be 
 
 * Strictly speaking, E should be taken as denoting the resilience 
 for sudden applications of stress so sudden that there is not time for 
 changes of temperature produced by the stress to be sensibly 
 
6 4 
 
 UNITS AND PHYSICAL CONSTANTS. [CHAP. 
 
 different for different specimens of the same material 
 Employing the values given in the Table ( 64), we have 
 
 
 Values of E. 
 
 Values 
 ofD. 
 
 ,E 
 Values of v D ' 
 
 or velocity. 
 
 Glass, first specimen, 
 ,, second specimen,.. 
 Brass, . 
 
 6*03 x io 11 
 574 
 
 I'O7C X 1 1 12 
 
 2-942 
 
 2*935 
 8 "47 1 
 
 4-53 xio 5 
 
 4-42 
 
 Steel, 
 
 2'I^Q 
 
 
 T22 
 
 Iron, wrought, 
 
 1*96^ 
 
 7'677 
 
 , 
 5'06 
 
 , , cast, 
 
 I "34.Q 
 
 
 
 Copper, . 
 
 1-234 ,, 
 
 8*84^ 
 
 374 
 
 
 
 
 
 81. If the density of a specimen of red pine be *5, and 
 its modulus of longitudinal elasticity be 1*6 x io 6 pounds 
 per square inch at a place where g is 981, compute the 
 velocity of sound in the longitudinal direction. 
 
 By the table at the commencement of the present 
 volume, a pound per square inch (g being 981) is 6-9 
 x io 4 dynes per square centim. Hence we have for the 
 required velocity 
 
 . > '5 
 centims. per second. 
 
 82. The following numbers, multiplied by io 5 , are the 
 velocities of sound through the principal metals, as 
 determined by Wertheim : 
 
 diminished by conduction. This remark applies to both 79 and 
 80. For the amount of these changes of temperature, see a later 
 section under Heat. 
 
VII.] 
 
 VELOCITY OF SOUND. 
 
 
 At 20 C. 
 
 At 100 C. 
 
 At 200 C. 
 
 Lead, 
 
 1*23 
 
 I'2O 
 
 
 Gold 
 
 I "16. 
 
 I '72 
 
 I "71 
 
 Silver, 
 
 2'6l 
 
 2'64. 
 
 2-4.8 
 
 Copper,. ., 
 
 3*6 
 
 3 '20 
 
 2'Q^ 
 
 Platinum, . 
 
 J O w 
 
 2-60 
 
 2'C7 
 
 y -> 
 
 2*4.6 
 
 Iron, 
 
 C'l? 
 
 C'3O 
 
 4.'72 
 
 Iron wire (ordinary) , . . . 
 Cast steel,. 
 Steel wire (English), .. 
 
 4-92 
 4'99 
 
 is 
 
 5'10 
 4-92 
 5^4 
 
 TOI 
 
 479 
 5-00 
 
 The following velocities in wood are from the ob- 
 servations of Wertheim and Chevandier, ' Comptes 
 Rendus,' 1846, pp. 667 and 668 : 
 
 
 Along 
 Fibres. 
 
 Radial 
 Direction. 
 
 Tangential 
 Direction. 
 
 Pine, 
 
 3*32 x IO 5 
 
 2*83 x IO 5 
 
 I '^9 x IO 5 
 
 Beech 
 
 V34. 
 
 v67 
 
 2'87 
 
 Witch-elm, . 
 
 VQ2 
 
 ^'4.1 
 
 * O 5 
 
 2"?Q 
 
 Birch 
 
 4. '4.2 
 
 2 '14. 
 
 7 Q'7 
 
 Fir, 
 
 4*64. 
 
 2'67 
 
 o u o >' 
 I '^7 
 
 Acacia, 
 
 4'7I 
 
 
 1 j/ >' 
 
 Aspen, 
 
 To8 
 
 
 
 
 
 
 
 83. Musical Strings. 
 
 Let M denote the mass of,a string per unit length, 
 F stretching force, 
 
 L length of the vibrating portion ; 
 
 then the velocity with which pulses travel along the 
 string is /F 
 
 VM' 
 
 and the number of vibrations made per second is 
 
 v 
 
 n = 
 
 2L' 
 
66 UNITS AND PHYSICAL CONSTANTS. 
 
 Example. 
 
 For the 4 strings of a violin the values of M in grammes 
 per centimetre of length are 
 
 00416, '00669. '0106, '0266. 
 The values of n are 
 
 660, 440, 293!, 1955*-: 
 
 and the common value of L is 33 centims. Hence the 
 values of v or 2Ln are 
 
 43560, 29040, 19360, 12910 
 
 centims. per second ; and the values of F or Mz> 2 , in 
 dynes, are 
 
 7-89 x io 6 , 5-64 x io 6 , 3-97 x 10, 4-43 x io 6 . 
 
 84. Faintest Audible Sound. 
 
 Lord Rayleigh (' Proc. R.S..' 1877, vol. xxvi. p. 248), 
 from observing the greatest distance at which a whistle 
 giving about 2730 vibrations per second, and blown by 
 water-power, was audible without effort in the middle of 
 a fine still winter's day, calculates that the maximum 
 velocity of the vibrating particles of air at this distance 
 from the source was '0014 centims. per second, and that 
 the amplitude was 8'i x io~ 8 centims., the calculation 
 being made on the supposition that the sound spreads 
 uniformly in hemispherical waves, and no deduction being 
 made for dissipation, nor for waste energy in blowing. 
 
CHAPTER VIII. 
 
 LIGHT. 
 
 85. ALL kinds of light have the same velocity in vacuo. 
 According to the most recent experiments by Cornu (see 
 'Nature,' February 4, 1875) this velocity is 3*004 x io ](> 
 centims. per second. Foucault's determination was 
 2-98 x io 10 . 
 
 The velocity of light of given refrangibility in any 
 medium is the quotient of the velocity in vacuo by the 
 absolute index of refraction for light of the given refrangi- 
 bility in that medium. If then ^ denote this index, the 
 velocity will be 3 . 004 x IQ io 
 
 J* 
 
 Light of given refrangibility is light of given wave- 
 frequency. Its wave-length in any medium is the quotient 
 of the velocity in that mediurn by the wave-frequency. If 
 ;/ denote the wave-frequency (that is to say, the number 
 of waves which traverse a given point in one second), the 
 wave-length will be 3-004 x i 10 
 
 */i 
 
 86. The following are the wave-lengths adopted by 
 Angstrom for the principal Fraunhofer lines in air at 760 
 millims. pressure (at Upsal) and i6C. : 
 
68 UNITS AND PHYSICAL CONSTANTS. [CHAP. 
 
 Centims. 
 
 A 7-604 x io- 5 
 
 B 6-867 
 
 C 6-56201 ,, 
 
 Mean of lines D 5-89212 ,, 
 
 E 5-26913 ,, 
 
 F 4-86072 ,, 
 
 H 1 ......... 3-96801 
 
 These numbers will be approximately converted into 
 the corresponding wave-lengths in vacuo by multiplying 
 them by 1-00029. 
 
 87. The formula established by the experiments of 
 Biot and Arago for the index of refraction of air was 
 
 p - i = >0002 943 . _^_ 
 i + at 760 
 
 / denoting the temperature Centigrade, a the coefficient 
 of expansion "00366, and h the pressure in millims. of 
 mercury at zero. As the pressure of 760 millims. of such 
 mercury at Paris is 1-0136 x io 6 dynes per square centim., 
 the general formula applicable to all localities alike will 
 be _ i = -0002943 P 
 
 I + '00366 / 1*0136 x 10' 
 
 where P denotes the pressure in C.G.S. units. This can 
 be reduced to the form 
 
 -0002903 P_. () 
 
 i + -00366 / io 6 
 
 88. Adopting 3^Lllf^ that is 3-0033 x io 10 , as the 
 
 1*00029 
 
 velocity of light in air, and neglecting the difference of 
 velocity between the more and less refrangible rays, we 
 
VIIL] LIGHT. 69 
 
 obtain the following quotients of velocity in air by wave- 
 length : 
 
 Vibrations per second. 
 A 3-950 X IO U 
 
 B 4'373 
 
 c 4-577 
 
 D 5'97 ,. 
 
 E 5700 ,, 
 
 F 6-179 
 
 G 6-973 ,< 
 
 H 1 7-569 
 
 H 2 7-636 
 
 INDICES OF REFRACTION. 
 
 89. Dr. Hopkinson ('Proc. R. S.,' June 14, 1877,) has 
 determined the indices of refraction of the principal 
 varieties of optical glass made by Messrs. Chance, for the 
 fixed lines A, B, C, D, E, b, F, (G), G, A, H r By D is 
 to be understood the more refrangible of the pair of 
 sodium lines ; by b the most refrangible of the group of 
 magnesium lines ; by (G) the hydrogen line near G. 
 
 In connection with the results of observation, he 
 employs the empirical formula 
 
 //. - i = a {i + bx (i + ex) }, 
 
 where x is a numerical name for the definite ray of which 
 //. is the refractive index. In assigning the value of x, 
 four glasses hard crown, soft crown 1 , light flint, and dense 
 flint were selected on account of the good accord of 
 their results ; and the mean of their indices for any given 
 ray being denoted by /*, the value assigned to x for this 
 ray is /Z - /ip where ji? denotes the value of p for the 
 line F. 
 
70 UNITS AND PHYSICAL CONSTANTS. [CHAP. 
 
 The value of fi as a function of A, the wave-length in 
 io~ 4 centimetres, was found to be approximately 
 
 /* = 1-538414 4- 0-0067669 p - 0-0001734 - 4 
 
 + 0-000023 j - 
 
 The following were the results obtained for the different 
 specimens of glass examined : 
 
 Hard Crown, 1st specimen, density 2*48575. 
 a = 0-523145, 3=1-3077, ^-=-2-33. 
 
 Means of observed values of fi. 
 
 A 1-511755; B 1-513624; C 1-514571; D 1-517116; 
 
 E 1-520324; b 1-520962; F 1-523145; (G) 1-527996; 
 
 G i -528348 ; h i -530904 5 Hj i -532789- 
 
 Soft Crown, density 2 '55035. 
 
 a = 0-5209904, b = i -4034, c = - I -58. 
 
 Means of observed values of /u. 
 
 A 1-508956; B 1-510918; C 1-511910; D 1-514580; 
 E 1-518017; b 1-518678; F 1-520994; (01*526208; 
 G 1-526592; h 1-529360; H! 1-531415- 
 
 Extra Light Flint Glass, density 2-86636. 
 a = 0-549123, b= 17064, c= -0-198. 
 
 Means of observed values of /i. 
 
 A 1-534067; B 1-536450; C r537682; 
 
 D 1-541022; 1-545295; ^1-546169; 
 
 F i -549125 ; (G) i -555870 ; G i -556375 ; 
 
 h I-559992; H! 1-562760. 
 
 Light Flint Glass, density 3 '20609. 
 a = 0-583887, b = I -9605, c - 0-53. 
 
 Means of observed values of /u. 
 
 B 1-568558; C 1-570007; D 1-574013; 
 
 E 1-579227; b 1-580273; F 1-583881; 
 
 (G) 1-592184; G 1-592825; h r597332; 
 
 H! 1-600717. 
 
vin. J LIGHT. 71 
 
 Dense Flint, density 3*65865. 
 a 0*634744, b 2*2694, c 1*48. 
 
 Means of observed values of /*. 
 
 B 1*615704; C 1-617477; D 1*622411; 
 
 E 1*628882; b 1-630208; F 1*634748; 
 
 (0)1*645268; 01*646071; / 1-651830; 
 
 H! 1*656229. 
 
 Extra Dense Flint, density 3*88947. 
 
 a = 0*664226, b 2*4446, c 1*87. 
 
 Means of observed values of /x. 
 
 A 1-639143; B 1-642894; C 1-644871; 
 
 D 1*650374; E 1-657631; b 1-659108; 
 
 F 1-664246; (O) 1*676090; G 1-677020; 
 
 h 1-683575; H! 1-688590. 
 
 Double Extra Dense Flint, density 4-42162. 
 a 0*727237, b 2*7690, c 2*70. 
 
 Means of observed values of /t. 
 
 A 1*696531; B 1-701080; C 1*703485; 
 
 D 1-710224; E 1-719081; b 1-720908; 
 
 F 1-727257; (0)1742058; 01*743210; 
 
 h 1751485. 
 
 INDICES OF REFRACTION FOR LIQUIDS. 
 
 90. The following values of indices of refraction for 
 liquids are condensed from Fraunhofer's determinations, 
 as given by Sir John Herschel ('Enc. Met. Art./ Light, 
 
 P- 415): 
 
 Water > density I -ooo. 
 
 B 1*3309; C 1-3317; D 1-3336; 
 
 E I-3358; F 1-3378; 1-3413; 
 
 H 1-3442. 
 
 Oil of Turpentine, density 0*885. 
 
 B 1-4705; C I-47I5; D I-4744; E 1-4784; 
 F I -481 7; 01-4882; H 1-4939. 
 
72 UNITS AND PHYSICAL CONSTANTS. [CHAP. 
 
 91. The following determinations of the refractive 
 indices of liquids are from Gladstone and Dale's results, 
 as given in Watts' ' Dictionary of Chemistry/ iii., pp. 
 629-631 : 
 
 Sulphide of Carbon, at temperature 11. 
 A 1-6142; B 1-6207; C 1-6240; D 1*6333; 
 E 1-6465; F 1-6584; G 1-6836; H 17090. 
 
 Benzene, at temperature 10-5. 
 
 A 1-4879; B i '4913; C 1-4931; D i '4975; 
 E 1-5036; F 1-5089; G 1-5202; H 1-5305. 
 
 Chloroform, at temperature 10. 
 
 A 1-4438; B 1-4457; C 1-4466; D 1-4490; 
 E 1-4526; F 1-4555; G 1-4614; H 1-4661. 
 
 Alcohol, at temperature 15. 
 
 A 1-3600; B 1-3612; C 1-3621; D 1-3638; 
 E 1-3661; F 1-3683; G 1-3720; H 1-3751. 
 
 Ether, at temperature 15. 
 
 A 1-3529; B i -3545; C 1-3554; 01-3566; 
 E 1*3590; F 1-3606; G 1-3646; H 1-3683. 
 
 Water, at temperature 15. 
 
 A 1-3284; B 1-3300; C 1-3307; D 1-3324; 
 E i'3347; F i '3366; G 1-3402; H 1-3431- 
 
 DOUBLE REFRACTION. 
 
 92. The following indices of doubly refracting crystals 
 are from the table at the end of Lloyd's ' Light and 
 Vision ' : 
 
 Diamond, 2-439 to 2755 
 
 Chromate of lead (least refraction), - 2-500 
 
 (greatest ), - 2-950 
 
 Zircon (least refraction), - - - 1*961 
 
 (greatest ), - - - 2-015 
 
VIII.] 
 
 LIGHT. 
 
 73 
 
 Carbonate of lead (least refraction), 
 ,, (greatest ,, ), 
 
 Brazilian topaz (ordinary index), 
 
 ,, (extraordinary index), 
 
 Quartz (ordinary index), 
 
 ,, (extraordinary index), 
 Arragonite (extraordinary index), 
 (ordinary ,, ), 
 
 Sulphate of copper (least refraction), 
 ,, (greatest ,, ), 
 
 Iceland spar (least refraction), 
 
 ,, (greatest ), - 
 
 Nitre (least refraction), 
 ,, (greatest ,, ), 
 
 1-813 
 2-084 
 632 
 640 
 548 
 558 
 '535 
 693 
 
 531 
 552 
 
 w 
 665 
 
 335 
 514 
 
 93. The two following tables are from Watts' 'Dic- 
 tionary of Chemistry/ vol. iii., p. 615. The indices given 
 are for the yellow rays, except Wollaston's, which are for 
 the extreme red : 
 
 Indices of Refraction of Solids. 
 
 Index. Observer. 
 
 Chromate of lead, - 2*50 to 2 '97 Brewster. 
 
 Diamond, - - - 2*47 to 275 Brewster ; Rochon. 
 
 Phosphorus, - - 2^224 Brewster. 
 
 Glass of antimony, 2 '216 ,, 
 
 Sulphur (native), 2 '115 ,, 
 
 Zircon, - - - 1*95 Wollaston. 
 
 Nitrate of lead, - - I '866 Herschel. 
 
 Carbonate of lead, - i'8i to 2*08 Brewster. 
 
 Ruby, 1779 
 
 Felspar, - - - ^764 ,, 
 
 Tourmalin,- - - i'668 ,, 
 
 Topaz, colourless, - i'6io Biot. 
 
 Beryl, I -598 Brewster. 
 
 Tortoise-shell, - - 1-591 ,, 
 
 Emerald, - - - 1*585 ,, 
 
 Flint glass, - - - I -57 to I -58 Brewster ; Wollaston. 
 
74 
 
 UNITS AND PHYSICAL CONSTANTS. [CHAP 
 
 
 Index. 
 
 Observer. 
 
 Rock-crystal, 
 
 1*547 
 
 Wollaston. 
 
 Rock-salt, 
 
 1-545 
 
 Newton. 
 
 Apophyllite, 
 
 1-543 
 
 Brewster. 
 
 Colophony, 
 
 1-543 
 
 Wollaston. 
 
 Sugar, 
 
 1-535 
 
 ,, 
 
 Phosphoric acid, 
 
 1-534 
 
 Brewster. 
 
 Sulphate of copper, 
 
 1-531 to 1-552 
 
 >? 
 
 Canada balsam, - . * 
 
 I-532 
 
 Young. 
 
 Citric acid, 
 
 1-527 
 
 Brewster. 
 
 Crown glass, 
 
 i -525 to 1-534 
 
 ,, 
 
 Nitre, 
 
 i"5H 
 
 ,, 
 
 Plate glass, - - 
 
 1-514 to 1-542 
 
 
 
 Spermaceti, 
 
 l'503 
 
 Young. 
 
 Crown glass, 
 
 1*500 
 
 Wollaston. 
 
 Sulphate of potassium, 
 
 1*500 
 
 Brewster. 
 
 Ferrous sulphate, 
 
 1-494 
 
 ,, 
 
 Tallow ; wax, 
 
 1-492 
 
 Young. 
 
 Sulphate of magnesium, 
 
 1-488 
 
 Brewster. 
 
 Iceland spar, 
 
 1-654 
 
 Malus. 
 
 Obsidian, - - - 
 
 1-488 
 
 Brewster. 
 
 Gum, 
 
 1-476 
 
 Newton. 
 
 Borax, ^ - - 
 
 1-475 
 
 Brewster. 
 
 Alum, - - - 
 
 1*457 
 
 Wollaston. 
 
 Fluorspar, - - - 
 
 1-436 
 
 Brewster. 
 
 Ice, - - - . 
 
 1-310 
 
 Wollaston. 
 
 Tabasheer, - - 
 
 1-1115 
 
 Brewster. 
 
 Indices of Refraction of Liquids. 
 
 Sulphide of carbon, 
 
 Oil of cassia, 
 
 Bitter almond oil, 
 
 Nut oil, 
 
 Linseed oil, 
 
 Oil of naptha, - 
 
 Rape oil, - 
 
 Olive oil, - 
 
 Oil of turpentine, 
 
 I '678 Brewster. 
 
 1-031 Young. 
 
 "603 Brewster. 
 
 5oo 
 
 485 Wollaston. 
 
 475 Young. 
 
 -475 Brewster; Youn< 
 
 470 Brewster. 
 
 -470 Wollaston. 
 
VIII.] 
 
 LIGHT. 
 
 75 
 
 Oil of almonds, - 
 
 Oil of lavender, - 
 
 Sulphuric acid (sp. gr. 17), 
 
 Nitric acid (sp. gr. I "48), - 
 
 Solution of potash (sp. gr. I '410), 
 
 Hydrochloric acid (concentrated), 
 
 Sea-salt (saturated), - 
 
 Alcohol (i-ectified), 
 
 Ether, 
 
 Alum (saturated), 
 
 Human blood, - 
 
 White of egg, - - - 
 
 Vinegar (distilled), - - 
 
 Saliva, - 
 
 Water, .... 
 
 Index. 
 
 Observer. 
 
 1-469 
 
 Wollaston. 
 
 1-457 
 
 Brewster. 
 
 1-429 
 
 Newton. 
 
 1-410 
 
 Young ; Wollaston. 
 
 1-405 
 
 Fraunhofer. 
 
 1-410 
 
 Biot. 
 
 1-372 
 
 Herschel. 
 
 1-358 
 
 Wollaston. 
 
 1-356 
 
 Herschel. 
 
 1-354 
 
 Young. 
 
 1-351 
 
 Enler, jun. 
 
 1-372 
 
 Herschel. 
 
 1-339 
 
 Young. 
 
 1-336 
 
 Wollaston; Brewster. 
 
 INDICES FOR GASES. 
 
 94. The following indices of refraction of gases are 
 from the determinations of Dulong. They are for the 
 temperature oC, and the pressure of 76 c.m. of mercury 
 .at Paris : 
 
 Vacuum, - - 
 Hydrogen, 
 Oxygen, 
 
 Atmospheric air, 
 Nitrogen, 
 Nitric oxide, 
 Carbonic oxide, 
 Ammonia, 
 Carbonic acid gas, - 
 Nitrous oxide, 
 Sulphurous acid gas, 
 Chlorine, 
 
 oooooo 
 
 000138 
 
 000272 
 
 000294 
 
 000300 
 
 000303 
 
 000340 
 
 i -000385 
 
 i -000449 
 
 i -000503 
 
 i -000665 
 
 i -000772 
 
76 
 
 UNITS AND PHYSICAL CONSTANTS. [CHAP. 
 
 DISPERSION IN GASES. 
 
 95. Croullebois ('Ann. de. Chim.', 1870, vol. xx., p. 
 185) has made the following determinations of the 
 indices of refraction of gases for the rays corresponding 
 to the fixed lines C, E, and G : 
 
 Indices. 
 
 Dispersion. 
 
 
 f 
 C 
 
 E G 
 
 G-C 
 
 Air, - --..'- 4 
 
 1-0002578 
 
 I -000305 1 I "0003 147 
 
 0000569 
 
 Nitrogen, 
 
 I -000258 
 
 1-000302 1-000321 
 
 000063 
 
 Oxygen, - - - 
 
 I '000255 
 
 i -000294 i -0003 1 5 
 
 000060 
 
 Hydrogen, - 
 
 i '000129 
 
 1*000140 
 
 000153 
 
 000024 
 
 Carbonic acid, 
 
 i -000395 
 
 i -000456 
 
 000496 
 
 oooioi 
 
 Chlorine, 
 
 I '000699 
 
 i -000792 
 
 000840 
 
 000141 
 
 Cyanogen, 
 
 I '000804 
 
 i -000834 
 
 000895 
 
 -000091 
 
 Sulphuretted hydrogen, 
 
 1-000599 
 
 i -000647 
 
 000691 
 
 000092 
 
 Ammonia, - 
 
 1-000374 
 
 i -000399 
 
 000444 
 
 000070 
 
 Carbonic oxide, 
 
 I -000301 
 
 i -000350 
 
 000391 
 
 000090 
 
 Olefiant gas, 
 
 1-000652 
 
 i -000694 
 
 000722 
 
 000070 
 
 Marsh gas, - 
 
 1-000412 
 
 1-000471 
 
 000502 
 
 000090 
 
 The " dispersive powers," as computed by the formula 
 G C 
 
 
 klj^ *W**WTT*U M*V*^ 
 
 
 of refraction for white 
 
 light (as determined 
 
 by M. Croulle- 
 
 bois) are appended. 
 
 
 
 
 Dispersive 
 
 Index for 
 
 
 Power. 
 
 White Light. 
 
 Air, * 
 
 1864 
 
 I -0002943 
 
 Nitrogen, - 
 
 2086 
 
 1-0003019 
 
 Oxygen, 
 
 2040 
 
 I -OOO27O 
 
 Hydrogen, - - 
 
 1714 
 
 I-OOOI37 
 
 Carbonic Acid, - 
 
 2214 
 
 I '000440 
 
 Chlorine, - - 
 
 - - -1780 
 
 I -000774 
 
 Cyanogen, - 
 
 - -1091 
 
 I '000829 
 

 viii.] 
 
 LIGHT. 
 
 77 
 
 Sulphuretted hydrogen, 
 Ammonia, - 
 Carbonic oxide, - - 
 Olefiant gas, - - 
 Marsh gas, - - - 
 
 Dispersive 
 Power. 
 
 '1421 
 '1754 
 '2571 
 '1008 
 '1910 
 
 Index for 
 White Light. 
 
 '000639 
 "000390 
 '000344 
 '000669 
 '000449 
 
 96. The following very different determinations of the 
 indices of refraction of air for the principal fixed lines 
 were obtained by Ketteler (' Pogg. Ann./ vol. cxxiv., p. 
 390 ; ' Phil. Mag.,' 1866, vol. xxxii., p. 336) : 
 
 A 1-00029286 
 
 B 1*00029350 
 
 C 1-00029383 
 
 D 1-00029470 
 
 E 1-00029584 
 
 F 1-00029685 
 
 G 1-00029873 
 
 H 1-00030026 
 
CHAPTER IX. 
 
 HEAT. 
 
 97. THE unit of heat is usually defined as the quantity 
 of heat required to raise, by one degree, the temperature 
 of unit mass of water, initially at a certain standard tem- 
 perature. The standard temperature usually employed is 
 o C; but this is liable to the objection that ice may be 
 present in water at this temperature. Hence 4 C. has 
 been proposed as the standard temperature ; and another 
 proposition is to employ as the unit of heat one hundredth 
 part of the heat required to raise the unit mass of water 
 from o to 1 00 C. 
 
 98. According to Regnault (' Mem. Acad. Sciences/ 
 xxi. p. 729) the quantity of heat required to raise a given 
 mass of water from o to t C. is proportional to 
 
 / + '00002 t~ + '0000003 / 3 (T) 
 
 The mean thermal capacity of a body between two stated 
 temperatures is the quantity of heat required to raise it 
 from the lower of these temperatures to the higher, 
 divided by the difference of the temperatures. The mean 
 thermal capacity of a given mass of water between o C. 
 and f is therefore proportional to 
 
 I + 'OOOO2 / + '0000003 / 2 . . . .,.'. (2) 
 
HEAT. 79 
 
 The thermal capacity of a body at a stated temperature 
 is the limiting value of the mean thermal capacity as the 
 range is indefinitely diminished. Hence the thermal 
 capacity of a given mass of water at t is proportional to 
 the differential coefficient of (i), that is to 
 
 i + '00004 / + -0000009 t- (3) 
 
 Hence the thermal capacities at o and 4 are as i to 
 1*000174 nearly; and the thermal capacity at o is to the 
 mean thermal capacity between o and 100 as i to 
 1-005. 
 
 99. If we agree to adopt the capacity of unit mass of 
 water at a stated temperature as the unit of capacity, the 
 unit of heat must be defined as n times the quantity of 
 heat required to raise unit mass of water from this initial 
 
 temperature through - - of a degree when n is inde- 
 finitely great. 
 
 Supposing the standard temperature and the length of 
 the degree of temperature to be fixed, the units both of 
 heat and of thermal capacity vary directly as the unit of 
 mass. 
 
 In what follows, we adopt as the unit of heat (except 
 where the contrary is stated) the heat required to raise a 
 gramme of pure water through i C. at a temperature 
 intermediate between o and 4. This specification is 
 sufficiently precise for the statement of any thermal 
 measurements hitherto made. 
 
 100. The thermal capacity of unit mass of a substance 
 at any temperature is called the specific heat of the sub- 
 stance at that temperature. 
 
 The following determinations of specific heat by Dulong 
 
8o UNITS AND PHYSICAL CONSTANTS. [CHAP. 
 
 and Petit agree very well with later determinations by 
 Regnault and other experimenters, except in the case of 
 platinum : 
 
 Mean Specific Mean Specific 
 
 Heat between Heat between 
 
 o and 100. o and 300. 
 
 Iron, .... -1098 . . -1218 
 
 Copper, - t49 - - -1013 
 
 Zinc, - '0927 - - -1015 
 
 Silver, - -0557 - - -06 11 
 
 Antimony, - - '0507 - - '0549 
 
 Platinum, - '0355 - -0355 
 
 Glass, .... -J770 - ... -1990 
 
 According to Pouillet's experiments, the mean specific 
 heat of platinum between 
 
 o and 1 00 is '0335 
 ,, 300 ,, -0343 
 500 ,, -0352 
 ,, 700 -0360 
 1000 ,, -0373 
 
 ,, 1200 ,, -0382 
 
 101. Specific heat is of zero dimensions in length, 
 mass, and time. It is in fact the ratio 
 
 increment of heat in the substance 
 increment of heat in water 
 
 for a given increment of temperature, the comparison 
 being between equal masses of the substance at the actual 
 temperature and of water at the standard temperature. 
 The numerical value of a given concrete specific heat 
 merely depends upon the standard temperature at which 
 the specific heat of water is called unity. 
 
 102. The thermal capacity of unit volume of a substance 
 
ix.] HEAT. 8 1 
 
 is another important element : we shall denote it by c. 
 Let s denote the specific heat, and d the density of the 
 substance ; then c is the thermal capacity of d units 
 of mass, and therefore c = sd. The dimensions of c in 
 length, mass, and time are the same as those of d, namely 
 
 M 
 
 . Its numerical value will not be altered by any 
 
 L 3 
 
 change in the units of length, mass, and time which 
 leaves the value of the density of water unchanged. 
 . In the C.G.S. system, since the density of water 
 between o and 4 is very approximately unity, the 
 thermal capacity of unit volume of a substance is the 
 value of the ratio 
 
 increment of heat in the substance 
 increment of heat in water 
 
 for a given increment of temperature, when the compari- 
 son is between equal volumes. 
 
 103. The following- table (from Miller's 'Chemical 
 Physics,' p. 313, fourth edition) exhibits the specific heats 
 of most of the elementary bodies, also their atomic 
 weights, and the product of the two : 
 
 Specific Atomic p , 
 Heat. Weight. Product - 
 
 Diamond, - '1468 48 6*0464 
 
 Graphite, - /- - - * - -2018 33 6*6594 
 
 Wood Charcoal, - - - "2415 
 
 Silicon, fused, - - - '1750 35 6*125 
 
 ,, crystallized, - - '1767 
 
 Boron, crystallized, - - - '250 
 
 Sulphur, native, ... -17760 32 5 '6832 
 
 Selenium, .... -08370 79-5 6*6541 
 
 Tellurium, .... '04737 129 6*1107 
 
 Magnesium, - - - - '2499 24 5 -9976 
 
UNITS AND PHYSICAL CONSTANTS. [CHAP. 
 
 Zinc, 
 
 Cadmium, 
 
 Aluminium, 
 
 Iron, 
 
 Nickel, - 
 
 Cobalt, - 
 
 Manganese, 
 
 Tin, 
 
 Tungsten, . -. 
 
 Molybdenum, - 
 
 Copper, - % - 
 
 Lead, 
 
 Mercury, solid, 
 
 ,, liquid, 
 Platinum, 
 Palladium, 
 Rhodium, 
 Osmium, 
 Iridium, - 
 Iodine, - 
 Bromine, solid, 
 
 ,, liquid, 
 Potassium, 
 Sodium, - .-. 
 Lithium, 
 Phosphorus, 
 Arsenic, - 
 Antimony, 
 Bismuth, 
 Thallium, 
 Silver, 
 Gold, 
 
 104. Variation of Specific Heat with Temperature. 
 Bede's results (' Mem. couronnes de 1'Acad. de Brtix- 
 elles,' xxvii. i) have been summed up in the following 
 
 Specific 
 Heat. 
 
 Atomic 
 Weight. 
 
 Product. 
 
 -09555 
 
 65 
 
 6-2108 
 
 05669 
 
 112 
 
 6-3482 
 
 2143 
 
 27-5 
 
 5^730 
 
 II379 
 
 56 
 
 6-3722 
 
 10863 
 
 59 
 
 6 '4090 
 
 10696 
 
 59 
 
 63106 
 
 1217 
 
 55 
 
 6-6934 
 
 05623 
 
 118 
 
 6-6356 
 
 03342 
 
 184 
 
 6-1492 
 
 07218 
 
 96 
 
 6-931 
 
 09515 
 
 63-5 
 
 6-0419 
 
 03140 
 
 207 
 
 6-4999 
 
 03192 
 
 200 
 
 6-3840 
 
 03332 
 
 200 
 
 6 -6640 
 
 03243 
 
 197-2 
 
 6-3952 
 
 05927 
 
 1 06 -4 
 
 6-3072 
 
 05803 
 
 104-4 
 
 6-0582 
 
 '03063 
 
 198-8 
 
 6-0892 
 
 03259 
 
 197-2 
 
 6-4266 
 
 05412 
 
 127 
 
 6-8732 
 
 08430 
 
 So 
 
 67440 
 
 10600 
 
 80 
 
 8-4800 
 
 16956 
 
 39 
 
 6-6128 
 
 -29340 
 
 23 
 
 6-7480 
 
 9408 
 
 7 
 
 6-5856 
 
 18870 
 
 31 
 
 5 "8497 
 
 08140 
 
 75 
 
 6*1050 
 
 05077 
 
 122 
 
 6-1939 
 
 03084 
 
 2IO 
 
 6-4764 
 
 03255 
 
 2O4 
 
 6 '6402 
 
 05701 
 
 108 
 
 6-1570 
 
 03244 
 
 I96-6 
 
 6-3777 
 
IX.] 
 
 HEAT. 
 
 formulae by Prof. G. C. Foster, who has communicated 
 them to the editor of this work : 
 
 Specific Heats at t. 
 
 Iron, - - - '1040 + "000144^ 
 
 Copper, - - - '0892 + 'oooo65/ 
 
 Tin, - - - '0512 + "000063^ 
 
 Zinc, - - - '08595 + "000084^ 
 
 Lead, - - - '0283 + 000036^ 
 
 According to Violle, the specific heat of platinum 
 at f is '0317 + "000012 /, its latent heat of fusion 
 27*2, its melting point i775"5, an( ^ tne melting point of 
 silver 954. 
 
 According to H. F. Weber, the true specific heat of 
 diamond at t is 
 
 0947 + "000994 / "00000036 / 2 . 
 
 105. The following table (from Miller's 'Chemical 
 Physics,' p. 307) exhibits the specific heats of certain 
 substances in the solid form, as determined by Regnault, 
 along with the specific heats of the same substances 
 in the liquid form, as determined by Person : 
 
 Solid. 
 
 Liquid. 
 
 
 Sp. heat. 
 
 Temperature 
 between. 
 
 Sp. heat. 
 
 Temperature 
 between. 
 
 Ice, 
 
 5050 
 
 - 30 and 
 
 o 
 
 I 'OOOO 
 
 o and 20 
 
 Sodic nitrate, - 
 
 2782 
 
 j, 
 
 100 
 
 "4130 
 
 3 20 430 
 
 Potassic nitrate, 
 
 2387 
 
 o ,, 
 
 IOO 
 
 3318 
 
 35 >, 435 
 
 Sulphur, - 
 
 "2O26 
 
 o ,, 
 
 IOO 
 
 "2340 
 
 120 ,, 150 
 
 Phosphorus, 
 
 1788 
 
 -14 
 
 7 
 
 2045 
 
 50 ,, loo 
 
 Bromine, - 
 
 0843 
 
 -78 
 
 -20 
 
 "1 060 
 
 -12 ,, 4 8 
 
 Tin, 
 
 0562 
 
 o ,, 
 
 IOO 
 
 0637 
 
 250 350 
 
 Iodine, 
 
 0541 
 
 o ,, 
 
 IOO 
 
 1082 
 
 Not stated. 
 
 Lead, 
 
 0314 
 
 o ,, 
 
 IOO 
 
 0402 
 
 350 and 450 
 
 Bismuth, - 
 
 "0308 
 
 o 
 
 IOO 
 
 0363 
 
 280 380 
 
 Mercury, - 
 
 0319 
 
 ... 
 
 ... 
 
 0333 
 
 ,, IOO 
 
8 4 
 
 UNITS AND PHYSICAL CONSTANTS. [CHAP. 
 
 1 06. The following table (from Miller's ' Chemical 
 Physics/ p. 308) contains the results ofRegnault's experi- 
 ments on the specific heat of gases. The column headed 
 " equal weights " contains the specific heats in the sense 
 in which we have defined that term. The column headed 
 " equal volumes" gives the relative thermal capacities of 
 equal volumes at the same pressure and temperature : 
 
 Thermal Capacities of Gases and Vapours. 
 
 
 Enua.1 
 
 
 Equal. 
 
 Gets or Vapour, 
 
 J -*H u - cti> 
 
 Gcis or Vapour. 
 
 
 
 Vols. 
 
 Weights. 
 
 
 Vols. 
 
 Weights 
 
 Air, 
 Oxygen, 
 
 2375 
 2405 
 
 2375 
 2175 
 
 Hydrochloric ) 
 acid. \ 
 
 2352 
 
 1842 
 
 Nitrogen, . - 
 Hydrogen, 
 Chlorine, . 
 
 2 3 68 
 
 2359 
 2964 
 
 2438 
 3-4090 
 'I2IO 
 
 Sulphuretted ) 
 hydrogen, i 
 Water, - 
 
 2857 
 2989 
 
 2432 
 
 4805 
 
 Bromine, 
 
 3040 
 
 0555 
 
 Alcohol, 
 
 7171 
 
 '4534 
 
 Nitrous oxide, 
 
 '3447 
 
 **** 
 
 2262 
 
 Wood spirit, - 
 
 5063 
 
 4580 
 
 Nitric oxide, - 
 
 '2406 
 
 2317 
 
 Ether, - 
 
 I -2266 
 
 4796 
 
 Carbonic ) 
 oxide, \ 
 
 2370 
 
 2450 
 
 Ethyl chloride, 
 ,, bromide, 
 
 6096 
 7026 
 
 2738 
 1896 
 
 Carbonic } 
 anhydride, \ 
 
 3307 
 
 2163 
 
 ,, disul- ) 
 phide, j 
 
 I -2466 
 
 4008 
 
 Carbonic di- ) 
 sulphide, \ 
 
 4122 
 
 1569 
 
 Ethyl cyanide, 
 Chloroform, - 
 
 8293 
 6461 
 
 4261 
 1566 
 
 Ammonia, 
 
 '2996 
 
 5084 
 
 Dutch liquid, 7911 
 
 2293 
 
 Marsh gas, 
 
 3277 
 
 5929 
 
 Acetic ether, 
 
 I '2184 
 
 4008 
 
 Olefiant gas, - 
 
 4106 
 
 4040 
 
 Benzol, - 
 
 I 'OII4 
 
 "3754 
 
 Arsenious ) 
 chloride, \ 
 Silicic chloride, 
 
 7013 
 
 7778 
 
 '1122 
 1322 
 
 Acetone, ^ 
 Oil of tur- ) 
 pentine, \ 
 
 8341 
 
 2-3776 
 
 4125 
 5061 
 
 Titanic , , 
 Stannic ,, 
 
 8564 
 86 39 
 
 I29O 
 0939 
 
 Phosphorus ) 
 chloride, \ 
 
 6386 
 
 1347 
 
 Sulphurous | 
 
 
 
 
 
 
 anhydride, \ 
 
 '34 1 
 
 ' r S4 
 
 
 
 
 107. E. Wiedemann ('Pogg. Ann.' 1876, No. i, p. 39) 
 has made the following determinations of the specific 
 heats of gases : 
 
ix.] HEAT. 85 
 
 
 Specific 
 
 Heat 
 
 
 
 
 
 
 
 -p . 
 
 
 At o. 
 
 At 100. 
 
 At 200. 
 
 Density. 
 
 Air, - - - 
 
 0-2389 
 
 
 ... 
 
 I 
 
 Hydrogen, 
 
 3-410 
 
 
 
 0*0692 
 
 Carbonic oxide, 
 
 0-2426 
 
 ... 
 
 ... 
 
 0*967 
 
 Carbonic acid, 
 
 0-1952 
 
 0-2169 
 
 0*2387 
 
 I*529 
 
 Ethyl, - 
 
 0-3364 
 
 0*4189 
 
 0*5015 
 
 0*9677 
 
 Nitric oxide, - 
 
 0*1983 
 
 0*2212 
 
 0-2442 
 
 1-5241 
 
 Ammonia, 
 
 0*5009 
 
 0-53I7 
 
 0*5629 
 
 0*5894 
 
 Multiplying the specific heat by the relative density, 
 he obtains the following values of 
 
 Thermal Capacity of Equal Volumes. 
 
 At o. At 100. At 200. 
 
 Air, - - - - 0-2389 
 
 Hydrogen, - - - 0*2359 
 
 Carbonic oxide, ' - 0*2346 
 
 Carbonic acid, *- - 0*2985 0*3316 0-3650 
 
 .Ethyl, - - ,- - 0-3254 0-4052 0-4851 
 
 Nitri' oxide, - - - 0-3014 0-3362 0*3712 
 
 Ammonia, - - 0*2952 0*3134 0*3318 
 
 108. The same author (' Pogg. Ann.' 1877, New Series, 
 vol. ii., p. 195) has made trje following determinations of 
 specific heats of vapours at temperature f : 
 
 Vapour. 
 
 Range of Temp, 
 in Experiments. 
 
 Specific Heat. 
 
 Chloroform, 
 
 26 -9 to 1 89 *8 
 
 1341 + -0001354^ 
 
 Bromic ethyl, 
 
 27*9 to 189*5 
 
 1354 + -003560^ 
 
 Benzine, 
 
 34-1 to 115-1 
 
 2237 + "OOIO228/ 
 
 Acetone, - 
 
 26-2 to 179-3 
 
 2984 + -0007738^ 
 
 Acetic ether, 
 
 32*9 to 113-4 
 
 2738 + *ooo87oo/ 
 
 Ether, 
 
 25-4 to 1 88 -8 
 
 '37 2 5 + 'ooo8536/ 
 
36 UNITS AND PHYSICAL CONSTANTS. [CHAP. 
 
 Regnault's determinations for the same vapours were 
 as follows : 
 
 Mean Specific Heat for this Range. 
 
 Vapour. 
 
 Temperature. 
 
 According to 
 Regnault. 
 
 According 
 Wiedemani 
 
 Chloroform, 
 
 117 to 228 
 
 1567 
 
 1573 
 
 Bromic ethyl, - 
 
 777 to 196-5 
 
 1896 
 
 1841 
 
 Benzine, - 
 
 116 to 218 
 
 '3754 
 
 3946 
 
 Acetone, - 
 
 129 to 233 
 
 4125 
 
 3946 
 
 Acetic ether, 
 
 U5 to 219 
 
 4008 
 
 4190 
 
 Ether, 
 
 70 tO 220 
 
 '4797 
 
 '4943 
 
 Regnault's determinations for the specific heats of the 
 liquids corresponding to some of these vapours are as 
 follows : 
 
 Chloroform, - .- '23235 + '000101432^ 
 
 Acetone, - - - - - - '5064 + "000793^ 
 
 Acetic ether, - - * - - '52741 + '0010464^ 
 
 Regnault has also determined the mean specific heat 
 of bisulphide of carbon vapour between 80 and 147 to 
 be '1534, and between 80 and 229 to be '1613, and 
 has found for the specific heat of liquid bisulphide of 
 carbon the expression 
 
 23523 + -00016303^. 
 
 Schuller has found the specific heat of liquid benzine 
 to be 
 
 37980 + -OOI44/. 
 
 All these results are quoted by E. Wiedemann in the 
 paper above referred to. 
 
 109. The following approximate table of melting points 
 is based on that given in the second supplement to 
 Watts' ' Dictionary of Chemistry/ pp. 242, 243. 
 
HEAT. 
 
 Platinum, - 
 
 2000 
 
 I Tin, - - - 2 3o 
 
 Palladium, 
 Gold, 
 
 195 
 1 200 
 
 ! Selenium, - - 217 
 | Cane sugar, - 160 
 
 Cast iron, - 
 Glass, 
 
 1 200 
 
 IIOO 
 
 Sulphur, - II1[ 
 ! Sodium, - - 9 
 
 Copper, - 
 Silver, 
 
 1090 
 
 IOOO 
 
 1 Wax, - . - 68 
 I Potassium, - 
 
 Borax, 
 
 1000 
 
 Paraffin, _ - - 54 
 
 Antimony, 
 Zinc, 
 
 43 2 
 360 
 
 ! Spermaceti, 44 
 ! Phosphorus, - - 43 
 
 Lead, 
 
 33 
 
 j Water, - 
 
 Cadmium, - 
 
 320 
 
 j Bromine, 
 
 Bismuth, - 
 
 265 
 
 I Mercury, - 4 
 
 TTO. The folk 
 
 nvins: tal 
 
 )le (from Miller's ' Chemical 
 
 Physics/ p. 344) exhibits the change of volume of certain 
 substances in passing from the liquid to the vaporous 
 conditionunder the pressure of one atmosphere : 
 
 i volume of water yields 1696 volumes of vapour. 
 ,, alcohol 528 ,, ,, 
 
 ,, ether 298 ,, ,, 
 
 ,, oil of turpentine 193 ,, 
 
 in. The following table (from Watts' 'Dictionary of 
 Chemistry/ vol. iii., p. 77) exhibits the latent heats of 
 fluidity of certain substances, together with their melting 
 
 rnmtc 
 
 Melting Latent 
 Point. Heat. 
 
 Tin, - - - 235 14-25 
 Silver, - - 1000 21 -I 
 
 Zinc, - - - 433 28>I 
 Chloride of calcium 
 
 (CaC1.3H 2 0), 28-5 40-7 
 Nitrate of potassium, 339 47 '4 
 Nitrate of sodium, 310-5 63*0 
 
 The latent heat of fluidity of water was found by Reg- 
 nault, and by Provostaye and Desains, to be 79. Bunsen, 
 by means of his ice-calorimeter (' Pogg. Ann./ vol. cxli., 
 p. 30) has obtained the value 80-025. He finds the 
 specific gravity of ice to be '9167. 
 
 1 1 2. The following table of latent heats of vaporization 
 
 1 
 
 Melting 
 
 Latent 
 
 
 Point. 
 
 Heat. 
 
 Mercury, 
 
 - -39 
 
 2-82 
 
 Phosphorus, 
 
 - 44 
 
 5'0 
 
 Lead, - 
 
 - 332 
 
 5 '4 
 
 Sulphur, 
 Iodine, 
 
 - H5 
 - 107 
 
 9 '4 
 ii>' 
 
 Bismuth, 
 
 - 270 
 
 12-6 
 
 Cadmium, - 
 
 - 320 
 
 13-6 
 
 
UNITS AND PHYSICAL CONSTANTS. [CHAP. 
 
 at atmospheric pressure 
 Physics/ p. 342 : 
 
 is from Miller's ' Chemical 
 
 Latent Heat for 
 
 Unit Mass. 
 
 Equal Volumes. 
 
 Xj 
 
 
 
 Steam = icx 
 
 DO. 
 
 Water, - 
 
 536-67 
 
 1000 
 
 Regnault. 
 
 ,, - 
 
 535'90 
 
 
 Andrews. 
 
 Wood spirit, 
 
 26370 
 
 872-9 
 
 
 
 Alcohol, - .- r 
 
 202*40 
 
 963-1 
 
 i> 
 
 Fousel oil, 
 
 121-37 
 
 1104-7 
 
 Favre & Silbermann. 
 
 Formic acid, 
 
 120-72 
 
 574-4 
 
 11 
 
 Methyl formiate, 
 
 117*10 
 
 726-6 
 
 Andrews. 
 
 Butyric acid, 
 
 114-67 
 
 1043-8 
 
 F. & S. 
 
 Methyl acetate, - 
 
 110-20 
 
 843-5 
 
 Andrews. 
 
 Formic ether, - 
 
 105-30 
 
 806-0 
 
 
 
 Valeric acid, 
 
 I03-52 
 
 1092-0 
 
 F. &S. 
 
 Acetic acid, 
 
 lOI-pI 
 
 632-3 
 
 ii 
 
 Acetic ether, 
 
 105-80 
 
 963-0 
 
 j> 
 
 .; " 
 
 92-68 
 
 843-5 
 
 Andrews. 
 
 Ether, 
 
 90*45 
 
 692-3 
 
 
 
 > ~ 
 
 91-11 
 
 695-4 
 
 F. & S. 
 
 Methyl butyrate, 
 
 8733 
 
 921-5 
 
 it 
 
 Carbonic disulphide, - 
 
 86-67 
 
 681-4 
 
 Andrews. 
 
 Oil of lemons, - 
 
 70-02 
 
 986-1 
 
 F. &S. 
 
 " 
 
 80-00 
 
 1125-6 
 
 Brix. 
 
 Oil of turpentine, 
 
 74-00 
 
 1040-5 
 
 ,, 
 
 ,, 
 
 6873 
 
 966-9 
 
 F. & S. 
 
 Terebene, 
 
 67-21 
 
 945-0 
 
 
 
 Oxalic ether, 
 
 7272 
 
 1097-5 
 
 Andrews. 
 
 Amylic ether, - 
 
 69-40 
 
 1134*0 
 
 F. & S. 
 
 Ethal, - 
 
 58-44 
 
 1452-0 
 
 ,, 
 
 Phosphorus chloride, 
 
 51-42 
 
 752-9 
 
 Andrews. 
 
 Ethyl iodide, - 
 
 46-87 
 
 756-8 
 
 
 
 Methyl iodide, - 
 
 46-07 
 
 671-8 
 
 > 
 
 Bromine, - 
 
 45-60 
 
 754-1 
 
 
 
 Stannic chloride, 
 
 30-53 
 
 820-0 
 
 ,, 
 
 Iodine, ... 
 
 23-95 
 
 627-9 
 
 F. & S. 
 
IX.] 
 
 HEAT. 
 
 89 
 
 113. Regnault's approximate formula for what he calls 
 "the total heat of steam at /," that is, for the heat 
 required to raise unit mass of water from o to f in the 
 liquid state and then convert it into steam at /, is 
 
 606*5 + '305 /. 
 
 If the specific heat of water were the same at all tem- 
 peratures, this would give 
 
 606*5 - '695 / 
 
 as the heat of evaporation at t. But since, according 
 to Regnault, the heat require^ to raise the water from 
 o to f is 
 
 / + 'OO002 t 2 + "0000003 ^ 3 , 
 
 the heat of evaporation will be the difference between 
 this and the " total heat," that is, will be 
 
 606*5 - '695 / - '00002 ft - '0000003 / 3 , 
 
 which is- accordingly the value adopted by Regnault as 
 the heat of evaporation of water at t. 
 
 114. According to Regnault, the increase of pressure 
 at constant volume, and increase of volume at constant 
 pressure, when the temperature increases from o to 100, 
 have the following values for the gases named : 
 
 At Constant 
 
 At Constant 
 
 Gas. 
 
 Volume. 
 
 Pressur 
 
 Hydrogen, - . *' 
 
 3667 
 
 3 66l 
 
 Air, - - 
 
 3665 
 
 3670 
 
 Nitrogen, -, . .- 
 
 3668 
 
 ... 
 
 Carbonic oxide, 
 
 3667 
 
 3669 
 
 Carbonic acid, 
 
 '3688 
 
 3710 
 
 Nitrous oxide, 
 
 3676 
 
 3719 
 
 Sulphurous acid, 
 
 3845 
 
 3903 
 
 Cyanogen, 
 
 3829 
 
 3877 
 
UNITS AND PHYSICAL CONSTANTS. [CHAP. 
 
 00366957 
 
 00367430 
 
 00365620 
 
 0036677 
 
 0037060 
 
 0037067 
 
 Jolly has obtained the following values for the co- 
 efficient of increase of pressure at constant volume : 
 
 Air, 
 
 Oxygen, - 
 Hydrogen, 
 Nitrogen, 
 Carbonic acid, - 
 Nitrous oxide, - 
 
 Mendelejeff and Kaiander have determined the co- 
 efficient of expansion of air at constant pressure to be 
 0036843. 
 
 115. The following table, showing the pressure of 
 aqueous vapour near the ordinary boiling point, is based 
 on Regnault's determinations, as revised by Moritz 
 (Guyot's tables, second edition, collection D, table 
 xxv.) : 
 
 Temperature. 
 
 o 
 
 99 -o 
 
 99*i 
 
 99*2 
 99 '3 
 99 '4 
 99 '5 
 99-6 
 
 997 
 99-8 
 
 99 "9 
 
 lOO'O 
 lOO'I 
 IOQ'2 
 I00'3 
 1 00 '4 
 
 100-5 
 
 iOQ'6 
 
 Centims. of 
 Mercury 
 at Paris. 
 
 73-3I9 
 
 74-382 
 74-650 
 74-9I8 
 75-I87 
 75*457 
 757 2 8 
 76*000 
 76-273 
 76-546 
 76-820 
 
 77-095 
 
 77-371 
 77-647 
 
 Dynes per 
 sq. cm. 
 
 9779 
 9-814 
 
 x io 5 
 
 9-885 
 
 9-920 
 
 9-956 
 
 9*992 
 
 i '0028 x 
 0064 
 oioo 
 
 -0136 
 0173 
 
 0209 
 
 0245 
 0282 
 
 0319 
 
 0356 
 
ix.] HEAT. 91 
 
 iperature. 
 
 Centims. of 
 Mercury 
 at Paris. 
 
 Dynes per 
 sq. cm. 
 
 I00'7 
 
 77-925 
 
 1-0393 x io 
 
 100-8 
 
 78-203 
 
 1-0430 
 
 100-9 
 
 78-482 
 
 1-0467 
 
 lOI'O 
 
 78-762 
 
 I - 0505 
 
 1 1 6. Regnault's results as to the departures from 
 Boyle's law are given in the form 
 
 Mi = j A (m - i) B (m - i)*, 
 v o-to 
 
 V l denoting the volume at the pressure P b V the volume 
 at atmospheric pressure P , and m the ratio -. 
 
 For air, the negative sign is prefixed to A and the posi- 
 tive sign to B, and we have 
 
 log A - 3-0435120, 
 logB = 5-2873751. 
 
 For nitrogen, the signs are the same as for air, and we 
 have 
 
 log A = _4 '8399375* 
 
 log B = 6*8476020. 
 
 For carbonic acid, the negative sign is to be prefixed 
 both to A and B, and we have 
 
 log A = 3*9310399, 
 log B = 6'862472i. 
 
 For hydrogen, the positive sign is to be prefixed both to 
 A and B, and we have 
 
 log A = 4-7381736, 
 log B = 6-9250787. 
 
92 UNITS AND PHYSICAL CONSTANTS. [CHAP. 
 
 117. The following determinations of the specific heat 
 of the same substance in different states are from Reg- 
 nault's experiments (' Mem. Acad. Sciences/ xxvi., pp. 
 327-332) : 
 
 Ice, mean sp. heat from - 78 to o, - - 0*474 
 
 ,, - 20 to o, - - 0-504 
 
 Water, at temperatures below 100, - - I * 
 
 Steam, mean sp. heat between 128 and 220, 0-4805 
 Bromine, solid, mean sp. heat from 77 '8 
 
 to - 25, - - - - 0-0833 
 
 ,, liquid, mean from 7 -3 to 10, - 0*1060 
 
 ,, gaseous, mean from 83 to 228, - 0*0555 
 
 Alcohol, liquid, at - 20, .... -5053 
 
 o, - '5475 
 
 + 20, ---- -5951 
 
 ,, 40, - >6 479 
 
 60, .--. -7060 
 
 80, ... - 7694 
 
 ,, gaseous, mean from 105 to 220, - "4534 
 
 Ether, liquid, at - 30, - - - - -5113 
 
 o, -5290 
 
 + 30> - -5467 
 
 ,, gaseous, mean from 70 to 220, - '4797 
 
 Sulphide of carbon, liquid, at 30, - - '2303 
 
 o, - - -2352 
 
 + 3> - - '2401 
 
 45, - - "2426 
 ,, gaseous, mean from 73 to 
 
 192, .... -1570 
 
 Benzene, liquid, mean from 20 to 71, - '43^0 
 
 gaseous, Ii6to2i8, - -3754 
 
 Oil of turpentine, liquid, at o, - - -- "4106 
 
 40, - -4538 
 
 120, - - - -5019 
 
 160, - - - -5068 
 
 ,, gaseous, mean from 179 
 
 to 249, - - - -5061 
 
ix.] HEAT. 93 
 
 Boiling Points. 
 
 1 1 8. The following table gives the temperatures (by 
 air thermometer) at which according to Regnault's ex- 
 periments ('Mem. Acad. Sciences/ xxvi., 658,) the 
 vapours of the liquids named exert a pressure equal to 
 that of 76 c.m. of mercury at Paris: 
 
 Nitrous oxide, 
 Carbonic anhydride, 
 Sulphydric acid, - 
 Ammonia, - - - 
 Chlorine, 
 Sulphurous anhydride, 
 Ether, .... 
 
 -87-90 
 -78-2 
 -61-8 
 -38-5 
 -33'6 
 10*08 
 
 34 '97 
 
 Sulphide of Carbon, 
 Chloroform, - 
 Alcohol, 
 Benzene, 
 Oil of turpentine, - 
 Mercury, 
 
 - 46 "2O 
 - 60-16 
 - 78-26 
 - 80-36 
 - I59-I5 
 - 357 >2 5 
 
 119. Change of volume in melting, from Kopp's ex- 
 periments (Watt's ' Diet.,' art. Heat, p. 78): 
 
 Phosphorus. Calling the volume at o unity, the volume at the 
 
 melting point (44) is 1*017 in the solid, and 1*052 in the 
 
 liquid, state. 
 Sulphur. Volume at o being I, volume at the melting point (115) 
 
 is 1*096 in the solid, and 1*150 in the liquid, state. 
 Wax. Volume at o being I, volume at melting point (64) is 1*161 
 
 in solid, and 1*166 in liquid, state. 
 Stearic acid. Volume at o being I, volume at melting point (70) 
 
 is 1*079 i n solid, and 1*198 in liquid, state. 
 Rose's fusible metal (2 parts bismuth, I tin, I lead). Volume at o 
 
 being I, volume at 59 is a maximum, and is 1*0027. Volume 
 
 at melting point (between 95 and 98) is greater in liquid 
 
 than in solid state by 1*55 per cent. 
 
 120. Temperature of evaporation and dew-point 
 (Glaisher's Tables, second edition, page iv). The fol- 
 lowing are the factors by which it is necessary to mul- 
 tiply the excess of the reading of the dry thermometer 
 
94 UNITS AND PHYSICAL CONSTANTS. [CHAP. 
 
 over that of the wet, to give the excess of the tempera- 
 ture of the air above that of the dew-point : 
 
 Reading of 
 
 Dry Bulb Factor. 
 Therm. 
 
 - 10 C. = I4F. 876 
 
 - 5 23 7-28 
 
 o 32 3-32 
 
 + 5 41 2-26 
 
 + 10 50 2 '06 
 
 Reading of 
 
 Dry Bulb Factor. 
 
 Therm. 
 
 o o 
 
 20 68 1 79 
 
 25 77 170 
 
 30 86 i -65 
 
 35 95 i '60 
 
 Conductivity. 
 
 121. By the thermal conductivity of a substance 
 at a given temperature is meant the value of k in the 
 expression 
 
 where Q denotes the quantity of heat that flows, in time 
 /, through a plate of the substance of thickness x, the area 
 of each of the two opposite faces of the plate being A, 
 and the temperatures of these faces being respectively 
 i\ and z> 2 , each differing but little from the given temper- 
 ature. The lines of flow of heat are supposed to be 
 normal to the faces, or, in other words, the isothermal 
 surfaces within the plate are supposed to be parallel 
 to the faces ; and the flow of heat is supposed to be 
 steady, in other words, no part of the plate is to be gaining 
 or losing heat on the whole. 
 
 Briefly, and subject to these understandings, con- 
 ductivity may be denned as the quantity of heat that passes 
 in unit time, through unit area of a plate whose thickness is 
 unity, when its opposite faces differ in temperature by on? 
 degree. 
 
ix.j HEAT. 95 
 
 122. Dimensions of Conductivity. From equation (i) 
 we have 
 
 *=^ * (2> 
 
 v 9 -f>t A/ 
 
 The dimensions of the factor -^ are simply M, since 
 
 z> 2 - v^ 
 
 the unit of heat varies jointly as the unit of mass and the 
 length of the degree. The dimensions of the factor 
 
 - X - are ; hence the dimensions of k are - . This is 
 Ar J_< 1 J_/ 1 
 
 on the supposition that the unit of heat is the heat 
 required to raise unit mass of water one degree. In 
 calculations relating to conductivity it is perhaps more 
 usual to adopt as the unit of heat the heat required to 
 raise unit volume of water one degree. The dimensions of 
 
 O L 2 
 
 ^ will then be L 8 , and the dimensions of k will be 
 
 v^-v T 
 
 These conclusions may be otherwise expressed by 
 saying that the dimensions of conductivity are when 
 
 \ j L 
 
 the thermal capacity of unit mass of water is taken as 
 
 L 2 
 unity, and are - when the capacity of unit volume 
 
 of water is taken as unity. In the C.G.S. system the 
 capacities of unit mass and unit volume of water are 
 practically identical. 
 
 123. Let c denote the thermal capacity of unit volume 
 of a substance through which heat is being conducted. 
 
 Then - denotes a quantity whose value it is often neces- 
 sary to discuss in investigations relating to the transmis- 
 sion of heat. We have, from equation (2), 
 
96 UNITS AND PHYSICAL CONSTANTS. [CHAP. 
 
 k Q ; x 
 
 c 
 
 _ 
 A/' 
 
 where Q' denotes -. Hence - would be the numerical 
 
 c c 
 
 value of the conductivity of the substance, if the unit of 
 heat employed were the heat required to raise unit volume 
 of the substance one degree. Professor Clerk Maxwell 
 
 proposes to call - the thermometric conductivity, as dis- 
 
 tinguished from k, which is the thermal or calorimetric 
 conductivity. 
 
 Coefficient of Diffusion. 
 
 124. There is a close analogy between conduction and 
 diffusion. Let x denote the distance between two 
 parallel plane sections A and B to which the diffusion is 
 perpendicular, and let these sections be maintained in 
 constant states. Then, if we suppose one substance to be 
 at rest, and another substance to be diffusing through it, 
 the coefficient of diffusion K is denned by the equation 
 
 _ 
 
 where y denotes the thickness of a stratum of the mixture 
 as it exists at B, which would be reduced to the state 
 existing at A by the addition to it of the quantity which 
 diffuses from A to B in the line /. 
 
 When the thing diffused is heat, the states at A and B are 
 the temperatures z\ and z> 2 , and y denotes the thickness of 
 a stratum at the lower temperature which would be raised 
 to the higher by the addition of as much heat as passes 
 in the time /. This quantity of heat, for unit area, will be 
 kt , x 
 
ix.] HEAT. 97 
 
 which must therefore be equal to 
 
 whence we have 
 
 The "thermometric conductivity "- may therefore be re- 
 garded as the coefficient of diffusion of heat. 
 
 125. When we are dealing with the mutual inter- 
 diffusion of two liquids, or of two gases contained in a 
 closed vessel, subject in both cases to the law that the 
 volume of a mixture of the two substances is the sum of 
 the volumes of its components at the same pressure, the 
 quantity of one of the substances which passes any section 
 in one direction must be equal (in volume) to the quantity 
 of the other which passes it in the opposite direction, 
 since the total volume on either side of the section re- 
 mains unaltered ; and a similar equality must hold for the 
 quantities which pass across the interval between two 
 sections, provided that the absorption in the interval 
 itself is negligible. Let x as before denote the distance 
 between two parallel plane sections A and B to which the 
 diffusion is perpendicular. Let the mixture at A consist 
 of m parts by volume of the first substance to i - m of the 
 second, and the mixture at B consist of n parts of the 
 second to i - n of the first, m being greater than i - , and 
 therefore n greater than i - m. The first substance will 
 then diffuse from A to B, and the second in equal 
 quantity from B to A. Let each ol these quantities be 
 such as would form a stratum of thickness z (the vessel 
 being supposed prismatic or cylindrical, and the sections 
 
 G 
 
98 UNITS AND PHYSICAL CONSTANTS. [CHAP. 
 
 considered being normal sections), then z will be pro- 
 portional to 
 
 m - ( i - n) . ^, ^ - m + n - i . 
 
 - : - tf. that is, to -- f, 
 
 x x 
 
 and the coefficient of interdiffusion K is defined by 
 the equation 
 
 rr Ml + 11 I . i \ 
 
 z = K - -/. (2) 
 
 The numerical quantity m + ni may be regarded as 
 measuring the difference of states of the two sections 
 A and B. 
 
 Ify now denote the thickness of a stratum in the con- 
 dition of B which would be reduced to the state existing 
 at A by the abstraction of a thickness z of the second 
 substance, and the addition of the same thickness of the 
 first, we have (in)y + z as the expression for the 
 quantity of the first substance in the stratum after the 
 operation. This is to be equal to my. Hence we have 
 
 . ' ''**' * (3) 
 
 and substituting for z its value in (2) we have finally 
 
 which is of the same form as equation (i), y now denot- 
 ing the thickness of a stratum of the mixture as it exists 
 at B, which would be reduced to the state existing at A 
 by the addition to it of the quantity of one substance 
 which diffuses from A to B in the time /, and the removal 
 from it of the quantity of the other substance which 
 diffuses from B to A in the same time. 
 
 126. The following values of K in terms of the centi- 
 metre and second are given in Professor Clerk Maxwell's 
 
ix.] HEAT. 99 
 
 ' Theory of Heat,' 4th edition, p. 332, on the authority of 
 Professor Loschmidt of Vienna. 
 
 Coefficients of Interdiffusion of Gases. 
 
 Carbonic Acid and Air, .... "1423 
 
 ,, ,, Hydrogen, - ... '5614 
 
 ,, ,, Oxygen, ... '1409 
 
 ,, ., Marsh Gas, - - - '1586 
 
 ,, ,, Carbonic Oxide, - - '1406 
 
 ,, ,, Nitrous Oxide, - - '0982 
 
 Oxygen and Hydrogen, - - - - 7214 
 
 ,, ,, Carbonic Oxide, - - - '1802 
 
 Carbonic Oxide and Hydrogen, ... '6422 
 
 Sulphurous Acid and Hydrogen, - - '4800 
 
 k 
 
 127. These may be compared with the value of - for 
 
 air, which, according to Professor J. Stefan of Vienna, is 
 256. 
 
 The value of k for air, according to the same authority, 
 is 5-58 x io~ 5 , and is independent of the pressure. Pro- 
 fessor Maxwell, by a different method, calculates its value 
 at 5*4 x io~ 5 . 
 
 Results of Experiments on Conductivity of Solids. 
 
 128. Principal Forbes' results for the conductivity of 
 iron (Stewart on Heat, p. 261, second edition) are ex- 
 pressed in terms of the foot and minute, the cubic foot 
 of water being the unit of thermal capacity. Hence the 
 value of Forbes' unit of conductivity, when referred to 
 
 C.G.S., is , or 15-48; and his results must be 
 
 60 
 
 multiplied by 15*48 to reduce them to the C.G.S. scale. 
 His observations were made on two square bars ; the 
 side of the one being ij inch, and of the other an 
 
loo UNITS AND PHYSICAL CONSTANTS. [CHAP. 
 
 inch. The results when reduced to C.G.S. units are as 
 follows : 
 
 Temp. 
 
 Cent. ij-inch bar. i-inch bar. 
 
 o 
 
 o - - -207 - - -1536 
 
 25 - - -1912 - - -1460 
 50 - , - -1771 - - -1399 
 
 75 . - ; - -1656 - - -1339 
 
 loo - - ... '1567 - - '1293 
 
 125 - ; - '1496 -' - -1259 
 
 150 - - "1446 - - '1231 
 175 - ; . - ' '1399 - - -1206 
 
 200 -- \ - V I3S6 T - "1183 
 
 225 - - '1317 - - '1160 
 250 - - ' -1279 - - -1140 
 
 275 - - '1240 - - 'II2I 
 
 129. Neumann's results ('Ann. de. Chim.' vol. Ixvi., p. 
 185) must be multiplied by '000848 to reduce them to 
 our scale. They then become as follows : 
 
 Conductivity. 
 
 Copper, .... i-ioS 
 Brass, - .- - - ^ ** "302 
 Zinc, - - - - '307 
 
 Iron, '164 
 
 German silver, ... -109 
 
 Ice, i - - - '0057 
 
 In the same paper he gives for the following substances 
 
 k k 
 
 the values of - or ; that is, the quotient of conductivity 
 sa c 
 
 by the thermal capacity of unit volume. These require 
 the same reducing factor as the values of k, and when 
 reduced to our scale are as follows : 
 
ix.] ,-. HEAT. ioi 
 
 Values of -. 
 c 
 Coal, '00116 
 
 Melted sulphur, - - - '00142 
 
 Ice, *oi 14 
 
 Snow, .... -00356 
 
 Frozen mould, - - - '00916 
 
 Sandy loam, - - - '0136 
 
 Granite (coarse), - - - '0109 
 
 Serpentine, .... '00594 
 
 130. Sir W. Thomson's results, deduced from observa- 
 tions of underground thermometers at three stations at 
 Edinburgh ('Trans. R. S. E.,' 1860, p. 426), are given in 
 terms of the foot and second, the thermal capacity of a 
 cubic foot of water being unity, and must be multiplied 
 by (3 0-48) 2 or 929 to reduce them to our scale. The 
 following are the reduced results : 
 
 k, or k 
 
 Conductivity, ~ c ' 
 
 Trap-rock of Calton Hill, - - - '00415 - '00786 
 Sand of experimental garden, - - '00262 - '00872 
 Sandstone of Craigleith Quarry, - - '01068 - '02311 
 
 k 
 My own result for the value of - from the Greenwich 
 
 underground thermometers ('Greenwich Observations,' 
 1860) is in terms of the French foot and the year. As 
 a French foot is 32-5 centifns., and a year is 31557000 
 seconds, the reducing factor is (32-5)% + 31557000; that 
 
 is> 3'347 x I0 ~ 5 - The result is 
 
 I 
 
 c 
 Gravel of Greenwich Observatory Hill, - '01249 
 
 Professors Ayrton and Perry ('Phil. Mag./ April, 1878) 
 determined the conductivity of a Japanese building stone 
 (porphyritic trachyte) to be '0059. 
 
102 
 
 UNITS AND PHYSICAL CONSTANTS. [CHAP. 
 
 131. Angstrom, in ' Pogg. Ann./vols. cxiv. (i86i)and 
 cxviii. (1863), employs as units the centimetre and the 
 minute ; hence his results must be divided by 60. These 
 results, as given at p. 429 of his second paper, will then 
 stand as follows : 
 
 Copper, first specimen, 
 
 ,, second specimen, - 
 Iron, 
 
 Value of *. 
 c 
 
 i '216 (i - '00214 t) 
 
 I'l63 (I ~ 'OOI5I9 /) 
 '224 (i -'002874 /) 
 
 He adopts for c the values 
 
 "84476 for copper ; "88620 for iron, 
 and thus deduces the following values of k : 
 
 Conductivity. 
 Copper, first specimen, - - i '027 (i - '00214 
 
 ,, second specimen, - - -983 (i - '001519 t] 
 
 Iron, -199 (i - '002874/5 
 
 132. In Professor George Forbes's paper on conduc- 
 tivity ('Proc. R. S. E.,' February, 1873) the units are the 
 centim. and the minute; hence his results must be divided 
 by 60. Thus reduced, they are : 
 
 Ice, along axis, - 
 
 00223 
 
 Kamptulikon, - 
 
 oooi i 
 
 Ice, perpendicular to / 
 axis, - - \ 
 
 00213 
 
 Vulcanized india- } 
 rubber, - - * ] 
 
 000089 
 
 Black marble, 
 
 00177 
 
 Horn, 
 
 000087 
 
 White marble, - -\ 
 
 '00115 
 
 Beeswax, - 
 
 000087 
 
 Slate, - - \- 
 
 00081 
 
 Felt, 
 
 000087 
 
 Snow, - - 
 
 00072 
 
 Vulcanite, 
 
 0000833 
 
 Cork, 
 
 000717 
 
 Haircloth, 
 
 0000402 
 
 Glass, - 
 
 0005 
 
 Cotton-wool, divided, 
 
 0000433 
 
 Pasteboard, - - 
 
 000453 
 
 ,, pressed, 
 
 0000335 
 
 Carbon, - - 
 
 000405 
 
 Flannel, - 
 
 0000355 
 
 Roofing-felt, 
 
 000335 
 
 Coarse linen, 
 
 0000298 
 
 Fir, parallel to fibre, - 
 
 0003 
 
 Quartz, along axis, - 
 
 000922 
 
 Fir, across fibre and \ 
 
 '000088 
 
 ,, 
 
 00124 
 
 along radius, - \ 
 
 
 > 
 
 00057 
 
 Boiler-cement, - 
 
 000162 
 
 
 
 00083 
 
 Paraffin, - 
 
 00014 
 
 Quartz, perpendicular ) 
 
 
 Sand, very fine, - 
 
 000131 
 
 to axis, - - \ 
 
 '0040 
 
 Sawdust, ... 
 
 000123 
 
 ,, 
 
 0044 
 
ix.] HEAT. 103 
 
 Professor Forbes quotes a paper by M. Lucien De la 
 Rive ('Soc. de Ph. et d'Hist. Nat. de Geneve/ 1864) in 
 which the following result is obtained for ice, 
 Ice, ... "00230. 
 
 M. De La Rive's experiments are described in 
 * Annales de Chimie,' ser. 4, torn, i., pp. 504-6. 
 
 133. A Committee, consisting of Professors Herschel 
 and Lebour, and Mr. J. "F. Dunn, appointed by the British 
 Association to determine the thermal conductivities of 
 certain rocks, have obtained results, of which the follow- 
 ing (communicated to me in May, 1877, by Professor 
 Herschel) with some additions, within brackets, of later 
 determinations, are deemed the most reliable : 
 
 Substance. ^I'S" [f ' 
 
 Iron pyrites, more than - - '01 more than '0170 
 
 Rock salt, rough crystal, - - "0113 '0288 
 
 Fluorspar, rough crystal, - - '00963 '0156 
 
 Quartz, opaque crystal [and 
 
 quartzites], .... '0080 to '0092 '017510-0190 
 
 [Siliceous sandstones (slightly wet), '00641 to '00854] '0130 to "0230 
 
 Galena, rough crystal, [inter- 
 spersed with quartz,] - - '00705 '0171 
 
 Sandstone and hard grit, dry, - '00545 to '00565 '0120 
 
 Sandstone and hard grit, thor- 
 oughly wet, - <- - -* '00590 to '00610 -oioo 
 
 [Micaceous flagstone, along the 
 
 cleavage, - i - - '00632 '0116 
 
 [Micaceous flagstone, across cleav- 
 age, '00441] '0087 
 
 Slate, along cleavage, - - '00550 to "00650 '0102 
 
 Do., across cleavage, - - '00315 to "00360 "0057 
 
 Granite [various specimens, about "00510 to "00550 "oiooto"Oi2O 
 
 Marbles, limestone, calcite, and 
 
 compact dolomite, - - - "00476 to '00560 "0085 to "0095 
 
io 4 
 
 UNITS AND PHYSICAL CONSTANTS. [CHAP. 
 
 Substance. 
 
 Red serpentine (Cornwall), 
 [Caen stone (building limestone), 
 Whinstone, trap rock [and mica 
 
 schist], - - - *. 
 Clay slate (Devonshire), 
 [Tough clay- (sun-dried), 
 
 Do., soft (with one-fourth 
 
 of its weight of water, 
 Chalk, - 
 
 [Calcareous sandstone (firestone), 
 Plate-glass [German and] English, 
 [German glass toughened, - 
 Heavy spar, opaque rough crystal, 
 Fire-brick, - 
 Fine red brick, "* * 
 
 Fine plaster of Paris, dry plate, 
 
 Do., thoroughly wet, 
 
 [White sand, dry, - 
 
 Do., saturated with 
 
 water, about - * '* 
 House coal and cannel coal, 
 Pumice stone, - 
 
 Conductivity in 
 C.G.S. Units. 
 
 e- 
 
 - '00441 
 
 0065 
 
 ), '004331 
 
 0089 
 
 ci __ ^ 
 
 - -00280 to -00480 
 
 0055 to '0095 
 
 - -00272 
 
 0053 
 
 - -00223 
 
 0048 
 
 - -00310] 
 
 0035 
 
 - 'OO2OO tO 'OO33O 
 
 0046 to "0059 
 
 ), 'OO2II 
 
 0049 
 
 h, [-00198 to] '00234 
 
 00395 to ? 
 
 - -00185] 
 
 00395 
 
 .1, -00177 
 
 ... 
 
 - -00174 
 
 0053 
 
 - -00147 
 
 0044 
 
 - -00120 
 
 '0060 ) . 
 
 :, -00160 
 
 > about 
 0025 ] 
 
 - -00093 
 
 '0032 
 
 - "00700] 
 
 0120 about 
 
 - '00057 to '00113 
 
 0012 to '0027] 
 
 - -00055 
 
 ... 
 
 134. Peclet in ' Annales de Chimie,' ser. 4, torn, ii., p. 
 114 [1841], employs as the unit of conductivity the trans- 
 mission, in one second, through a plate a metre square 
 and a millimetre thick, of as much heat as will raise a 
 cubic decimetre (strictly a kilogramme) of water one 
 degree. Formula (2) shows that the value of this conduc- 
 tivity, in the C.G.S. system, is 
 
 10000 
 
 100 
 

 ix.] HEAT. 
 
 His results must accordingly be divided by 100 ; and 
 they then become : 
 
 Gold, , - 
 
 Conductivity. 
 2128 
 
 Marble, - 
 
 Conductivity. 
 0048 
 
 Platinum, - 
 
 2095 
 
 Baked earth, 
 
 - ' -0023 
 
 Silver, 
 
 
 
 
 Copper, - 
 
 I9II 
 
 
 
 Iron, 
 
 0795 
 
 
 
 Zinc, 
 
 0774 
 
 
 
 Lead, 
 
 - ' -0384 
 
 
 
 The value given for lead was from direct experiment. 
 The values given for the other metals were not from 
 direct experiment, but were inferred from the value for 
 lead taken in conjunction with Despretz's results for the 
 relative conductivity of metals. 
 
 135. The same author published in 1853 a greatly 
 extended series of observations, in a work entitled 
 1 Nouveaux documents relatifs aux chauffage et a la 
 ventilation.' In this series, the conductivity which is 
 adopted as unity is the transmission, in one hour, through 
 a plate a metre square and a metre thick, of as much 
 heat as will raise a kilogramme of water one degree. 
 This conductivity, in C.G.S. units, is 
 
 1000 100 i , - . i 
 
 - . - - . - ; that is, -- 
 i 10000 3600 360 
 
 The results must therefore be divided by 360 ; and 
 they then become as follows : 
 
 Density. Conductivity. 
 Copper, - ... . ... - I7 8 
 
 Iron, - - - "x - i - ... -081 
 
 Zinc, ..... ... -078 
 
 Lead, - - - - - - ... -039 
 
 Carbon from gas-retorts, - - i'6l "0138 
 
io6 
 
 UNITS AND PHYSICAL CONSTANTS. 
 
 
 Density. 
 
 Conductivity. 
 
 Marble, fine-grained grey, - .- 
 
 2-68 
 
 0097 
 
 ,, sugar-white, coarse-grained, 
 
 277 
 
 0077 
 
 Limestone, fine-grained, 
 
 2-34 
 
 0058 
 
 >, - - 
 
 2-27 
 
 0047 
 
 55 " " 
 
 2-17 
 
 0035 
 
 Lias building-stone, coarse-grained 
 
 2-24 
 
 0037 
 
 f 
 
 2 '22 
 
 0035 
 
 Plaster of Paris, ordinary, made up, 
 
 
 00092 
 
 ,, very fine, ,, 
 
 1-25 
 
 00144 
 
 ,, for casts, very fine, 
 made up, 
 
 H 
 
 '00122 
 
 Alum paste (marble cement), ,, - 
 
 173 
 
 00175 
 
 Terra-cotta, 
 
 1-98 
 
 00192 
 
 
 
 1-85 
 
 00142 
 
 Fir, across fibres, 
 
 48 
 
 00026 
 
 along fibres, 
 
 48 
 
 00047 
 
 Walnut, across fibres, - 
 
 
 00029 
 
 ,, along fibres - 
 
 ... 
 
 00048 
 
 Oak, across fibres 
 
 ... 
 
 00059 
 
 Cork, 
 
 '22 
 
 00029 
 
 Caoutchouc, - 
 
 ... 
 
 00041 
 
 Gutta percha, .... 
 
 
 '00048 
 
 Starch paste, 
 
 I'OI7 
 
 00118 
 
 Glass, 
 
 2 '44 
 
 002 1 
 
 > ~ 
 
 2'55 
 
 0024 
 
 Sand, quartz, .... 
 
 i'47 
 
 00075 
 
 Brick, pounded, coarse-grained, 
 
 I'O 
 
 00039 
 
 , , passed through 
 silk sieve, 
 
 (.76 
 
 00046 
 
 Fine brickdust, obtained by decan- 
 
 J--55 
 
 00039 
 
 Chalk, powdered, slightly damp, - 
 
 ) 
 92 
 
 00030 
 
 ,, washed and dried, 
 
 8 5 
 
 00024 
 
 ,, washed, dried, and 
 
 | 
 
 
 compressed, - 
 
 jl'02 
 
 00029 
 
 Potato-starch, 
 
 71 
 
 00027 
 
 Wood-ashes, - - 
 
 '45 
 
 00018 
 
 Mahogany sawdust, ... 
 
 3i 
 
 00018 
 
IX.] 
 
 HEAT. 
 
 107 
 
 25 
 
 Density. 
 
 Wood charcoal, ordinary, powdered, "49 
 Bakers' breeze, in powder, passed 
 
 through silk sieve, 
 Ordinary wood charcoal, in powder, 
 
 passed through silk sieve, - 
 Coke, powdered, - - - '77 
 
 Iron filings, - - - - 2*05 
 
 Binoxide of manganese, - - I '46 
 
 Conductivity. 
 00022 
 
 OOOIQ 
 
 C00225 
 
 00044 
 '00044 
 00045 
 
 Woolly Substances. 
 
 Cotton Wool of all densities, - 
 Cotton swansdown (molleton de 
 
 coton), of all densities, 
 Calico, new, of all densities, - 
 Wool, carded, of all densities, 
 Woollen swansdown (molleton de 
 
 laine), of all densities, 
 Eider-down, - - - - ... 
 
 Hempen cloth, new ... -54 
 old --- -58 
 Writing-paper, white - - - '85 
 Grey paper, unsized ... -48 
 
 oooi 1 1 
 000139 
 
 000122 
 
 000067 
 
 000108 
 000144 
 000119 
 000119 
 000094 
 
 Emission and Surface Conduction. 
 
 136. Mr. D. M'Farlane has published (' Proc. Roy. Soc/ 
 1872, p. 93) the results of experiments on the loss of heat 
 from blackened and polished copper in air at atmospheric 
 pressure. They need no reduction, the units employed 
 being the centimetre, gramme, and second. The general 
 result is expressed by the formulae 
 
 x -'000238 + 3'o6 x io~ 6 / 2'6 x io~ 8 / 2 
 for a blackened surface, and 
 
 x ='000168 + 1*98 x io~ 6 / 1*7 x io~ 8 / 2 
 for polished copper, x denoting the quantity of heat lost 
 
joS 
 
 UNITS AND PHYSICAL CONSTANTS. [CHAP. 
 
 per second per square centim. of surface of the copper, 
 per degree of difference between its temperature and that 
 of the walls of the enclosure. These latter were blackened 
 internally, and were kept at a nearly constant temperature 
 of 14 C. The air within the enclosure was kept moist 
 by a saucer of water. The greatest difference of tempera- 
 ture employed in the experiments (in other words, the 
 highest value of /) was 50 or 60 C. 
 
 The following Table contains the values of x calculated 
 from the above formulae, for every fifth degree, within the 
 limits of the experiments. 
 
 
 Value 
 
 of jr. 
 
 
 Difference of 
 
 
 
 Rcitio. 
 
 Temperature. 
 
 
 
 
 
 Polished Surface. 
 
 Blackened Surface. 
 
 
 o 
 
 
 
 
 5 
 
 000178 
 
 OOO252 
 
 707 
 
 10 
 
 OOOI86 
 
 OOO266 
 
 699 
 
 15 
 
 OOOI93 
 
 000279 
 
 692 
 
 20 
 
 000201 
 
 000289 
 
 695 
 
 25 
 
 OOO2O7 
 
 000298 
 
 6 94 
 
 30 
 
 000212 
 
 000306 
 
 693 
 
 35 
 
 OOO2I7 
 
 000313 
 
 6 93 
 
 40 
 
 'OOO22O 
 
 000319 
 
 693 
 
 45 
 
 000223 
 
 000323 
 
 6 9 o 
 
 5o 
 
 OOO225 
 
 '000326 
 
 690 
 
 55 
 
 000226 
 
 000328 
 
 690 
 
 60 
 
 OOO226 
 
 000328 
 
 690 
 
 137. Professor Tait has published ('Proc. R. S. E.' 
 1869-70, p. 207) observations by Mr. J. P. Nichol on the 
 loss of heat from blackened and polished copper, in air, 
 at three different pressures, the enclosure being blackened 
 internally and surrounded by water at a temperature of 
 
IX.] 
 
 HEAT. 
 
 109 
 
 approximately 8 C.* Professor Tait's units are the grain- 
 degree for heat, the square inch for area, and the hour for 
 time. The rate of loss per unit of area is 
 
 heat emitted 
 area x time 
 
 The grain-degree is -0648 gramme-degree. 
 The square inch is 6 '45 14 square centims. 
 The hour is 3600 seconds. 
 Hence Professor Tait's unit rate of emission is 
 
 0648 
 
 = 279 
 
 10 
 
 6-4514 x 3600 
 
 of our units. Employing this reducing factor, Professor 
 Tait's Table of Results will stand as follows : 
 
 Pressure 1*014 * Io6 [7^o millims. of mercury]. 
 Blackened. 
 
 -LJlclClVtl 
 
 Temp. Cent. 
 
 Loss per sq. cm. 
 
 Temp. Cent. 
 
 -LJ1 It'lll. 
 
 Loss per sq. en 
 
 6l''2 - J / 
 
 per second. 
 - -01746 
 
 63'8 - 
 
 per second. 
 - -00987 
 
 50-2 - \- 
 
 - '01360 
 
 57'i - 
 
 - ; - -00862 
 
 41-6 - 
 
 - -01078 
 
 50-5 - 
 
 - -00736 
 
 34*4 - ) - 
 
 - -00860 
 
 44-8 - 
 
 - -00628 
 
 273 - 
 
 - '00640 
 
 40-5 - 
 
 00562 
 
 20-5 - 
 
 - '00455 
 
 34'2 - 
 
 - -00438 
 
 
 
 29-6 - 
 
 - -00378 
 
 
 
 23-3 - 
 
 - -00278 
 
 
 
 iS'6 - 
 
 4T 
 
 - -00210 
 
 Pressure 
 
 1-36 X io 5 [102 millims. 
 
 of mercury]. 
 
 62-5 - - 
 
 - -01298 
 
 67-8 - 
 
 - -00492 
 
 57'5 - 
 
 "01158 
 
 61-1 - 
 
 - "00433 
 
 53'2 - 
 
 - -01048 
 
 55 
 
 - - - -00383 
 
 47'5 - ' 
 
 - -00898 
 
 497 - 
 
 - -00340 
 
 43 
 
 - -00791 
 
 44 -9 - 
 
 - -00302 
 
 28-5 - - 
 
 - -00490 
 
 40-8 - 
 
 - -00268 
 
 * This temperature is not stated in the "Proceedings," but has 
 been communicated to me by Professor Tait. 
 
no UNITS AND PHYSICAL CONSTANTS. [CHAP. 
 
 Pressure 1-33 X lO 1 [10 millims. of mercury]. 
 
 Blackened. 
 Temp. Cent. Loss per sq. cm. 
 per second. 
 
 Bright. 
 Temp. Cent. Loss per sq. m. 
 o per second. 
 
 62-5 - - 
 
 01182 
 
 65 - 
 
 - -00388 
 
 S7'5 ' - 
 
 01074 
 
 60 
 
 - -00355 
 
 54'2 
 
 . ' 
 
 01003 
 
 5o 
 
 - -00286 
 
 417 
 
 
 00726 
 
 40 
 
 00219 
 
 37*5 
 
 " 
 
 00639 
 
 30 
 
 - -00157 
 
 341 
 
 * 
 
 00569 
 
 23-5 - - - -00124 
 
 27-5 
 
 < .- '00446 
 
 
 24-2 
 
 - -00391 
 
 
 Mechanical Equivalent of Heat. 
 
 138. The value originally deduced by Joule from his 
 experiments on the stirring of water was 772 foot-pounds 
 of work (at Manchester) for as much heat as raises a 
 pound of water through i Fahr. This is 1389-6 foot- 
 pounds for a pound of water raised i C, or 1389-6 foot- 
 grammes for a gramme of water raised i C. As a foot 
 is 30-48 centims., and the value of g at Manchester is 
 981-3, this is 1389-6 x 30-48 x 981-3 ergs per gramme- 
 degree ; that is, 4-156 x io 7 ergs per gramme-degree. 
 
 A later determination by Joule (* Brit. Assoc. Report/ 
 1867, pt. i., p. 522, or ' Reprint of Reports on Electrical 
 Standards,' p. 186) is 25187 foot-grain-second units of 
 work per grain-degree Fahr. This is 45337 of the same 
 units per grain-degree Centigrade, or 45337 foot-gramme- 
 second units of work per gramme-degree Centigrade; 
 that is to say, 
 
 45337 x (3'48) 2 = 4*212 x io 7 
 ergs per gramme-degree Centigrade. 
 
 At the meeting of the Royal Society, January, 1878 
 (' Proceedings,' vol. xxvii., p. 38), an account was given by 
 Joule of experiments recently made by him with a view 
 
ix.] HEAT. in 
 
 to increase the accuracy of the results given in his former 
 paper. ('Phil. Trans., 1850.') The result he has now 
 arrived at from the thermal effects of the friction of water, 
 is, that taking the unit of heat as that which can raise a 
 pound of water, weighed in vacuo, from 60 to 61 of the 
 mercurial Fahrenheit thermometer ; its mechanical equiva- 
 lent, reduced to the sea-level at the latitude of Greenwich, 
 is 77 2 '55 foot-pounds. 
 
 To reduce this to water at o C. we have to multiply 
 by 1*00089,* giving 773-24 ft. Ibs., and to reduce to ergs 
 per gramme-degree Centigrade we have to multiply by 
 
 Q 
 
 981-17 x 30*48 x . 
 The product is 4*1624 x io 7 . 
 
 139. Some of the best determinations by various experi- 
 menters are given (in gravitation measure) in the following 
 list, extracted from Watt's ' Dictionary of Chemistry,' 
 Supplement 1872, p. 687. The value 429*3 in this list 
 corresponds to 4*214 x io 7 ergs : 
 
 Him, - 432 - - Friction of water and brass. 
 
 , , * 433 - - Escape of water under pressure. 
 
 - 441*6 - - Specific heats of air. 
 
 ,, * 4 2 5' 2 - Crushing of lead. 
 
 Joule, - 429*3 - - | He^produced by an electric 
 
 Violle, 
 
 435*2 (copper) - ' 
 434*9 (aluminium) 
 435 '8 (tin) 
 
 Heat produced by induced 
 
 currents. 
 
 437*4 (lead) - 
 Regnault, 437 - - Velocity of sound. 
 
 We shall adopt 4*2 x io 7 ergs as the equivalent of 
 
 * This factor is found by giving / the value 15*8 (since the tem- 
 perature 60*5 Fahr. is 15*8 Cent.) in formula (3) of art. 98. 
 
 
ii2 UNITS AND PHYSICAL CONSTANTS. [CHAP. 
 
 i gramme-degree ; that is, employing J as usual to denote 
 Joule's equivalent, we have 
 
 J = 4'2 x io 7 = 42 millions. 
 140. Heat and Energy of Combination with Oxygen. 
 
 i gramme of 
 
 Compound 
 
 Gramme- 
 degrees of heat 
 
 Equivalent 
 Energy, in 
 
 
 
 produced. 
 
 ergs. 
 
 Hydrogen, - 
 Carbon, 
 
 H 2 O 
 CO 2 
 
 34000 A F 
 8000 A F 
 
 1-43 x io 12 
 3-36 x io 11 
 
 Sulphur, - - . 
 
 SO 2 
 
 2300 A F 
 
 9-66 x io 10 
 
 Phosphorus, 
 
 P-2Q5 
 
 5747 A 
 
 2 '41 X IO 11 
 
 Zinc, - 
 
 ZnO 
 
 1301 A 
 
 5-46 x io 10 
 
 Iron, ... 
 
 Fe 3 O 
 
 1576 A 
 
 6-62 x io 10 
 
 Tin, - 
 
 SnO 2 
 
 1233 A 
 
 5-18 
 
 Copper, 
 
 CuO 
 
 602 A 
 
 2'53 
 
 Carbonic oxide, - 
 
 CO 2 
 
 2420 A 
 
 I'O2 X IO 11 
 
 Marsh-gas, - 
 
 C0 2 andH 2 
 
 13100 A F 
 
 5'5 
 
 Olefiant gas, 
 
 ?> 
 
 11900 A F 
 
 S'oo . 
 
 Alcohol, - 
 
 
 
 6900 A F 
 
 2 '9 j > 
 
 Combustion in Chlorine. 
 
 Hydrogen, - 
 
 HC1 
 
 23000 F T 
 
 9 '66 X io 11 
 
 Potassium, - 
 
 KC1 
 
 2655 A 
 
 I'I2 
 
 ' Zinc, - 
 
 ZnCP 
 
 1529 A 
 
 6-42 x io 10 
 
 Iron, - - 
 
 Fe 2 Cl 
 
 1745 A 
 
 7 '33 
 
 Tin, - 
 
 SnCl 4 
 
 1079 A 
 
 4'53 
 
 Copper, 
 
 CuCl 2 
 
 961 A 
 
 4*04 
 
 The numbers in the last column are the products of 
 the numbers in the preceding column by 42 millions. 
 
 The authorities for these determinations are indicated 
 by the initial letters A (Andrews), F (Favre and Silber- 
 mann), T (Thomsen). Where two initial letters are 
 given, the number adopted is intermediate between those 
 obtained by the two experimenters. 
 
ix.] HEAT. 113 
 
 141. Difference between the two specific heats of a gas. 
 Let S-L denote the specific heat of a given gas at 
 
 constant pressure, 
 
 s 2 the specific heat at constant volume, ' 
 a the coefficient of expansion per degree, 
 v the volume of i gramme of the gas in cubic 
 
 centim. at pressure/ dynes per square centim. 
 When a gramme of the gas is raised from o to i 
 at the constant pressure /, the heat taken in is $i t the 
 increase of volume is az>, , and the work done against 
 external resistance is <wp (ergs). This work is the 
 equivalent of the difference between s 1 and s z that is, we 
 have 
 
 s 1 - s 2 = y^, where J - 4-2 x IO T . 
 
 For dry air at o the value of vp is 7*838 x io 8 , and a 
 is "003665. Hence we find si s z = '0684. The value of 
 Ji, according to Regnault, is '2375. Hence the value of 
 s 2 is '1691. 
 
 The value of -^ % , or ^-, for dry air at o and a 
 
 megadyne per square centim. is 
 
 *lJi= 26*4 = 8726x10-*; 
 
 v 783-8 Y 
 
 and this is also the value of 2 for any other gas (at 
 
 the same temperature and pressure) which has the same 
 coefficient of expansion. 
 
 142. Change of freezing point due to change si pressure. 
 Let the volume of the substance in the liquid state be 
 
 to its volume in the solid state, as i to i + e. ,. 
 
 H 
 
ii4 UNITS AND PHYSICAL CONSTANTS. [CHAP. 
 
 When unit volume in the liquid state solidifies under 
 pressure P+/, the work done by the substance is, the 
 product of P +p by the increase of volume e, and is there- 
 fore ~Pe+_pe. 
 
 If it afterwards liquefies under pressure P, the work 
 done against the resistance of the substance is ~Pe ; and if 
 the pressure be now increased to P+/, the substance 
 will be in the same state as at first. 
 
 Let T be the freezing temperature at pressure P, 
 T + 1 the freezing temperature at pressure P +/, 
 / the latent heat of liquefaction, 
 d the density of the liquid. 
 
 Then d is the mass of the substance, and Id is the heat 
 taken in at the temperature of melting T. Hence, by 
 thermodynamic principles, the heat converted into me- 
 chanical effect in the cycle of operations is 
 
 T+2 73 
 But the mechanical effect is^e. Hence we have 
 
 , 
 
 m J 
 
 t_ <-(T + 2 73 ) . , > 
 
 J* JU 
 
 - - is the lowering of the freezing-point for an additional 
 / 
 
 pressure of a dyne per square centim. ; and x io 6 
 
 will be the lowering of the freezing-point for each addi- 
 tional atmosphere of io 6 dynes per square centim. 
 
ix.] HEAT. 115 
 
 For water we have 
 
 e = -087, /= 79-25, T = o, d= i 
 
 / K "087 x 27^? 
 
 x io b = - ^ = '00714. 
 
 / 42 x 79-25 
 
 Formula (3) shows that is opposite in sign to e. 
 P 
 
 Hence the freezing point will be raised by pressure if the 
 substance contracts in solidifying. 
 
 143. Change of temperature produced by adiabatic com- 
 pression of a fluid ; that is, by compression under such 
 circumstances that no heat enters or leaves the fluid. 
 
 Let a cubic centim. of fluid at the initial temperature 
 t C. and pressure p dynes per square centim. be sub- 
 jected to the following cycle of four operations : 
 
 1. Increase of pressure, adiabatically, from p to/ + TT, 
 TT being small. 
 
 2. Addition of heat, at constant pressure / + TT, till the 
 temperature rises by the amount dt. 
 
 3. Diminution of pressure, adiabatically, from p + TT 
 to/. 
 
 4. Subtraction of heat, at constant pressure/, till the 
 temperature falls to t. 
 
 Let T denote the increase of temperature and v the 
 diminution of volume in (i) ; and let* denote the expan- 
 sion per degree at constant pressure. 
 
 Then, neglecting small quantities of the second order, 
 the changes of pressure, temperature, and volume are as 
 shown in the following tabular statement : 
 
UNITS AND PHYSICAL CONSTANTS. [CHAP. 
 
 Operation. 
 
 Pressure. 
 
 Temperature. 
 
 Change of 
 volume. 
 
 I 
 
 p iop + TT 
 
 t to t+T 
 
 V 
 
 2 
 
 P+TT 
 
 t + T to f + T + rt"/ edt 
 
 3 
 
 p+TT tO/ 
 
 t + T + dt to /4-rtV 
 
 V 
 
 4 
 
 p 
 
 t+dttot 
 
 -edt 
 
 The work done by the fluid in the operations (i) and 
 (3), taken together, is zero. 
 
 The work done by the fluid in the operations (2) and- 
 (4), taken together, is iredt. 
 
 The heat taken in by the fluid in (2) is, as far as small 
 quantities of the first order are concerned, equal to that 
 given out in (4), and is Cdt, C denoting the thermal 
 capacity of a cubic centim. of the substance at constant 
 pressure. 
 
 If T denote the absolute temperature, or 273 + /, the 
 
 heat converted into mechanical effects is Cdt ; and this 
 
 must be equal to 
 
 rr 
 
 . We have, therefore, I C = ~, 
 
 or T = ^ , where T denotes the increase of temperature 
 
 J^ 
 produced by the increase TT of pressure. 
 
 144. Resilience as affected by heat of compression. 
 The expansion due to the increase of temperature T, 
 
 r-p 9 
 
 above calculated, is re ; that is, - ; and the ratio of 
 
 J^ 
 
 this expansion to the contraction ^, which would be pro- 
 
 1 
 
 duced at constant temperature (E denoting the resilience 
 
ix.] HEAT. 117 
 
 ETV 2 
 of volume at constant temperature), is : i. Putting 
 
 J^ 
 m for -^ ^-, the resilience for adiabatic compression will be 
 
 -p 
 
 ; or, if m is small, E (i + m) : and this value is to 
 i - m 
 
 be used instead of E in calculating the change of volume 
 due to sudden compression. 
 
 The same formula expresses the value of Young's 
 modulus of resilience for sudden extension or compression 
 of a solid in one direction, E now denoting the value of 
 the modulus at constant temperature. 
 
 Examples. 
 
 For compression of water between 10 and 11 we 
 have 
 
 E = 2'i x io 10 , T 283, e '000092, C = i ; 
 
 hence 
 
 ET* 8 
 
 -jc = 
 
 For longitudinal extension of iron at 10 we have 
 E = 1*96 x io 12 , T = 283, e^ '0000122, C = '109 x 77 ; 
 hence 
 
 -~ 3, 
 
 Thus the heat of compression increases the volume- 
 resilience of water at this temperature by about \ per 
 cent., and the longitudinal resilience of iron by about 
 J per cent. 
 
n8 UNITS AND PHYSICAL CONSTANTS, [CHAP. 
 
 For dry air at o and a megadyne per square centim., 
 we have 
 
 E = io 6 , T = 273, e - JL, C = '2375 x -001276, 
 273 
 
 El> 2 i 
 
 m TTT- = '288, - 1-404. 
 
 JC ' i - m 
 
 145. Expansions of Volumes per degree Cent, (abridged 
 from Wattes ( Dictionary of Chemistry] Article Heat, pp. 
 67, 68, 71). 
 
 Glass, "00002 to "00003 
 
 Iron, - .- -,--- - '000035 -000044 
 
 Copper, - '- - - "000052 -000057 
 
 Platinum, .... -000026 '000029 
 
 Lead, -000084 '000089 
 
 Tin, '000058 '000069 
 
 Zinc, - - - - -oooo87 '000090 
 
 Gold, '000044 "000047 
 
 Brass, '000053 '000056 
 
 Silver, -000057 '000064 
 
 Steel, -000032 -000042 
 
 Cast Iron, - - about '000033 
 
 These results are partly from direct observation, and 
 partly calculated from observed linear expansion. 
 
 Expansion of Mercury, according to Regnault ( Wattes 
 1 Dictionary] p. 56). 
 
 Te m p.=,. Volume at/. *%%&. 
 
 o - - i '000000 -00017905 
 
 io - > 1*001792 . '00017950 
 20 - - 1-003590 -0001800 i 
 
 30 -- - 1*005393 -00018051 
 
 50 - - ' I '009013 -00018152 
 
 70 - - 1-012655 -00018253 
 
 loo - - 1-018153 '00018405 
 
 The temperatures are by air-thermometer. 
 

 IX.] 
 
 HEAT. 
 
 119 
 
 Expansion of Alcohol and Ether, according to Kopp ( Watts s 
 t Dictionary] p. 62). 
 
 Temp. 
 
 o 
 O 
 
 10 
 20 
 
 30 
 40 
 
 Volume. 
 
 Alcohol. 
 
 Ether. 
 
 I'OOOO 
 
 I'OOOO 
 
 I -0105 
 
 I '0152 
 
 I-02I3 
 
 I '0312 
 
 I -0324 
 
 I '0483 
 
 I '0440 
 
 I '0667 
 
 UHI7EESIT7 
 
CHAPTER X. 
 
 MAGNETISM. 
 
 146. The unit magnetic /<?&, or the pole of unit strength, 
 is that which repels an equal pole at unit distance with 
 unit force. In the C.G.S. system it is the pole which 
 repels an equal pole, at the distance of i centimetre, with 
 a force of i dyne. 
 
 If P denote the strength of a pole, it will repel an equal 
 
 P 2 
 
 pole at the distance L with the force j^- Hence we have 
 
 the dimensional equations 
 
 P 2 L~ 2 = force = MLT~ 2 , P 2 -ML 3 T- J , P = M*L*T~ 1 ] 
 that is, the dimensions of a pole (or the dimensions of 
 strength of pole] are M^L^T- 1 . 
 
 147. The intensity of a magnetic field is the force which 
 a unit pole will experience when placed in it. Denoting 
 this intensity by I, the force on a pole P will be IP. 
 Hence 
 
 IP . force = MLT- 2 , 1 - MLT~ 2 . M^L^T = M'lT^T' 1 ; 
 
 that is, the dimensions vt field-intensity are M 2 L 2 T~ . 
 
 148. The moment of a magnet is the product of the 
 
MAGNETISM. 121 
 
 strength of either of its poles by the distance between 
 them. Its dimensions are therefore LP; that is, _M*L Z T~ 1 . 
 
 Or, more rigorously, the moment of a magnet is a 
 quantity which, when multiplied by the intensity of a 
 uniform - field, gives the couple which the magnet ex- 
 periences when held with its axis perpendicular to the 
 lines offeree in this field. It is therefore the quotient of 
 
 a couple ML 2 T~ 2 by a field-intensity M^L^T" 1 ; that 
 is, it is M^L^T" 1 as before. 
 
 149. If different portions be cut from a uniformly mag- 
 netized substance, their moments will be simply as their 
 volumes. Hence the intensity of magnetization of a 
 uniformly magnetized body is defined as the quotient of 
 its moment by its volume. But we have 
 
 . L~ 3 = M^L" -T" 1 . 
 
 volume 
 
 Hence intensity of magnetization has the same dimensions 
 as intensity otjield. When a magnetic substance (whether 
 paramagnetic or diamagnetic) is placed in a magnetic 
 field, it is magnetized by induction ; and each substance 
 has its own specific coefficient of magnetic induction (con- 
 stant, or nearly so, when -the field is not excessively 
 intense), which expresses the ratio of the intensity of the 
 induced magnetization to the intensity of the field. For 
 paramagnetic substances (such as iron) this coefficient is 
 positive ; for diamagnetic substances (such as bismuth) it 
 is negative ; that is to say, the induced polarity is reversed 
 end for end as compared with that of a paramagnetic sub- 
 stance placed in the same field. 
 
122 UNITS AND PHYSICAL CONSTANTS. [CHAP. 
 
 150. The work required to move a pole P from one 
 point to another is the product of P by the difference of 
 the magnetic potentials of the two points. Hence the 
 dimensions of magnetic potential are 
 
 = M'L'T" 1 . 
 
 Examples. 
 
 i. To find the multiplier for reducing magnetic 
 intensities from the foot-grain-second system to the C.G.S. 
 system. 
 
 The dimensions of the unit of intensity are M^Lr^T" 1 . 
 In the present case we have M = -0648, L = 30-48, T = i, 
 since a grain is '0648 gramme, and a foot is 30*48 centim. 
 
 Hence M'lT- 1 = A - -0461 1 ; that is, the foot- 
 
 v 3'4 8 
 
 grain-second unit of intensity is denoted by the number 
 0461 1 in the C.G.S. system. This number is accordingly 
 the required multiplier. 
 
 2. To find the multiplier for reducing intensities from 
 the millimetre-milligramme-second system to the C.G.S. 
 system, we have 
 
 . 
 
 IOOO 10 IOOO 10 
 
 Hence is the required multiplier. 
 10 
 
 3. Gauss states (Taylor's ' Scientific Memoirs,' vol. ii., 
 p. 225) that the magnetic moment of a steel bar-magnet, 
 of one pound weight, was found by him to-be 100877000 
 millimetre-milligramme-second units. Find its moment 
 in C.G.S. units. 
 
x.J MAGNETISM. 123 
 
 Here the value of the unit moment employed is, in 
 terms of C.G.S. units, M^IJT- 1 , where M is io~ 3 , L is 
 lo" 1 , and T is i ; that is, its value is io~* . io~* = io~ 4 . 
 Hence the moment of the bar is 100877 C.G.S. units. 
 
 4. Find the mean intensity of magnetization of the bar, 
 assuming its specific gravity to be 7*85, and assuming that 
 the pound mentioned in the question is the pound avoir- 
 dupois of 453-6 grammes. 
 
 Its mass in grammes, divided by its density, will be 
 its volume in cubic centimetres ; hence we have 
 
 53_ 57 -78 = volume of bar. 
 7'5 
 
 T , .. f *: * moment 10088 ^ 
 
 Intensity of magnetization = = 1 74*6. 
 
 volume 5778 
 
 5. Kohlrausch states (' Physical Measurements/ p. 195, 
 English edition) that the maximum of permanent mag- 
 netism which very thin rods can retain is about 1000 
 millimetre-milligramme-second units of moment for each 
 milligramme of steel. Find the corresponding moment 
 per gramme in C.G.S. units, and the corresponding in- 
 tensity of magnetization. 
 
 For the moment of a milligramme we have 1000 
 x io~ 4 = io -1 . 
 
 For the volume of a milligramme we have (7'8s)~ l 
 x io~ 3 , taking 7^85 as the density of steel. 
 
 Hence the moment per gramme is lo" 1 x io 3 = 100, 
 and the intensity of magnetization is 100 x 7-85 = 785. 
 
 6. The maximum intensity of magnetization for speci- 
 mens of iron, steel, nickel, and cobalt has been deter- 
 mined by Professor Rowland ('Phil. Mag.,' 1873, vol. 
 xlvi., p. 157, and November, 1874) that is to say, the 
 
i2 4 UNITS AND PHYSICAL CONSTANTS. [CHAP: 
 
 limit to which their intensities of magnetization would 
 approach, if they were employed as the cores of electro- 
 magnets, and the strength of current and number of con- 
 volutions of the coil were indefinitely increased. Professor 
 Rowland's fundamental units are the metre, gramme, 
 
 and second ; hence his unit of intensity is of the C.G.S. 
 
 10 
 
 unit. His values, reduced to C.G.S. units, are 
 
 At I2C. At 220C. 
 
 Iron and Steel, " * - - 1390 1360 
 
 Nickel, - - - - - 494 380 
 
 Cobalt, - ... -. - - 800 (?) 
 
 7. Gauss states (loc. tit.) that the magnetic moment of 
 the earth, in millimetre-milligramme-second measure, is 
 
 3-3092 R 3 , 
 
 R denoting the earth's radius in millimetres. Reduce 
 this value to C.G.S. 
 
 Since R 3 is of the dimensions of volume, the other 
 factor, 3-3092, must be of the dimensions of intensity. 
 Hence, employing the reducing factor lo" 1 above found, 
 we have '33092 as the corresponding factor for C.G.S. 
 measure ; and the moment of the earth will be 
 
 33092 R 3 , 
 
 R denoting the earth's radius in centimetres that is 
 6-37 x io 8 . 
 
 We have 
 
 33092 X (6'37 X Io8) 3 = 8-55 X I0 
 
 for the earth's magnetic moment in C.G.S. units. 
 
 8. From the above result, deduce the intensity of mag- 
 netization of the earth regarded as a uniformly magnetized 
 body. 
 
x.] MAGNETISM. 125 
 
 We have 
 
 . L . L moment 8'^ x io 25 
 intensity = = J -^ - - = "0790. 
 volume 1*083 x icr' 
 
 This is about of the intensity of magnetization of 
 
 2 2OO 
 
 Gauss's pound magnet; so that 2*2 cubic decimetres of 
 earth would be equivalent to i cubic centim. of strongly 
 magnetized steel, if the observed effects of terrestrial mag- 
 netism were due to uniform magnetization of the earth's 
 substance. 
 
 9. Gauss, in his papers on terrestrial magnetism, em- 
 ploys two different units of intensity, and makes mention 
 of a third as " the unit in common use." The relation 
 between them is pointed out in the passage above referred 
 to. The total intensity at Gottingen, for the i9th of 
 July, 1834, was 4*7414 when expressed in terms of one 
 of these units the millimetre-milligramme-second unit ; 
 was 1357 when expressed in terms of the other unit em- 
 ployed by Gauss, and 1*357 in terms of the "unit in 
 common use." In C.G.S. measure it would be "47414. 
 
 151. A first approximation to the distribution of mag- 
 netic force over the earth's surface is obtained by assuming 
 the earth to be uniformly magnetized, or, what is mathe- 
 matically equivalent to this, by assuming the observed 
 effects to be due to a small magnet at the earth's centre. 
 The moment of the earth on the former supposition, or 
 the moment of the small magnet on the latter, must be 
 
 33092 R 3 , 
 
 R denoting the earth's radius in centims. The magnetic 
 poles, on these suppositions, must be placed at 
 77 5 of. north lat, 296 29' east long., 
 and at 77 50' south lat, 116 29' east long. 
 
126 UNITS AND PHYSICAL CONSTANTS. [CHAP. 
 
 The intensity of the horizontal component of terrestrial 
 magnetism, at a place distant A from either of these 
 poles, will be 
 
 33092 sin A 
 the intensity of the vertical component will be 
 
 66184 cos A; 
 
 and the tangent of the dip will be 
 2 cotan A. 
 
 The magnetic potential, on the same suppositions, 
 will be 
 
 33092 R cos A; 
 
 see Maxwell, ' Electricity and Magnetism,' vol. ii., p. 8. 
 Gauss's approximate expression for the potential and 
 intensity at an arbitrary point on the earth's surface con- 
 sists of four successive approximations, of which this is 
 the first. 
 
 152. The mean value of the intensity of horizontal mag- 
 netic force at Greenwich was 
 
 1716 in the year 1848, 
 1776 1867; 
 
 and its rate of increase in successive years is sensibly 
 uniform. 
 
 The place of greatest horizontal intensity is in lat. o, 
 long. 259 E., where the value is '3733. 
 
 In 1843 the dip at Greenwich was about 69 i 7 ; it has 
 diminished, with a rate continually accelerating, till in 
 1868 it was 67 56'. The total intensity, as computed 
 from the dip and horizontal intensity, was 
 
 4791 in the year 1848, 
 4740 1866. 
 
x.] MAGNETISM. 127 
 
 The place of greatest total intensity is in South Victoria, 
 about 70 S., 160 E., where its value is 7898. 
 
 The place of least total intensity is near St. Helena, in 
 lat. 16 S., long. 355 E., where its value is -2828. 
 
 We have extracted these data from 'Airy on Magnetism/ 
 PP- 74, 93, 94, 97, and 98. 
 
 152*. The following mean values of the magnetic 
 elements at Greenwich have been kindly furnished by 
 the Astronomer Royal : 
 
 West declination, - - 19 12"! -(t- 1876) x 7"38. 
 Horizontal force, - - 0-1797 + (/- 1876) x '00027. 
 Dip, - - - - 674o'-3-(;'-i876)x2'-04. 
 
 Vertical force, - - - 0-4375 -(*- 1876) x '00008. 
 = Horizontal force x tan. dip. 
 
 Each of these formulae gives the mean of the entire 
 year /. 
 
 Magneto-Optic Rotation. 
 
 153. According to J. E. H. Gordon (' Phil. Trans.,' 
 1877), the rotation of the plane of polarization between 
 points, one centimetre apart, whose magnetic potentials 
 (in C.G.S. measure) differ by unity, is (in circular 
 measure) 3-04763 x io~ 5 
 
 in bisulphide of carbon, for the principal green thallium 
 ray, and is 4'496^x io~ 6 
 
 in distilled water, for white light. 
 
 Mr. Gordon infers from Becquerel's experiments ('Comp. 
 Rend.' March 31, 1879) that it is about 
 
 6 x io~ 9 
 for coal gas. 
 
128 
 
 CHAPTER XI. 
 
 ELECTRICITY. " * " 
 
 Electrostatics. 
 
 154. IF q denote the numerical value of a quantity of 
 electricity in electrostatic measure, the mutual force be- 
 tween two equal quantities q at the mutual distance / will 
 
 ff 2 
 
 be j r In the C.G.S. system the electrostatic unit of 
 
 electricity is accordingly that quantity which would repel 
 an equal quantity at the distance of i centim. with a force 
 of i dyne. 
 
 Since the dimensions of force are -, we have, as 
 regards dimensions, 
 
 f ml ? ^M/- 1 
 
 72 = 7?, whence ? = -p,g = in I t . 
 
 155. The work done in raising a quantity of electricity 
 q through a difference of potential v is qv. 
 
 Hence we have 
 
 In the C.G.S. system the unit difference of potential is 
 that difference through which a unit of electricity must be 
 raised that the work done may be i erg. 
 
ELECTRICITY. 129 
 
 Or, we may define potential as the quotient of quantity 
 of electricity by distance. This gives 
 
 v = ;A V 1 . /- 1 = 0/V*/- 1 , as before. 
 
 156. In the C.G.S. system the unit of potential is the 
 potential due to unit quantity at the distance of i centim. 
 
 The capacity of a conductor is the quotient of the 
 quantity of electricity with which it is charged by the 
 potential which this charge produces in it. Hence we 
 have 
 
 capacity = = m*t~ l . m~t = /. 
 v 
 
 The same conclusion might have been deduced from 
 the fact that the capacity of an isolated spherical conductor 
 is equal (in numerical value) to its radius. The C.G.S. 
 unit of capacity is the capacity of an isolated sphere of 
 i centim. radius. 
 
 157. The numerical value of a current (or the strength 
 of a current) is the quantity of electricity that passes in 
 unit time. 
 
 Hence the dimensions of current are-^ ; that is, nflflt~*. 
 
 The C.G.S. unit of current is that current which con- 
 veys the above defined unit of quantity in i second. 
 
 158. The dimensions of resistance can be deduced from 
 Ohm's law, which asserts that the resistance of a wire is 
 the quotient of the difference of potential of its two ends, 
 by the current which passes through it. Hence we have 
 
 resistance = m^r 1 . m~*r*t* = /~V. 
 
 Or, the resistance of a conductor is equal to the time 
 required for the passage of a unit of electricity through it, 
 
130 UNITS AND PHYSICAL CONSTANTS. [CHAP. 
 
 when unit difference of potential is maintained between 
 its ends. Hence 
 
 resistance = foe x potential =t . m \ft^. ,,HH; = /-'/. 
 quantity 
 
 159. As the force upon a quantity q of electricity, in a 
 field of electrical force of intensity /, is iq, we have 
 
 The quantity here denoted by / is commonly called the 
 *' electrical force at a point." 
 
 Electromagnetics. 
 
 1 60. A current C (or a current of strength C) flowing 
 along a circular arc, produces at the centre of the circle an 
 intensity of magnetic field equal to C multiplied by length 
 of arc divided by square of radius. Hence C divided by 
 a length is equal to a field-intensity, or 
 
 C - length x intensity = L . M^L^T' 1 = lAM*!*- 1 . 
 
 1 6 1. The quantity of electricity Q conveyed by a cur- 
 rent is the product of the current by the time that it lasts. 
 The dimensions of Q are therefore L*M^ 
 
 162. The work done in urging a quantity Q through a 
 circuit, by an electromotive force E, is EQ ; and the work 
 done in urging a quantity Q through a conductor, by 
 means of a difference of potential E between its ends, is 
 EQ. Hence the dimensions of electromotive force, and 
 also the dimensions of potential, are ML 2 T~ 2 . L~*M~^, or 
 
XL] 
 
 ELECTRICITY. 
 
 163. The capacity of a conductor is the quotient of 
 quantity of electricity by potential. Its dimensions are 
 therefore 
 
 M*L*. M-^L-^T 2 ; that is, L^T 2 . 
 p 
 
 164. Resistance is ; its dimensions are therefore 
 
 O 
 
 M*L*T- f . M-*L-*T ; that is, LT- 1 . 
 
 165. The following table exhibits the dimensions of 
 each electrical element in the two systems, together with 
 their ratios : 
 
 
 Dimensions in 
 electrostatic 
 system. 
 
 Dimensions in 
 electromagnetic 
 system. 
 
 Dimensions in E.S. 
 
 Dimensions in E. M. 
 
 Quantity, - - 
 
 M^V 1 
 
 M*L* 
 
 LT- 1 
 
 Current, - - 
 
 M*L^T- 2 
 
 M^L^T- 1 
 
 LT- 1 
 
 Capacity, - - 
 
 L 
 
 L - 1T2 
 
 L 2 T~ 2 
 
 Potential and ) 
 electronic- > 
 tive force, ) 
 
 *Vr* 
 
 ^ 
 
 L-T 
 
 Resistance - - 
 
 L-T 
 
 LT- 1 
 
 L- 2 T 2 
 
 1 66. The heat generated in time T by the passage of a 
 current C through a wire of resistance R (when no other 
 
 C 2 RT 
 
 work is done by the current in the wire) is gramme 
 
 degrees, J denoting 4-2 x io 7 ; and this is true whether C 
 and R are expressed in electromagnetic or in electrostatic 
 units. 
 
132 UNITS AND PHYSICAL CONSTANTS. [CHAP. 
 
 Ratios of the tivo sets of Electric Units. 
 
 167. Let us consider any general system of units 
 based on 
 
 a unit of length equal to L centims., 
 a unit of mass equal to M grammes, 
 a unit of time equal to T seconds. 
 
 Then we shall have the electrostatic unit of quantity 
 equal to 
 
 MFL^T- 1 C.G.S. electrostatic units of quantity, 
 and the electromagnetic unit of quantity equal to 
 M^L2 C.G.S. electromagnetic units of quantity. 
 
 It is possible so to select L and T that the electrostatic 
 unit of quantity shall be equal to the electromagnetic 
 unit. We shall then have (dividing out by M^l3) 
 
 LT" 1 C.G.S. electrostatic units 
 
 = i C.G.S. electromagnetic unit ; 
 or the ratio of the C.G.S. electromagnetic unit to the 
 
 C.G.S. electrostatic unit is -. 
 
 Now is clearly the value in centims. per second of 
 
 that velocity which would be denoted by unity in the 
 new system. This is a definite concrete velocity; and its 
 numerical value will always be equal to the ratio of the 
 electromagnetic to the electrostatic unit of quantity, 
 whatever units of length, mass, and time are employed. 
 
 1 68. It will be observed that the ratio of the two units 
 of quantity is the inverse ratio of their dimensions ; and 
 the same can be proved in the same way of the other 
 
XL] ELECTRICITY. 133 
 
 four electrical elements. The last column of the above 
 table shows that M does not enter into any of the ratios, 
 and that L and T enter with equal and opposite indices, 
 showing that all the ratios depend only on the velocity 
 L 
 T' 
 
 Thus, if the concrete velocity ^ be a velocity of v 
 
 centims. per second, the following relations will subsist 
 
 between the C.G.S. units : 
 
 i electromagnetic unit of quantity = v electrostatic units. 
 
 i current =v 
 
 i capacity = e; 2 
 
 v electromagnetic units of potential = i electrostatic unit. 
 
 v 1 ,, resistance = i 
 
 169. Weber and Kohlrausch, by an experimental 
 comparison of the two units of quantity, determined the 
 value of v to be 
 
 3 - io74 x i o 10 centims. per second. 
 Sir W. Thomson, by an experimental comparison of 
 the two units of potential, determined the value of v 
 to be 
 
 2-825 x io 10 - 
 
 Professor Clerk Maxwell,' by an experiment in which 
 an electrostatic attraction was balanced by an electro- 
 dynamic repulsion, determined the value of v to be 
 
 2-88 x io 10 . 
 
 Professors Ayrton and Perry, by measuring the capacity 
 of an air-condenser both electromagnetically and statically 
 (' Nature,' Aug. 29, 1878, p. 470), obtained the value 
 2-98 x io 10 . 
 
134 UNITS AND PHYSICAL CONSTANTS. [CHAP. 
 
 All these determinations differ but little from the 
 velocity of light in vacuo, which, according to Foucault's 
 determination, is 
 
 2-98 x io 10 , 
 
 and according to the recent experiments of Cornu (see 
 'Nature,' February 4, 1875, p. 274) is 
 
 3*004 x io 10 . 
 We shall adopt the round number 
 
 as the value of v. 
 
 170. The dimensions of the electric units are rather 
 simpler when expressed in terms of length, density, and 
 time. 
 
 Putting D for density, we have M = L 3 D. Making 
 this substitution for M, in the expressions above obtained 
 ( 165), we have the following results: 
 
 Electrostatic. Electromagnetic. 
 
 Quantity, 
 Current, - 
 Capacity, 
 Potential, - 
 Resistance, - - - L~ J T - - - LT- 1 
 
 It will be noted that the exponents of L and T in these 
 expressions are free from fractions. 
 
 Specific Inductive Capacity. 
 
 171. The specific inductive capacity of an insulating 
 substance is the ratio of the capacity of a condenser in 
 which this substance is the dielectric to that of a con- 
 denser in other respects equal and similar in which air is 
 
XL] 
 
 ELECTRICITY. 
 
 
 the dielectric. It is of zero dimensions, and its value 
 exceeds unity for all solid and liquid insulators. 
 
 Messrs. Gibson and Barclay, by experiments performed 
 in Sir W.Thomson's laboratory ('Phil. Trans.,' 1 87 1, p. 5 7 3), 
 determined the specific inductive capacity of solid paraffin 
 to be i -97 7. 
 
 Dr. J. Hopkinson (' Phil. Trans./ 1877, p. 23) gives the 
 following results of his experiments on different kinds of 
 flint glass : 
 
 Kind of 
 Flint Glass. 
 
 Density. 
 
 Specific 
 Inductive 
 Capacity. 
 
 Quotient 
 by 
 Density. 
 
 Index of 
 Refraction 
 for D line. 
 
 Very light, 
 
 2-8 7 
 
 6-57 
 
 2-29 
 
 1*541 
 
 Light, 
 
 3 -2 
 
 6-85 
 
 2-14 
 
 I'S74 
 
 Dense, - 
 
 3'66 
 
 7 '4 
 
 2'02 
 
 1-622 
 
 Double extra ) 
 dense, - - \ 
 
 4'5 
 
 IO'I 
 
 2-25 
 
 1710 
 
 Wiillner ( ' Sitzungsber. konigl. bayer. Akad.,' March 
 1877) finds the following values of specific inductive 
 capacity : 
 
 Paraffin, - 1*96 Shellac, - 2*95 to 373 
 
 Ebonite, - 2-56 Sulphur, - 2 '88 to 3-21 
 
 Plate glass, - 6'io 
 
 Boltzmann ('Carl's Repertorium,' x., 92 165) finds 
 the following values : 
 
 Paraffin, - 2*32 Colophonium, - 2*55 
 
 Ebonite, - 3-15 Sulphur, - - 3-84 
 
 Schiller ( 'Pogg. Ann.,' clii., 535, 1874) finds : 
 
 Paraffin, - 1-85 to 2 '47 Caoutchouc, - - 2*12 to 2'^ 
 Ebonite, - 2'2i to 276 Do., vulcanized, 2*69 to 2-94 
 
 Plate glass, 5*83 to 6-34 
 
136 UNITS AND PHYSICAL CONSTANTS. [CHAP. 
 
 Silow ('Pogg. Ann.,' clvi. and clviii.) finds the following 
 values for liquids : 
 
 Oil of turpentine, . ' .- 2-155 to 2*221 
 Benzene, - - - - - - 2*199 
 
 Petroleum, - - - - - - 2*039 to 2*071 
 
 Boltzmann ('Wien. Akad. Ber.' (2), Ixx., 342, 1874) 
 finds for sulphur in directions parallel to the three prin- 
 cipal axes, the values 
 
 4773- 3'97o- 3-811. 
 
 J. E. H. Gordon (Proc. Roy. Soc.,' 1878-79), by a 
 method in which the electrification is reversed some 
 12,000 times per second, finds the following values : 
 
 Glass, double extra dense flint, - - - - 3*1639 
 
 ,, extra dense flint, "3'536 
 
 ,, light flint, - . ... 3-0129 
 
 ,, hard crown, 3*1079 
 
 ,, common plate, mean of two specimens, - - 3*2431 
 
 Ebonite, mean of four specimens, .... 2*2^38 
 
 Gutta percha, best quality, 2*4625 
 
 Chatterton's compound, ------ 2*5474 
 
 Indian Rubber, black, 2.2200 
 
 ,, vulcanized, - 2*4969 
 
 Solid paraffin, mean of six specimens, ) -0016 
 
 sp. gr. at ii C, 0*9109, ! 
 
 Shellac, - - -"*--.--- - 2-7470 
 
 Sulphur, - - * ' -- - - 2*5793 
 
 Bisulphide of carbon (observation doubtful), - - I "8096 
 
 Baluta, mean of three specimens, - - - - 2 '4849 
 
 According to Professor Clerk Maxwell's electro-mag- 
 netic theory of light, the square root of the specific 
 inductive capacity is equal to the index of refraction. 
 
XI.] 
 
 ELECTRICITY. 
 
 137 
 
 The following comparison is given in Mr. Gordon's 
 paper : 
 
 Square root of Nearest value Ray to which 
 pecific inductive of index of this value 
 
 specific inductive 
 capacity. 
 
 17783 
 
 refraction. belongs. 
 
 7460) Band in X - 
 '* ' treme violet 
 
 Glass, double extra dense 
 
 flint, 
 Glass, extra dense flint, 
 
 ,, light flint, - 
 
 ,, hard crown, 
 
 plate, 
 
 Paraffin, 
 
 Sulphur, - - - 
 Bisulphide of Carbon, 
 
 172. Professors Ayrton and Perry have found the 
 following values of the specific inductive capacities of 
 gases, air being taken as the standard : 
 
 7474 
 7343 
 7629 
 
 i -6757 > 
 i-5ii3 
 i -59207 
 
 in magne- 
 sium spark 
 spectrum. 
 
 8009 
 
 1-543 
 
 
 
 ( R 
 
 ays of infinite 
 
 4119 
 
 1-422 j 
 
 wave length. 
 
 6060 
 
 2-115 
 
 
 3456 
 
 1-6114 
 
 
 Air, . --. 
 Vacuum, - - - 
 Carbonic acid, - 
 
 i -oooo 
 0-9985 
 i -0008 
 
 Hydrogen, - - 0*9998 
 Coal gas, - - - I '0004 
 Sulphurous acid, i '0037 
 
 Ohm, Volt, &c. 
 
 173. The unit of resistance employed by practical 
 electricians is the Ohm, which is defined by certain 
 standard coils, each of which is to be taken at a stated 
 temperature. The resistance of each of these coils at its 
 proper temperature is intended to be 
 
 io 9 C.G.S. electromagnetic units of resistance. . 
 
 We shall therefore speak of io 9 C.G.S. units as the 
 theoretical ohm. The practical ohm was constructed 
 under the direction of a Committee of the British 
 Association, its construction being based upon experi- 
 ments in which the resistance of a certain coil of wire 
 was determined in electromagnetic measure. 
 
138 UNITS AND PHYSICAL CONSTANTS. [CHAP. 
 
 F. Kohlrausch has since conducted experiments (see 
 'Phil. Mag.,' 1874, vol. xlvii.) from which he infers that the 
 practical ohm (as defined by the standard coils) is 
 1-0196 x i o 9 C.G.S. units. 
 
 Still more recently Lorenz (' Pogg. Ann./ 1873, vol. 
 cxlix., p. 251) has made a determination of the absolute 
 value of Siemens' unit (the resistance at o C. of a column 
 of pure mercury, a metre long and a square millimetre in 
 section), and has found it to be '9337 x io 9 C.G.S. units. 
 Assuming with Kohlrausch that the practical ohm is 
 equal to i '0493 Siemens, it would follow from Lorenz's 
 determination that the practical ohm is 
 '9797 x I0 C.G.S. units. 
 
 H. F. Weber ('Phil. Mag./ March, 1878) finds for 
 Siemens' unit the following absolute values : 
 
 C From 1 8 series of experiments in which the 
 '9545 x io 9 < variable currents generated by magneto- 
 
 ( induction were employed. 
 
 ( From 24 series of experiments in which the 
 '9554 x io 9 ^ variable currents called forth by sudden 
 
 ( voltaic induction were employed. 
 
 C From 5 series of experiments in which the 
 '955 OXI 9 ^ heat-production of stationary galvanic 
 
 ( currents was used. 
 
 Employing the general mean "9550 x io 9 , and taking 
 the practical ohm as 1*0493 Siemens, we have for the 
 value of the practical ohm 
 
 ro493 x '9550 x io 9 = 1*002 x io 9 C.G.S. units. 
 174. The practical unit of electromotive force is the 
 Volt. Its theoretical value is 
 
 i Volt = io 8 C.G.S. units of potential. 
 
XL] ELECTRICITY. 139- 
 
 The practical unit of quantity of electricity is the quan- 
 tity conveyed in one second by a current due to an 
 electromotive force of i volt working through a resistance 
 of i ohm. It is called the Weber. Its theoretical 
 
 value is j 
 
 i weber= of C.G.S. unit of quantity. 
 
 TO 
 
 The' practical unit of current is the current due to an 
 electromotive force of i volt working through a resistance 
 of i ohm. It is called a current of i weber per second \ 
 Its theoretical value is 
 
 i weber per second = T of C.G.S. unit of current. 
 10 
 
 In the "Testing Instructions" of the Indian Telegraph 
 Department it is called the (Erstedt. 
 
 . The capacity of a condenser which holds i weber 
 when charged to a potential of i volt is called a farad, 
 Its theoretical value is 
 
 i farad = 1^ = io~ 9 C.G.S. units of capacity. 
 io 8 
 
 As the farad is much too large for practical convenience,, 
 its millionth part, called the microfarad, is practically 
 employed ; and standard condensers are in use which are 
 guaranteed to be of this Capacity. Their theoretical 
 value is 
 
 i microfarad = io~ 15 C.G.S. unit of capacity. 
 
 175. By way of assisting the memory, it is useful to 
 remark that the numerical value of the ohm is the same 
 as the numerical value of a velocity of one earth-quadrant 
 per second, since the length of a quadrant of the meridian 
 is io 9 centims. This equality will subsist whatever funda- 
 
I 4 
 
 UNITS AND PHYSICAL CONSTANTS. [CHAP. 
 
 mental units are employed, since the dimensions of 
 resistance are the same as the dimensions of velocity. 
 
 No special names have as yet been assigned to any 
 electrostatic units. 
 
 Electric Spark. 
 
 176. Sir W. Thomson has observed the length of spark 
 between two parallel conducting surfaces maintained at 
 known differences of potential, and has computed the 
 corresponding intensities of electric force by dividing (in 
 each case) the difference of potential by the distance, 
 since the variation of potential per unit distance measured 
 in any direction is always equal to the intensity of the 
 force in that direction. His results, as given on page 258 
 of " Papers on Electrostatics and Magnetism," form the 
 first two columns of the following table. 
 
 Distance 
 between 
 surfaces. 
 
 Intensity of 
 force in 
 electrostatic 
 units. 
 
 Difference of potential between surfaces. 
 
 In electrostatic 
 
 In electromagnetic 
 
 
 
 units. 
 
 units. 
 
 0086 
 
 26 7 -I 
 
 2-30 
 
 6'90X I0 10 
 
 0127 
 
 25 7 -0 
 
 3-26 
 
 97 8 
 
 0127 
 
 262*2 
 
 3'33 
 
 9-99 
 
 OlpO 
 
 224-2 
 
 4-26 
 
 1278 
 
 028l 
 
 2OO'6 
 
 5^4 
 
 16-92 
 
 0408 
 
 i5i'5 
 
 6-18 
 
 l8'54 
 
 0563 
 
 144-1 
 
 8-n 
 
 24'33 
 
 0584 
 
 139-6 
 
 8-15 
 
 24 '45 
 
 0688 
 
 140-8 
 
 9-69 
 
 29-07 
 
 0904 
 
 1 34 '9 
 
 12-20 
 
 36-60 
 
 1056 
 
 132-1 
 
 i3'95 
 
 4I-85 
 
 1325 
 
 131-0 
 
 17-36 
 
 52-08 
 
XL] 
 
 ELECTRICITY. 
 
 141 
 
 The numbers in the third column are the products of 
 those in the first and second. The numbers in the 
 fourth column are the products of those in the third by 
 3 x io 10 . 
 
 177. Dr. Warren De La Rue, and Dr. Hugo W. Miiller 
 ('Phil. Trans./ 1877) have measured the striking distance 
 between the terminals of a battery of chloride of silver 
 cells, the number of cells being sometimes as great as 
 1 1 ooo, and the electromotive force of each being 1-03 
 volt. Terminals of various forms were employed ; and 
 the results obtained with parallel planes as terminals have 
 been specially revised by Dr. De La Rue for the present 
 work. These revised results (which were obtained by 
 graphical projection of the actual observations on a larger 
 scale than that employed for the Paper in the Philosophi- 
 cal Transactions) are given below, together with the data 
 from which they were deduced : 
 
 DATA. 
 
 Striking 
 No of Cells 
 
 Distance. 
 
 
 In Inches. 
 
 In Centims. 
 
 
 | 
 
 
 1200 
 
 0'012 
 
 0*0305 
 
 2400 
 
 021 
 
 0533 
 
 3600 
 
 033 
 
 0838 
 
 4800 
 
 049 
 
 T245 
 
 5880 
 
 058 
 
 1473 
 
 6960 
 
 073 
 
 1854 
 
 8040 
 
 088 
 
 2236 
 
 9540 
 
 no 
 
 2794 
 
 1 1 OOO 
 
 133 
 
 3378 
 
 i 
 
 
142 UNITS AND PHYSICAL CONSTANTS. [CHAP. 
 
 DEDUCTIONS. 
 
 
 
 
 Intensity of 
 
 Force 
 
 Electromotive 
 Force in 
 
 Striking 
 Distance in 
 
 Volts per 
 
 InC.G.S. units. 
 
 Volts. 
 
 Centims. 
 
 v-'Cniiui, 
 
 
 
 
 
 
 Electromagnetic. 
 
 Electro- 
 static. 
 
 1000 
 
 0205 
 
 48770 
 
 4'88x io 12 
 
 I6 3 
 
 2OOO 
 
 0430 
 
 46500 
 
 4'65 
 
 !55 
 
 3000 
 
 0660 
 
 45450 
 
 4'55 
 
 152 
 
 4000 
 
 0914 
 
 43770 
 
 4-38 
 
 146 
 
 5000 
 
 1176 
 
 42510 
 
 4-25 
 
 142 
 
 6000 
 
 'H73 
 
 40740 
 
 4-07 
 
 136 
 
 7OOO 
 
 1800 
 
 38890 
 
 3-89 
 
 130 
 
 8000 
 
 2146 
 
 37280 
 
 373 
 
 124 
 
 9OOO 
 
 2495 
 
 36070 
 
 3-61 
 
 1 20 
 
 10000 
 
 2863 
 
 34920 
 
 3 '49 
 
 116 
 
 1 1000 
 
 3245 
 
 33900 
 
 3 '39 
 
 JI 3 
 
 11330 
 
 3378 
 
 33460 
 
 3 '35 
 
 112 
 
 178. The resistance of a wire (or more generally of 
 a prism or cylinder) of given material varies directly as 
 its length, and inversely as its cross section. It is there- 
 fore equal to 
 
 R length 
 
 section' 
 
 where R is a coefficient depending only on the material. 
 R is called the specific resistance of the material. Its 
 
 reciprocal is called the specific conductivity of the 
 R 
 
 material. 
 
 R is obviously the resistance between two opposite 
 faces of a unit cube of the substance. Hence in the 
 C.G.S. system it is the resistance between two opposite 
 faces of a cubic centim. (supposed to have the form 
 of a cube). 
 
XL] 
 
 ELECTRICITY. 
 
 143 
 
 The dimensions of specific resistance are resistance x 
 length ; that is, in electromagnetic measure, velocity x 
 length; that is, L^T" 1 . 
 
 RESISTANCE. 
 
 179. Table of Specific Resistances, in Electromagnetic 
 Measure (at o C. unless otherwise stated). 
 
 . 
 
 Specific resist- 
 ance. 
 
 Percentage 
 variation per 
 degree at 
 
 20 C. 
 
 Specific 
 gravity. 
 
 Silver, hard-drawn, - 
 
 1609 
 
 '377 
 
 IO'5O 
 
 Copper, ,, 
 Gold, 
 
 1642 
 2154 
 
 388 
 365 
 
 8'95 
 19-27 
 
 Lead, pressed, 
 Mercury, liquid, 
 
 19847 
 96146 
 
 387 
 072 
 
 II-39I 
 13*595 
 
 Gold 2, silver I, hard or / 
 annealed, - - \ 
 
 10988 
 
 06 5 
 
 I5-2I8 
 
 Selenium at 100 C.,crys- i 
 talline, - - \ 
 
 6 x io 13 
 
 I'OO 
 
 
 Water at 22 C, 
 
 7'i8x io 10 
 
 '47 
 
 
 , with '2percent. H 2 SO 4 
 
 4'47 , 
 
 "47 
 
 
 , 8-3 ,, 
 
 3'32Xio 9 
 
 653 
 
 
 , 20 ,, 
 
 i '44 . 
 
 799 
 
 
 , 35 
 
 1-26 
 
 1-259 
 
 
 , ,, 41 
 
 i '37 , 
 
 1-410 
 
 
 Sulphate of zinc and water ) 
 ZnSO 4 + 23H 2 O at 23 C. J 
 
 i -87 x io 10 
 
 
 
 Sulph. of copper and water ) 
 
 
 
 
 CuSO 4 + 45H 2 Oat22C. \ 
 
 V95 } 
 
 
 
 Glass at 200 C. 
 
 2-27X IO 16 
 
 
 
 250 
 
 i -39 x io 15 
 
 
 
 300 
 
 i -48 x ioi* 
 
 
 
 ,, 400 
 
 7'35x ioi3 
 
 
 
 Gutta percha at 24 C., 
 
 3'53xio 23 
 
 
 
 oC., - 
 
 7 x left 1 
 
 
 
 For the authorities for the above numbers see 
 
1 44 
 
 UNITS AND PHYSICAL CONSTANTS. [CHAP. 
 
 Maxwell, ' Electricity and Magnetism/ vol. i., last 
 chapter. 
 
 1 80. The following table of specific resistances of 
 metals at o C. is reduced from Table IX. in Jenkin's 
 Cantor Lectures. It is based on Matthiessen's experi- 
 ments. 
 
 Silver, annealed, 
 ,, hard-drawn, 
 
 Copper, annealed, 
 ,, hard-drawn, - 
 
 Gold, annealed, - 
 ,, hard-drawn, 
 
 Aluminium, annealed, - 
 
 Zinc, pressed, 
 
 Platinum, annealed, 
 
 Iron, annealed, - 
 
 Nickel, annealed, 
 
 Tin, pressed, ... 
 
 Lead, pressed 
 
 Antimony, pressed, 
 
 Bismuth, pressed, 
 
 Mercury, liquid, - 
 
 Alloy, 2 parts platinum, I part 
 silver, by weight, hard or 
 annealed, 
 
 German silver, hard orannealed, 
 
 Alloy, 2 parts gold, I silver, 
 by weight, hard or an- 
 nealed, .... 
 
 Specific 
 resistance. 
 
 Percentage 
 of variation 
 for a degree 
 
 
 at 20C. 
 
 1521 
 
 '377 
 
 1652 
 
 
 1615 
 
 388 
 
 1652 
 
 
 2081 
 
 365 
 
 2118 
 
 
 2946 
 
 
 5690 
 
 365 
 
 9158 
 
 
 9827 
 
 
 12600 
 
 
 13360 
 
 365 
 
 19850 
 
 387 
 
 35900 
 
 389 
 
 132650 
 
 '354 
 
 96190 
 
 072 
 
 2466 
 
 21170 
 
 10990 
 
 031 
 
 044 
 
 065 
 
 Resistances of Conductors of Telegraphic Cables per nautical 
 
 mile, at 24C., in electromagnetic measure. 
 Red Sea, 7'94Xio 9 
 
 Malta- Alexandria, mean, - 3*49 ,, 
 
 Persian Gulf, mean, 6^284 ,, 
 
 Second Atlantic, mean, - 4'272 ,, 
 
XL] ELECTRICITY. 145 
 
 1 8 1. The following approximate values of the specific 
 resistance of insulators after several minutes' electrification 
 are given (with their authorities) in a paper by Professors 
 Ayrton and Perry (' Proc. Royal Society, March 21, 
 1878 ') 'On the Viscosity of Dielectrics ' : 
 
 Specific Temperature, . , 
 Resistance. Centigrade. >nty ' 
 
 Mica, - 
 
 8-4 x io 22 20 
 
 Ayrton and Perry. 
 
 Gutta-percha, 
 
 4-5 x io 23 24 
 
 \ Standard adopted by 
 ( Latimer Clark. 
 
 Shellac, - 
 
 9-0 x io 24 28 
 
 Ayrton and Perry. 
 
 Hooper's material, 
 
 i *5 x io 25 24 
 
 Recent cable tests. 
 
 Ebonite, 
 
 2'8x IO 25 46 
 
 Ayrton and Perry. 
 
 Paraffin, - 
 
 3 '4 x io 25 46 
 
 Do. 
 
 Glass, - '- . 
 
 Not yet measured 
 
 with accuracy, but greater 
 
 than any of the above. 
 Air, - - Practically infinite. 
 
 182. The following approximate determinations of the 
 resistance of water and ice at different temperatures are 
 contained in a paper by Professors Ayrton and Perry, dated 
 March, 1877 ('Proc. Phys. Soc., London,' vol.ii.,p. 178). 
 
 Temp. Specific 
 
 Cent. Resistance. 
 
 -12 4 - - - 2'240x i 10 18 
 
 6'2 - - - I"O23x*IO 18 
 
 5 '02 - - 9'486jx io 17 
 
 - 3-5 6-428 x 
 
 - 3-0 - - - 5^93 x 
 
 2-46 - S- - 4-844 x 
 
 - 1-5 - - - 3-876 x 
 
 0'2 - - - 2*840 x 
 
 + 0-75 - _--' i-i88x 
 
 about +2-2 - - - 2-48 x io 16 
 
 + 4-0 - - - 9-1 x io 15 
 
 + 775 - - - 5'4 x io 1 * 
 
 + II'02 - - 3'4 x IO 14 
 
 The values in the original are given in megohms, and we 
 have assumed the megohm = io 15 C.G.S. units. 
 
 K 
 
146 UNITS AND PHYSICAL CONSTANTS. [CHAP. 
 
 Electromotive Force. 
 
 183. The electromotive force of a Darnell's cell was 
 found by Sir W. Thomson (p. 245 of Papers on Electricity 
 and Magnetism) to be 
 
 00374 electrostatic unit, 
 
 from observation of the attraction between two parallel 
 discs connected with the opposite poles of a Daniell's 
 battery. 
 
 As i electrostatic unit is 3 x io 10 electromagnetic units, 
 this is -00374 x 3 x io 10 =- 1-12 x io 8 electromagnetic units, 
 or 1*12 volt. 
 
 According to Latimer Clark's experimental determina- 
 tions (' Journ. Soc. Tel. Eng.,' January, 1873), the elec- 
 tromotive force of a Daniell's cell is i-ii x io 8 , and the 
 electromotive force of a Grove's cell is 1-97 x io 8 . 
 
 According to the determination of F. Kohlrausch 
 ( Pogg. Ann./ vol. cxli. [1870], and Erganz., vol. vi. 
 [1874] p. 35), the electromotive force of a Daniell's cell 
 is 1*138 x io 8 , and that of a Grove's cell 1*942 x io 8 . 
 
 The electromotive force of Latimer Clark's standard 
 cell ('Phil. Mag.,' June, 1872, and 'Phil. Trans.,' 1873) 
 is 1-457 x io 8 . 
 
 For theoretical determinations see 188. 
 
 184. Professors Ayrton and Perry have made deter- 
 minations of the electromotive forces called out by the 
 contacts, two and two, of a great number of substances 
 measured inductively. The method of experimenting is 
 described in the Proceedings of the Royal Society for 
 March 21, 1878. The following abstract of their latest 
 results was specially prepared for this work by Professor 
 Ayrton in January, 1879. 
 
XL] 
 
 ELECTRICITY. 
 
 jo amp aqj ye ajn^iaduiax sSeaaAy 
 
 I! |i 
 
 tf" 5 G 
 
 1 g ^8 
 
 ssujg 
 
 * * 
 
 aers without an asterisk were obtained directly by experiment, those with an asterisk by ca 
 1-known assumption that in a compound circuit of metals all at the same temperature, th 
 force. 
 >ers in a vertical column below the name of a substance are the differences of potential, in volts, 
 t and the substance in the same horizontal row as the number, the two substances being ir 
 >ositive to copper, the electromotive force of contact being 0*542 volts. 
 s were those of commerce, and therefore only commercially pure. 
 
 H 00 VO ' tx 00 IX C-^ 01 
 
 ;* p p rf 01 co MO oo O 
 II II! 
 
 01117 
 
 * # * * 
 oo T}- rh tx in co >* N 
 
 O ON *<T tO 01 VO ^T" 01 
 
 IN oo IN co H ;4- JT oo 
 
 ^ 
 
 VO O O O H H "^ ON 
 
 ON tO O M OO OO rh tx 
 P Cx VD 01 ON C1 O M vo 
 
 M *i 
 
 uij. 
 
 t-o ^ co ^ Q. ^ co o 
 
 C s s T ^"CO^ x ^ ^ ^ ^CO 
 * * * ' * * 
 
 1 1 1 
 
 uimiiji3j f i 
 
 CO CO O^ M O w XO tx 
 H c< cotxC^ ONH 01 
 
 'i 'i '. i i 7 ', 
 
 p j Ba r i 
 
 OOOli-i wONOt^Ol 
 
 oo xn-J-O t^o 01 coTf 
 
 I [ 
 
 'UO.IJ 
 
 * *M V) * *rt- 
 'l "l *l "| 
 
 
 
 O MDOJOO'OOM-t^ 
 
 t^ ^Tj-coLoinONoo 
 co o . H Y~> w /* /N oo p 
 
 II l 1 1 1 
 
 
 * * * * 
 O vooo COLOVQCO *tf- 
 C^OO tf^H O\CAO H 
 
 O ^ ^t" "TO M tX O 0) Tt- 
 
 i i i i 'i 7 i 
 
 . 
 ; 
 
 i" 
 
 r~ ^ 
 
 1 1 1 1 . s a |l 1 
 
 U U 1 A J Ou H N^ N W 
 
 The num 
 using the we 
 electromotive 
 The numl 
 that substanc 
 Thus, lead is 
 The meta 
 
CONTACT DIFFERENCES OF POTENTIAL IN VOLTS. 
 
 
 
 
 
 
 
 g 
 
 
 
 
 j 
 
 1 
 
 
 d 
 
 J 
 
 
 
 
 3 
 
 EM 
 
 1 
 
 u 
 
 rt 
 
 C 
 
 
 
 U 
 
 o 
 
 
 M 
 
 s 
 
 H 
 
 
 Mercury, 
 
 092 
 
 308 
 
 502 
 
 .. 
 
 156 
 
 
 
 Distilled water, . . -j 
 
 01 to '17 
 depend- 
 ing on 
 
 '269 to 
 100 
 
 148 
 
 '171 
 
 285 
 
 to 
 
 177 
 
 
 \ 
 
 carbon. 
 
 
 
 
 '345 
 
 
 / 
 
 '' Alum saturated at 
 
 
 
 
 
 
 
 
 i6- 5 C 
 
 
 -127 
 
 -653 
 
 -'139 
 
 246 
 
 -'225 
 
 
 Copper sulphate solu- ) 
 tion, specific grav- [ 
 ity, ro87ati6'6C. j 
 
 
 103 
 
 
 -- 
 
 
 
 
 Copper sulphate, satu- 1 
 rated at 15 C. .. ) 
 
 
 '070 
 
 
 
 
 
 ' c/5 
 
 O 
 
 Sea salt, specific ) 
 gravity, i'i8 at > 
 
 
 -475 
 
 - -605 
 
 -'267 
 
 -856 
 
 -'334 
 
 P - 
 
 Sal-ammoniac, satu- ) 
 rated at 15 -5 C. ..) 
 
 .. 
 
 -396 
 
 -652 
 
 -'189 
 
 '057 
 
 -'364 
 
 1 
 
 Zinc sulphate solu- ~\ 
 tion, specific grav- V 
 
 
 
 
 
 
 
 
 ity i '125 at i6'9C. j 
 
 
 
 
 
 
 
 
 Zinc sulphate, satu- ^ 
 
 
 
 
 
 
 
 
 rated at 15. 3 C. .. ) 
 
 
 
 
 
 
 
 
 
 
 i Distilled water mixed ) 
 
 
 
 
 
 
 
 
 with 3 zinc sulphate, V 
 
 
 
 
 
 
 
 i 
 
 saturated solution. j 
 
 
 
 
 
 
 
 y / i 
 
 20 Distilled water, \ 
 
 
 
 
 
 
 
 2 f^l 
 
 i strong sulphuric > 
 
 
 
 
 
 
 
 P 1 
 
 acid, .. ..) 
 
 
 
 
 
 
 
 . L: 
 
 10 Distilled water, ) 
 i strong sulphuric > 
 
 about 
 
 
 
 
 
 
 t^ i > 
 
 acid, .. ..) 
 
 035 
 
 
 
 
 
 
 M^"\ -" 
 
 5 Distilled water, ) 
 
 
 
 
 
 
 
 H^ j^l 
 
 ) i strong sulphuric > 
 acid, .. . . j 
 
 
 
 
 
 
 
 
 ^ P 
 
 i Distilled water, | 
 
 
 
 
 
 
 
 s Ul 
 
 5 strong sulphuric > 
 acid, . . .. ) 
 
 '3 to 
 
 "OI 
 
 
 
 - '120 
 
 
 -'256 
 
 ' f 
 
 / 
 
 85 to-55 
 
 
 
 
 i *6oo 
 
 
 CONCEN-I 
 
 1 Sulphuric acid, . . J 
 
 depend- 
 ing on 
 carbon. 
 
 1-113 
 
 
 '720 to 
 
 I-252 
 
 to 
 
 I '300 
 
 
 TRATED 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 672 
 
 
 
 Mercurous sulphate ^ 
 
 
 
 
 
 
 
 
 paste, .. .. ) 
 
 
 
 
 
 
 
 
 Distilled water, with) 
 a trace of sulphuric / 
 
 
 
 
 
 
 
 
 acid, . . . . j 
 
 
 
 
 
 
 
 The average temperature at the time of experimenting was about 16 C. 
 All the liquids and salts employed were chemically pure ; the solids, however, 
 were only commercially pure. 
 
Solids with Liquids and Liquids with Liquids in Air. 
 
 
 
 g 
 
 malgamated Zinc. 
 
 
 i 
 <D 
 
 istillcd Water. 
 
 lum Solution, satu 
 rated at i6'sC. 
 
 opper Sulphate 
 Solution, satu- 
 rated at 150 C. 
 
 inc Sulphate Solu- 
 tion, Specific Grav- 
 ity 1*125 at i6'9C 
 
 2 
 
 % 
 
 II 
 
 IS* 
 
 ^ ~ '-r. 
 
 Distilled Water, 
 3 Zinc Sulphate. 
 
 trong Nitric Acid. 
 
 N 
 
 < 
 
 
 
 
 
 Q 
 
 < 
 
 O 
 
 N 
 
 NJ 
 
 M 
 
 f. 
 
 -'105 
 
 
 
 
 
 
 
 
 
 
 
 to 
 
 'ICO 
 
 231 
 
 
 
 .. 
 
 -'043 
 
 
 I6 4 
 
 
 
 + 156 
 
 
 
 
 
 
 
 
 
 
 
 -536 
 
 . * 
 
 -'014 
 
 
 
 
 
 "090 
 
 
 
 
 
 
 
 
 
 
 
 
 "OQr 
 
 
 
 -565 
 
 
 -'435 
 
 
 
 
 
 
 vyo 
 
 
 
 -637 
 
 
 -348 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 -'238 
 
 
 
 
 
 
 
 
 
 
 
 -'43 
 
 - -284 
 
 
 
 '200 
 
 
 -095 
 "102 
 
 
 
 
 
 ~ 444 
 
 -344 
 
 
 
 
 
 
 
 
 
 
 
 % 
 
 -358 
 
 
 
 
 
 
 
 
 
 
 
 - -429 
 
 
 
 
 
 
 
 
 
 
 
 
 016 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ~" 
 
 
 
 
 
 
 
 848 
 
 .. 
 
 .. 
 
 1*298 
 
 i '456 
 
 1*269 
 
 
 1-699 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 " 
 
 
 
 '475 
 
 
 
 
 
 
 
 
 "241 
 
 
 
 
 
 
 
 
 
 
 '078 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 Example of the above table : Lead is positive to distilled 
 contact difference of potentials is Q'IJI volts. 
 
 water, and the 
 
150 UNITS AND PHYSICAL CONSTANTS. [CHAP. 
 
 The authors point out that in all these experiments the 
 unknown electromotive forces of certain air contacts are 
 included. 
 
 From these tables we find we can build up the electro- 
 motive forces of some well-known cells. For example, 
 in a Daniell's cell there are four contact differences of 
 potential to consider, and in a Grove's cell five, viz. : 
 
 Daniel fs Cell. j- ff j ts 
 
 Copper and saturated copper sulphate, + 0*070 
 
 Saturated copper sulphate and saturated zinc sulphate, 0*095 
 
 Saturated zinc sulphate and zinc, + 0*430 
 
 Zinc and copper, + 0750 
 
 Groves Cell. 
 
 Copper and platinum, - - - - - - + 0*238 
 
 Platinum and strong nitric acid, ... + 0*672 
 
 Strong nitric acid and very weak sulphuric acid, - + 0*078 
 
 Very weak sulphuric acid and zinc, + 0*241 
 
 Zinc and copper, - - + 0*750 
 
 1*979 
 
 185. The electromotive force of a thermoelectric cir- 
 cuit is called Thermoelectric force. 
 
 The following table, showing the thermoelectric force 
 of a couple of which lead is one element, is from 
 Jenkin's * Electricity and Magnetism,' p. 176, except that 
 we have employed the multiplier 100 to reduce from 
 microvolts to C.G.S. electromagnetic units. It was 
 compiled from Matthiessen's experiments ; and the mean 
 temperature for which it is true may be taken at from 
 19 to 20 C. 
 
XL] 
 
 ELECTRICITY. 
 
 Table of Thermoelectric Forces, in electromagnetic units, for 
 i C. of difference of temperature of junctions, lead 
 being one element. 
 
 Bismuth, pressed com- ) 
 mercial wire, - \ 
 
 Bismuth, pure pressed ) 
 wire, - - - \ 
 
 Bismuth, crystal, axial, 
 ,, equatorial, - 
 
 Cobalt, - 
 
 German silver, - 
 
 Quicksilver, 
 
 Lead, ... 
 
 Tin, 
 
 Copper of commerce, 
 
 Platinum, 
 
 Gold, 
 
 Antimony, pressed \ 
 wire, - - - \ 
 
 + 9700 
 
 + 8900 
 
 + 6500 
 + 4500 
 
 + 22OO 
 
 + "75 
 
 + 41- 
 
 o 
 
 IO 
 
 10 
 
 - 90 
 
 120 
 
 - 280 
 
 Silver, pure hard, - 300 
 
 Zinc, pure pressed, - 370 
 Copper, galvano- ) 
 
 plastically precipi- ! 380 
 
 tated, - - - \ 
 
 Antimony, pressed } , 
 
 commercial wire, - \ 
 
 Arsenic, - 1356 
 
 Iron, pianoforte wire, - 1750 
 
 Antimony, axial, - - 2260 
 
 equatorial, - 2640 
 
 Phosphorus, red, - - 2970 
 
 Tellurium, - - - 50200 
 
 Selenium, - - 80700 
 
 1 86. The following table is based upon Professor 
 Tait's thermoelectric diagram ('Trans. Roy. Soc., Edin./ 
 vol. xxvii. [Dec., 1873]), joined with the assumption that 
 a Grove's cell has electromotive force 1-97 x io 8 : 
 
 Table of Thermoelectric Values referred to lead as zero. 
 
 Iron, '.. - > - ^ 
 
 Steel, - 
 
 Alloy, believed to be platinum iriclium, 
 Alloy, platinum 95 ; iridium 5 
 90; io 
 85; 15 - 
 85; 15 - 
 Soft platinum, 
 Alloy, platinum and nickel, 
 Hard Platinum, .... 
 
 Thermoelectric value in electro- 
 magnetic units (t denoting 
 temperature Centigrade). 
 
 -1734+ 4'87' 
 -1139+ 3'28/ 
 
 - 839 at all temperatures. 
 
 - 622+ '55 1 
 
 - 596+ 1-34* 
 
 - 709+ '63 ^ 
 
 - 577 at all temperatures^ 
 + 61+ i'io/ 
 
 - 544+ I-ICK* 
 
 - 260+ 75 / 
 
152 UNITS AND PHYSICAL CONSTANTS. [CHAP. 
 
 Thermoelectric value in electro- 
 magnetic units (t denoting 
 temperature Centigrade). 
 
 Magnesium, - - - - - - 224+ '95^ 
 
 German silver, -.. ' - - - +1207+ $'\2t 
 
 Cadmium, - - - - - 266- 4^29 1 
 
 Zinc, - - - - - - 234- 2-4.0 f 
 
 Silver, - - - - '.- - - 214- i'5o/ 
 
 Gold, - ' '- -, . - - .-. - 283- i'02/ 
 
 Copper, - - . i" - . - 136- -95 ^ 
 
 Lead, .... V . o 
 
 Tin, - +43- "55 ^ 
 
 Aluminium, "- - - - + 77- '39^ 
 
 Palladium, - - - - + 625+ 3 '59 / 
 
 Nickel to 175 C., - ,- +2204+ 5-12 / 
 
 ,, 250 to 310 C., - +8449-24'i/ 
 
 from 340 C., ^ . - - + 307+ 5'I2/ 
 
 The lower limit of temperature for the Table is - i8C. 
 for all the metals in the list. The upper limit is 416 C.. 
 with the following exceptions : Cadmium, 258 C. ; zinc, 
 373 C. ; German silver, 175 C. 
 
 The difference of the " thermoelectric values " of two 
 metals for a given temperature /, is the electromotive 
 force per degree of difference between the temperatures 
 of the junctions in a couple formed of these metals, when 
 the mean of the temperatures of the junctions is /. The 
 current through the hot junction is from the metal of 
 higher to that of lower " thermoelectric value." 
 
 Example i. 
 
 Required the electromotive force of a copper-iron 
 couple, the temperatures of the junctions being o C. and 
 100 C. 
 
XL] ELECTRICITY. 153 
 
 We have, for copper, - 136- '95 /; 
 
 iron, -I734 + 4-87/; 
 
 copper -iron, = 1598 -5-82 /. 
 
 The electromotive force per degree is 1598 - 5-82 x 50 
 = 1307 electromagnetic units, and the electromotive force 
 of the couple is 1307(100 - o) = 130700. 
 
 By the neutral point of two metals is meant the tempe- 
 rature at which their thermoelectric values are equal. 
 
 Example 2. 
 
 To find the neutral point of copper and iron we have 
 copper - iron = 1598 - 5^82 t = o, / 275 ; 
 
 that is, the neutral point is 275 C. When the mean of 
 the temperatures of the junctions is below this point, the 
 current through the warmer junction is from copper to 
 iron. The current ceases as the mean temperature 
 attains the neutral point, and is reversed in passing it. 
 
 Example 3. 
 
 F. Kohlrausch (' Pogg. Ann. Erganz.' vol. vi., p. 35 
 [1874]) states that, according to his determination, the 
 electromotive force of a couple of iron and German 
 silver is 24 x io 5 millimetre^nilligramme-second units for 
 i of difference of temperatures of the junctions at 
 moderate temperatures. Compare this result with the 
 above Table at mean temperature 100. 
 
 The dimensions of electromotive force are M*L^T~ 2 ; 
 hence the C.G.S. value of Kohlrausch's unit is 
 io~^ io~^ = io~ 3 , giving 2400 as the electromotive force 
 per degree of difference. 
 
154 UNITS AND PHYSICAL CONSTANTS. [CHAP. 
 
 From the above table we have 
 
 German silver - iron = 2941 + -25 /, 
 
 which, for /= 100, gives 2966 as the electromotive force 
 per degree of difference. 
 
 Electrochemical Equivalents. 
 
 187. The following are examples of electrolytic decom- 
 positions which require the same quantity of electricity 
 to effect them : 
 
 Substance Mass decom- 
 
 decomposed. 
 
 posed. 
 
 Water, - 
 
 18 
 
 Hydrochloric acid, 
 
 73 
 
 Potassium chloride, 
 
 149 
 
 Sodium ,, 
 
 117 
 
 Silver ,, 
 
 287 
 
 Potassium iodide 
 
 332 
 
 bromide, 
 
 238 
 
 Calcium chloride, 
 
 in 
 
 Zinc 
 
 136 
 
 Ferrous ,, 
 
 127 
 
 Ferric , , 
 
 io8J 
 
 Cuprous ,, 
 
 198 
 
 Cupric ,, 
 
 i34i 
 
 Mercuric ,, 
 
 271 
 
 Potassium sulphate 
 
 174 
 
 Zinc , , 
 
 163 
 
 Lead nitrate, 
 
 331 
 
 Silver 
 
 340 
 
 Stannous chloride, 
 
 189 
 
 Stannic ,, 
 
 130 
 
 Masses of products. 
 
 2 hydrogen, 16 
 
 2 ,, 71 
 
 78 potassium, 71 
 
 46 sodium, 71 
 
 216 silver, 71 
 
 78 potassium, 254 
 
 78 160 
 
 40 calcium, 7 1 
 
 65 zinc, 71 
 
 56 iron, 71 
 
 37i 71 
 
 127 copper, 71 
 
 634 M 71 
 
 200 mercury, 7 1 
 
 78 potassium. 
 
 65 zinc. 
 
 207 lead. 
 
 216 silver. 
 
 118 tin, 71 
 
 59 71 
 
 oxygen, 
 chlorine. 
 
 iodine, 
 bromine, 
 chlorine. 
 
 chlorine. 
 
 According to the experiments of F. Kohlrausch ('Pogg. 
 Ann.' vol. cxlix [1873]), the quantity of silver deposited 
 
XL] ELECTRICITY. 155 
 
 by the C.G.S. unit (electromagnetic) of electricity is 
 
 ?i6 
 
 011363 gramme. Hence or 19010 is the quan- 
 011363 
 
 tity of electricity required to produce the above effects if 
 the numbers are taken as denoting grammes. 
 
 1 88. Let W ergs be the chemical work done in a cell 
 of a battery for 65 grammes of zinc consumed (being the 
 heat of chemical combination multiplied by Joule's 
 
 W 
 
 equivalent : then will be the electromotive force of 
 
 19010 
 
 the cell, on the supposition that there is no wasteful 
 action. 
 
 According to a calculation made by Professor G. C. 
 Foster, for the first edition of this work, based on Julius 
 Thomsen's determinations of the heat of combination, 65 
 grammes of zinc consumed correspond to 
 
 40105 gramme-degrees in Smee's cell. 
 52347 Daniell's cell. 
 
 90162 Grove's cell. 
 
 Multiplying by - we obtain 
 19010 
 
 8'86 x io 7 as the electromotive force of Smee's cell. 
 1-156 x io 8 Daniell's cell. 
 
 1*991 x io 8 ,, ,, Grove's cell. 
 
 These results are slightly in excess of the values ob- 
 tained by direct observation (see 184). 
 
 189. Examples in Electricity. 
 
 i. Two conducting spheres, each of i centim. radius, 
 are placed at a distance of r centims. from centre to 
 centre, r being a large number ; and each of them is 
 
156 UNITS AND PHYSICAL CONSTANTS. [CHAP. 
 
 charged with an electrostatic unit of positive electricity. 
 With what force will they repel each other ? 
 
 Since r is large, the charge may be assumed to be 
 uniformly distributed over their surfaces, and the force 
 will be the same as if the charge of each were collected 
 
 at its centre. The force will therefore be \ of a dyne. 
 
 r- 
 
 2. Two conducting spheres, each of i centim. radius, 
 placed as in the preceding question, are connected one 
 with each pole of a Daniell's battery (the middle of 
 the battery being to earth) by means of two very 
 fine wires whose capacity may be neglected, so that 
 the capacity of each sphere when thus connected is 
 sensibly equal to unity. Of how many cells must the 
 battery consist that the spheres may attract each other 
 
 with a force of ~ of a dyne, r being the distance between 
 
 their centres in centims. ? 
 
 One sphere must be charged to potential i and the 
 other to potential - i. The number of cells required is 
 
 3. How many Daniell's cells would be required to 
 produce a spark between two parallel conducting surfaces 
 at a distance of '019 of a centim., and how many at a 
 distance of '0086 of a centim. (See 178, 184.) 
 
 A 4'26 2"IO ,. 
 
 ^j. _Z =1139; a = 61=5. 
 
 00374 -00374 
 
 4. Compare the capacity denoted by i farad with the 
 capacity of the earth. 
 
 The capacity of the earth in static measure is equal to 
 
XL] ELECTRICITY. 157 
 
 its radius, namely 6*37 x io 8 . Dividing by z/ 2 to reduce 
 to magnetic measure, we have 71 x io~ 12 , which is i 
 farad multiplied by -71 x io~ 3 , or is -00071 of a farad. 
 A farad is therefore 1400 times the capacity of the earth. 
 
 5. Calculate the resistance of a cell consisting of a 
 plate of zinc, A square centims. in area, and a plate of 
 copper of the same dimensions, separated by an acid solu- 
 tion of specific resistance io 9 , the distance between the 
 plates being i centim. 
 
 Ans. , or of an ohm. 
 A A 
 
 6. Find the heat developed in io minutes by the 
 passage of a current from io Daniell's cells in series 
 through a ,wire of resistance io 10 (that is, io ohms), 
 assuming the electromotive force of each cell to be 
 IT x io 8 , and the resistance of each cell to be io 9 . 
 
 Here we have 
 
 Total electromotive force = 1*1 x i 9 . 
 Resistance in battery = io 10 . 
 Resistance in wire -- io 10 . 
 
 1*1 X I O 
 
 Current = = 'cc x lo" 1 = 'o^. 
 
 2 x io 10 
 
 Heat developed in j _ (-05 5 2 ) x io 10 _ 
 wire per second j 4 - 2 x io 7 
 
 Hence the heat developed in io minutes is 432-14 
 gramme-degrees. 
 
 7. Find the electromotive force between the wheels on 
 opposite sides of a railway carriage travelling at the rate 
 of 30 miles an hour on a narrow-gauge line [4 feet 85 
 inches] due to cutting the lines of force of terrestrial 
 magnetism, the vertical intensity being '438. 
 
158 UNITS AND PHYSICAL CONSTANTS. [CHAP. 
 
 The electromotive force will be the product of the 
 velocity of travelling, the distance between the rails, and 
 the vertical intensity, that is, 
 
 (447 * 30) (2-54 x 56-5) (-438) - 84300 
 electromagnetic units. 
 
 This is about of a volt 
 
 1200 
 
 8. Find the electromotive force at the instant of passing 
 the magnetic meridian, in a circular coil consisting of 300 
 turns of wire, revolving at the rate of 10 revolutions per 
 second about a vertical diameter ; the diameter of the 
 coil being 30 centims., and the horizontal intensity of 
 terrestrial magnetism being '1794, no other magnetic 
 influence being supposed present. 
 
 The numerical value of the lines of force which go 
 through the coil when inclined at an angle 6 to the 
 meridian, is the horizontal intensity multiplied by the 
 area of the coil and by sin 6 ; say nH-n-a 2 sin #, where 
 H = '1794, a - 15, and n = 300. The electromotive 
 force at any instant is the rate at which this quantity 
 increases or diminishes ; that is, nHna 2 cos 6 . o>, if w 
 denote the angular velocity. At the instant of passing 
 the meridian cos 9 is i, and the electromotive force is 
 //H7iYZ 2 a>. With 10 revolutions per second the value of w 
 
 is 27T X 10. 
 
 Hence the electromotive force is 
 1794 x (3'i42) 2 x 225 x 20 x 300 = 2*39 x 10. 
 
 This is about of a volt. 
 42 
 
 190. To investigate the magnitudes of units of length, 
 
XL] ELECTRICITY. 159 
 
 mass, and time which will fulfil the three following 
 conditions : 
 
 1. The acceleration due to the attraction of unit mass 
 at unit distance shall be unity. 
 
 2. The electrostatic units shall be equal to the electro- 
 magnetic units. 
 
 3. The density of water at 4 C. shall be unity. 
 
 Let the 3 units required be equal respectively to L 
 centims., M grammes, and T seconds. 
 
 We have in C.G.S. measure, for the acceleration due 
 to attraction ( 72), 
 
 acceleration = C r ' where C = 6-48 x io~ 8 : 
 
 (distance) 2 
 
 and in the new system we are to have 
 
 acceleration = - , mass . . 
 (distance) 2 
 
 Hence, by division, 
 
 acceleration in C.G.S. units 
 
 acceleration in new units 
 
 P mass in C.G.S. units (distance in new units) 2 
 mass in new units ' (distance in C.G.S. units) 2 ' 
 
 that is, = eg 
 
 This equation expresses the first of the three conditions. 
 The equation = v expresses the second, v denoting 
 
 3 x lol - 
 
 The equation M = L 3 expresses the third. 
 
160 UNITS AND PHYSICAL CONSTANTS. [CHAP. 
 
 Substituting L 3 for M in the first equation, we find 
 -- . Hence, from the second equation, 
 
 and from the third, 
 
 Introducing the actual values of C and v, we have 
 approximately 
 
 T = 3928, L = 1-178 x io 14 , M = 1-63 x io 42 ; 
 that is to say, 
 
 The new unit of time will be about i h 5^ m ; 
 
 The new unit of length will be about 118 thousand 
 
 earth quadrants j 
 
 The new unit of mass will be about 2 '66 x io 14 times 
 the earth's mass. 
 
 Electrodynamics. 
 
 191. Ampere's formula for the repulsion between two 
 elements of currents, when expressed in electromagnetic 
 units, is 
 
 s ^ -(2 sin a sin a' cos 9 - cos a cos a'), 
 
 where c, c denote the strengths of the two currents ; 
 ds, ds the lengths of the two elements ; 
 a, a' the angles which the elements make with 
 
 the line joining them ; 
 r the length of this joining line ; 
 6 the angle between the plane of r, ds, and 
 
 the plane of r, ds. 
 For two parallel currents, one of which is of infinite 
 
XL] ELECTRICITY. 161 
 
 length, and the other of length /, the formula gives by 
 integration an attraction or repulsion, 
 
 2/ / 
 
 where D denotes the perpendicular distance between the 
 currents. 
 
 Example. 
 Find the attraction between two parallel wires a metre 
 
 long and a centim. apart when a current of 4- is passing 
 
 10 
 
 through each. 
 
 Here the attraction will be sensibly the same as if 
 one of the wires were indefinitely increased in length, 
 and will be 
 
 .2 
 
 that is, each wire will be attracted or repelled with a force 
 of 2 dynes, according as the directions of the currents are 
 the same or opposite. 
 
APPENDIX. 
 
 First Report of the Committee for the Selection and 
 Nomenclature of Dynamical and Electrical Units, 
 the Committee consisting of SIR W. THOMSON, F.R.S., 
 PROFESSOR G. C. FOSTER, F.R.S., PROFESSOR J. C. 
 MAXWELL, F.R.S., MR. G. J. STONEY, F.R.S.,* 
 PROFESSOR FLEEMING JENKIN, F.R.S., DR. SIEMENS, 
 F.R.S., MR. F. J. BRAMWELL, F.R.S., and PROFESSOR 
 EVERETT (Reporter). 
 
 WE consider that the most urgent portion of the task 
 intrusted to us is that which concerns the selection and 
 nomenclature of units of force and energy ; and under 
 this head we are prepared to offer a definite recommenda- 
 tion. 
 
 A more extensive and difficult part of our duty is the 
 selection and nomenclature of electrical and magnetic 
 units. Under this head we are prepared with a definite 
 recommendation as regards selection, but with only an 
 interim recommendation as regards nomenclature. 
 
 Up to the present time it has been necessary for every 
 person who wishes to specify a magnitude in what is 
 
 * Mr. Stoney objected to the selection of the centimetre as the 
 unit of length. 
 
1 64 APPENDIX. 
 
 called "absolute" measure, to mention the three funda- 
 mental units of mass, length, and time which he has 
 chosen as the basis of his system. This necessity will be 
 obviated if one definite selection of three fundamental 
 units be made once for all, and accepted by the general 
 consent of scientific men. We are strongly of opinion 
 that such a selection ought at once to be made, and to be 
 so made that there will be no subsequent necessity for 
 amending it. 
 
 We think that, in the selection of each kind of derived 
 unit, all arbitrary multiplications and divisions by powers 
 of ten, or other factors, must be rigorously avoided, and 
 the whole system of fundamental units of force, work, 
 electrostatic, and electromagnetic elements must be fixed 
 at one common level that level, namely, which is deter- 
 mined by direct derivation from the three fundamental 
 units once for all selected. 
 
 The carrying out of this resolution involves the adoption 
 of some units which are excessively large or excessively 
 small in comparison with the magnitudes which occur in 
 practice ; but a remedy for this inconvenience is provided 
 by a method of denoting decimal multiples and sub- 
 multiples, which has already been extensively adopted, 
 and which we desire to recommend for general use. 
 
 On the initial question of the particular units of mass, 
 length, and time to be recommended as the basis of the 
 whole system, a protracted discussion has been carried 
 on, the principal point discussed being the claims of the 
 gramme, the metre, and the second, as against the gramme, 
 the centimetre, and the second, the former combination 
 having an advantage as regards the simplicity of the name 
 metre, while the latter combination has the advantage of 
 
APPENDIX. 165 
 
 making the unit of mass practically identical with the 
 mass of unit- volume of water in other words, of making 
 the value of the density of water practically equal to unity. 
 We are now all but unanimous in regarding this latter 
 element of simplicity as the more important of the two ; 
 and in support of this view we desire to quote the 
 authority of Sir W. Thomson, who has for a long time 
 insisted very strongly upon the necessity of employing 
 units which conform to this condition. 
 
 We accordingly recommend the general adoption of the 
 Centimetre, the Gramme, and the Second as the three 
 fundamental units ; and until such time as special names 
 shall be appropriated to the units of electrical and mag- 
 netic magnitude hence derived, we recommend that they 
 be distinguished from " absolute " units otherwise derived, 
 by the letters " C.G.S." prefixed, these being the initial 
 letters of the names of the three fundamental units. 
 
 Special names, if short and suitable, would, in the 
 opinion of a majority of us, be better than the provisional 
 designations " C.G.S. unit of ... ." Several lists of 
 names have already been suggested ; and attentive con- 
 sideration will be given to any further suggestions which 
 we may receive from persons interested in electrical 
 nomenclature. 
 
 The u ohm," as represented by the original standard 
 coil, is approximately io 9 C.G.S. units of resistance; the 
 "volt" is approximately io 8 C.G.S. units of electromotive 
 
 force ; and the " farad " is approximately 9 of the C.G.S. 
 
 unit of capacity. 
 
 For the expression of high decimal multiples and sub- 
 multiples, we recommend the system introduced by Mr. 
 
1 66 APPENDIX. 
 
 Stoney, a system which has already been extensively 
 employed for electrical purposes. It consists in denoting 
 the exponent of the power of 10, which serves as multi- 
 plier, by an appended cardinal number, if the exponent 
 be positive, and by a prefixed ordinal number if the 
 exponent be negative. 
 
 Thus io 9 grammes constitute a gramme-nine ; -- 9 of a 
 
 gramme constitutes a ninth-gramme; the approximate 
 length of a quadrant of one of the earth's meridians is a 
 metre-seven, or a centimetre-nine. 
 
 For multiplication or division by a million, the prefixes 
 mega* and micro may conveniently be employed, according 
 to the present custom of electricians. Thus the megohm 
 is a million ohms, and the microfarad is the millionth part 
 of a farad. The prefix mega is equivalent to the affix six. 
 The prefix micro is equivalent to the prefix sixth. 
 
 The prefixes kilo, Jiecto, deca, deri, centi, milli can also be 
 employed in their usual senses before all new names of 
 units. 
 
 As regards the name to be given to the C.G.S. unit of 
 force, we recommend that it be a derivative of the Greek 
 Swa/us. The form dynamy appears to be the most satis- 
 factory to etymologists. Dynam is equally intelligible, 
 but awkward in sound to English ears. The shorter form, 
 dyne, though not fashioned according to strict rules of 
 etymology, will probably be generally preferred in this 
 country. Bearing in mind that it is desirable to construct 
 a system with a view to its becoming international, we 
 think that the termination of the word should for the 
 
 * Before a vowel, either meg or megal t as euphony may suggest, 
 may be employed instead of mega. 
 
APPENDIX. 167 
 
 present be left an open question. But we would earnestly 
 request that, whichever form of the word be employed, its 
 meaning be strictly limited to the unit of force of the 
 C.G.S. system that is to say, the force which, acting upon 
 a gramme of matter for a second, generates a velocity of a 
 centimetre per second. 
 
 The C.G.S. unit of work is the work done by this force 
 working through' a centimetre; and we propose to denote 
 it by some derivative of the Greek e/oyov. The forms 
 ergon, ergal, and erg have been suggested ; but the second 
 of these has been used in a different sense by Clausius. 
 In this case also we propose, for the present, to leave the 
 termination unsettled ; and we request that the word 
 ergon, or erg, be strictly limited to the C.G.S. unit of 
 work, or what is, for purposes of measurement, equivalent 
 to this, the C.G.S. unit of energy, energy being measured 
 by the amount of work which it represents. 
 
 The C.G.S. unit of power is the power of doing work at 
 the rate of one erg per second; and the power of an engine, 
 under given conditions of working, can be specified in 
 ergs per second. 
 
 For rough comparison with the vulgar (and variable) 
 units based on terrestrial gravitation, the following state- 
 ment will be useful : 
 
 The weight of a gramme, at any part of the earth's sur- 
 face, is about 980 dynes, or rather less than a kilodyne. 
 
 The weight of a kilogramme is rather less than a mega- 
 dyne, being about 980,000 dynes. 
 
 Conversely, the dyne is about 1*02 times the weight of 
 a milligramme at any part of the earth's surface; and 
 the megadyne is about 1*02 times the weight of a kilo- 
 gramme. 
 
1 68 APPENDIX. 
 
 The kilogrammetre is rather less than the ergon-eight, 
 being about 98 million ergs. 
 
 The gramme-centimetre is rather less than the kilerg, 
 being about 980 ergs. 
 
 For exact comparison, the value of g (the acceleration 
 of a body falling in vacuo) at the station considered must 
 of course be known". In the above comparison it is taken 
 as 980 C.G.S. units of acceleration. 
 
 One horse-power is about three quarters of an erg-ten 
 per second. More nearly, it is 7*46 erg-nines per second, 
 and one force-de-cheval is 7*36 erg-nines per second. 
 
 The mechanical equivalent of one gramme-degree 
 (Centigrade) of heat is 41*6 megalergs, or 41,600,000 
 ergs. 
 
 Second Report of the Committee for the Selection and 
 Nomenclature of Dynamical and Electrical Units, the 
 Committee consisting of PROFESSOR SIR W. THOMSON, 
 F.R.S., PROFESSOR G. C. FOSTER, F.R.S., PROFESSOR 
 J. CLERK MAXWELL, F.R.S., G. J. STONEY, F.R.S.. 
 PROFESSOR FLEEMING JENKIN, F.R.S., DR. C. W. 
 SIEMENS, F.R.S., F. J. BRAMWELL, F.R.S., PRO- 
 FESSOR W. G. ADAMS, F.R.S., PROFESSOR BALFOUR 
 STEWART, F.R.S., and PROFESSOR EVERETT 
 (Secretary). 
 
 THE Committee on the Nomenclature of Dynamical 
 and Electrical Units have circulated numerous copies of 
 their last year's Report among scientific men both at 
 home and abroad. 
 
 They believe, however, that, in order to render their 
 
APPENDIX. 169 
 
 recommendations fully available for science teaching and 
 scientific work, a full and popular exposition of the whole 
 subject of physical units is necessary, together with a 
 collection of examples (tabular and otherwise) illustrating 
 the application of systematic units to a variety of physical 
 measurements. Students usually find peculiar difficulty 
 in questions relating to units ; and even the experienced 
 scientific calculator is glad to have before him concrete 
 examples with which to compare his own results, as a 
 security against misapprehension or mistake. 
 
 Some members of the Committee have been preparing 
 a small volume of illustrations of the C.G.S. system 
 [Centimetre-Gramme-Second system] intended to meet 
 this want. 
 
 [The first edition of the present work is the volume of 
 illustrations here referred to]. 
 
INDEX. 
 
 The numerals refer to the pages. 
 
 Acceleration, 21. 
 
 Acoustics, 62-66. 
 
 Adiabatic compression, 115-118. 
 
 Air, density of, 35. 
 
 , expansion of, 89-90. 
 
 , specific heat of, 84-85, 113. 
 
 , thermal conductivity of, 99. 
 
 Ampere's formula-, 160. 
 Aqueous vapour, pressure of, 
 41, 90. 
 
 , density of, 41. 
 
 Astronomy, 57-61. 
 Atmosphere, standard, 34. 
 
 , its density upwards, 39. 
 
 Atomic weights, 81-82. 
 Attraction, constant of, 59-60. 
 
 at a point, 13. 
 
 Angle, 12. 
 
 , solid, 13. 
 
 Barometer, correction of, -for 
 
 capillarity, 43. 
 Barometric measurement of 
 
 heights, 39. 
 
 pressure, 33. 
 
 Batteries, 146, 150, 155. 
 Boiling points, 93. 
 
 of water near 100, 90. 
 
 Boyle's law, departures from, 91. 
 Bullet, melted by impact, 26. 
 
 Capacity, electrical, 129, 131. 
 
 Capacity, specific inductive, 134- 
 
 137- 
 
 , thermal, 78-86. 
 
 Capillarity in barometer, 43. 
 Cells, 146, 150, 155. 
 Centimetre, reason for selecting, 
 
 19, 164. 
 Centre of attraction, strength of, 
 
 13- 
 
 Centrifugal force, 26. 
 
 at equator, 28. 
 
 C.G.S. system, 19, 165. 
 Change of volume in evapora- 
 tion, 87. 
 
 in melting, 93. 
 
 Change-ratio, 5. 
 
 Chemical action, heat of, 112. 
 
 equivalents in electrolysis, 
 
 154- 
 
 Clark's standard cell, 146. 
 Cobalt, magnetization of, 124. 
 Coil, revolving, 158. 
 Combination, heat of, 1 1 2. 
 Combustion, heat of, 112. 
 Common scale, necessity for, 
 
 1 8. 
 Compressibility of liquids, 52-53. 
 
 of solids, 53-55. 
 
 Compression, adiabatic, 115-118. 
 Conductivity (thermal) defined, 
 
 94. 
 , thermometnc, 96. 
 
172 
 
 INDEX. 
 
 Conductivity of air, 99. 
 
 of various substances, 99- 
 
 107. 
 
 Contact electricity, 146-150. 
 Cooling, see Emission. 
 Current, heat generated by, 131. 
 Curvature, dimensions of, 13. 
 
 Daniell's cell, 146, 150, 155. 
 Day, sidereal, 58. 
 Decimal multiples, 20, 165. 
 Declination, magnetic, at Green- 
 wich, 127. 
 Densities, table of, 32. 
 
 of gases, 36. 
 
 of water, 30-31. 
 
 Density as a fundamental unit, 
 
 134- 
 
 Derived units, 1-2. 
 Dew-point from wet and dry 
 
 bulb, 94. 
 
 Diamagnetic substances, 121. 
 Diamond, specific heat of, 83. 
 Diffusion, coefficient of, 96-99. 
 Dimensional equations, 5- 
 Dimensions, 3-5. 
 Dip at Greenwich, 126-127. 
 Dispersive powers of gases, 76-77. 
 of glass and liquids, 
 
 70-72. 
 
 Diversity of scales, 18. 
 "Division," extended sense of, 6. 
 Double refraction, 72-73. 
 Dynamics, 11-13. 
 Dyne, 22, 166. 
 
 Earth, as a magnet, 124-126. 
 
 , size, figure, and mass of, 57. 
 
 Elasticity, 46-56. 
 
 as affected by heat of com- 
 pression, 1 1 6. 
 
 Electric units, tables of their di- 
 mensions, 131, 134. 
 Electricity, 128-161. 
 Electrochemical equivalents, 154. 
 Electrodynamics, 160-161. 
 Electromagnetic units, 130. 
 
 Electro-motive force, 146-155. 
 Electrostatic units, 128. 
 Emission of heat, 107-110. 
 Energy, 24. 
 
 , dimensions of, 12. 
 
 Equations, dimensional, 5. 
 -, physical, 8. 
 
 Equivalent, mechanical, of heat, 
 
 IIO-II2. 
 
 Equivalents, electrochemical, 154. 
 
 Erg, 24, 167. 
 
 Evaporation, change of volume 
 
 in, 87. 
 Examples in electricity, 155-160. 
 
 in general theory of units, 
 
 8-1 1. 
 
 in magnetism, 122-125. 
 
 Expansion of gases, 89-90. 
 of various substances, 118- 
 
 119. 
 
 Extended sense of " multiplica- 
 tion " and "division," 6. 
 
 Farad, 139. 
 
 compared with capacity of 
 
 earth, 157. 
 
 Field, intensity of, 121. 
 Films, tension in superficial, 42. 
 Foot-pound and foot-poundal, 25. 
 Force, 22. 
 , dimensions of, u. 
 
 at a point, 13. 
 
 , various units of, xv. 
 
 Freezing-point, change of, with 
 
 pressure, 113-115. 
 
 Frequencies of luminous vibra- 
 tions, 69. 
 
 Fundamental units, 3. 
 
 -, choice of, 15. 
 
 reduced to two, 60. 
 
 Gases, densities of, 36. 
 , expansion of, 89-90. 
 
 , indices of refraction of, 
 
 75-77- 
 -, inductive capacities of, 137. 
 
 , two specific heats of, 113. 
 
INDEX. 
 
 173 
 
 Gauss's expression for magnetic 
 potential, 126. 
 
 pound-magnet, 122. 
 
 units of intensity, 125. 
 
 Geometrical quantities, dimen- 
 sions of, ii. 
 
 Gottingen, total intensity at, 125. 
 Gramme-degree (unit of heat), 
 
 79- 
 
 Gravitation in astronomy, 58. 
 Gravitation measure of force, 23. 
 
 of work, 25. 
 
 Gravity, terrestrial, 21-22. 
 Greenwich, magnetic elements 
 
 at, 126-127. 
 Grove's cell, 146, 150, 155. 
 
 Heat, 78-119. 
 
 generated by current, 131. 
 
 , mechanical equivalent of, 
 
 110-112. 
 
 of combination, 112. 
 
 of compression, 115-118. 
 
 , unit of, 78-79. 
 
 , various units of, xvi. 
 
 Height, measured by barometer, 
 
 39- 
 Homogeneous atmosphere, height 
 
 of, 37-39- 
 Horse-power, 25. 
 Hydrostatics, 30. 
 Hypsometric table of boiling 
 
 points, 90. 
 
 Ice, specific gravity of, 87. 
 , electrical resistance of, 
 
 145- 
 
 Indices of refraction, 69-77. 
 
 . related to induc- 
 tive capacities, 136-137. 
 
 Inductive capacity, 134-137. 
 
 Induction, magnetic, coefficient 
 of, 121. 
 
 Insulators, resistance of, 143, 145. 
 
 Intel-diffusion, 97-99. 
 
 Joule's equivalent, 110-112. 
 
 Kilogramme and pound, XV. 
 Kinetic energy, 25. 
 Kupffer's determination of den- 
 sity of water, 30. 
 
 Large numbers, mode of ex- 
 pressing, 20, 165. 
 
 Latent heats, 87-89. 
 
 Latimer Clark's standard cell, 
 146. 
 
 Light, 67-77. 
 
 , velocity and wave-length 
 
 of,, 67-69. 
 
 Magnetic elements at Greenwich, 
 
 126-127. 
 
 Magnetic units, 1 20- 1 22, 
 Magnetism, 120-127. 
 
 , terrestrial, 124-127. 
 
 Magneto-optic rotation, 127. 
 Magnetization, intensity of, 12 1, 
 
 123, 125. 
 
 Mass, standards of, 16. 
 Mechanical equivalent of heat, 
 
 IIO-II2. 
 
 quantities, dimensions of, 1 1.- 
 
 units, 21. 
 
 Mega, as prefix, 34, 166. 
 Melting points, 87. 
 Metre and yard, xiv. 
 Micro, as a prefix, 166. 
 Microfarad, 139. 
 Moment of couple, 12. 
 of inertia, 12. 
 
 of magnet, 120. 
 
 of momentum, 12. 
 
 Momentum, u. 
 
 Moon, mass and distance of, 58. 
 "Multiplication," extended sense 
 of, 6. 
 
 Neutral point (thermoelectric), 
 
 153- 
 
 Nickel, magnetization of, 124. 
 Numerical value, I. 
 
 Oerstedt, 139. 
 
174 
 
 INDEX. 
 
 Ohm (unit of resistance), 137-138. 
 earth quadrant per second, 
 
 139- 
 
 Optics, 67-77. 
 
 Paramagnetic substances, 121. 
 Pendulum, seconds', 21-22. 
 " Per," meaning of, 6. 
 Platinum, specific heat of, 80, 83. 
 Poisson's ratio, 54. 
 Potential, electric, 128, 130. 
 
 , magnetic, 122, 126. 
 
 Poundal, 24. 
 
 Powers of ten as factors, 20. 
 
 Pressure, dimensions of, 12. 
 
 of liquid columns, 34. 
 
 , various units of, xvi. 
 
 Pressure-height, 38. 
 
 Quantity of electricity, 128, 130. 
 
 Radian, 12. 
 
 Radiation, 107-110. 
 
 Ratios of two sets of electric 
 units, 132-134. 
 
 Refraction, indices of, 69-77. 
 
 Reports of Units' Committee, 
 163-169. 
 
 Resilience, 46-56. 
 
 as affected by heat of com- 
 pression, 1 1 6. 
 
 Resistance, electrical, 142-145. 
 
 of a cell, 157. 
 
 Rigidity, simple, 52. 
 Rotating coil, 158. 
 
 Shear, 47-50. 
 Shearing stress, 50-52. 
 Siemens' unit, 138. 
 Sound, faintest audible, 66. 
 
 , velocity of, 62-65. 
 
 Spark, length of, 140-142. 
 Specific gravities, 32. 
 Specific heat, 79-86. 
 
 in different states, 92, 
 
 two, of gases, 113. 
 
 inductive capacity, 134-137. 
 
 Spring balance, 25. 
 Standards, comparison of French 
 and English, xiv. xv. 
 
 of length, 17. 
 
 - of mass, 1 6. 
 
 of time, 1 7. 
 
 Steam, pressure and density of, 
 41. 
 
 , total and latent heat of, 
 
 89. 
 
 Stoney's nomenclature for mul- 
 tiples, 165. 
 Strain, 44-45, 47-50. 
 -, dimensions of, 45. 
 
 Stress, 44, 46, 50-51. 
 
 - , dimensions of, 46. 
 Strings, musical, 65-66. 
 
 Sun's distance and parallax, 58. 
 Surface-conduction, 107-110. 
 Surface-tensions of liquids, 42. 
 
 Telegraphic cables, resistance of, 
 
 144. 
 
 Tenacities, table of, 55-56. 
 Tensions, superficial, of liquids, 
 
 42. 
 
 Thermodynamics, 1 1 0- 1 1 8. 
 Thermoelectricity, 150-154. 
 Time, standard of, 17. 
 Tortuosity, 13. 
 Two fundamental units sufficient, 
 
 60. 
 
 Unit, i. 
 
 Units, derived, 1-2. 
 
 - , dimensions of, 3-4. 
 
 - , special problems on, 61, 
 159. 
 
 Vapours, pressure of, 41, 42. 
 Velocity, 2-5. 
 
 - of light, 67. 
 
 - of sound, 62-65. 
 
 - , various units of, xv. 
 Vibrations per second of light, 
 
 69. 
 Volt and Weber, 138. 
 
INDEX. 
 
 175 
 
 Volume measured by weighing 
 in water, 32. 
 
 of a gramme of gas, 36. 
 
 , unit of, i. i' V*fi**W 
 
 Volume resilience, 47, 52-55. 
 , various units of, xiv. 
 
 Water, compressibility of, 52-53. 
 
 , density of, 30-31. 
 
 , expansion of, 31. 
 
 , specific heat of, 78-79. 
 
 , weighing in, 32. 
 
 Weber, 139. 
 
 Weight, force, and mass, 23. 
 
 , standards of, 16. 
 Weights and measures reduced 
 
 to C.G.S., xiv.-xvi. 
 Work, 24-25. 
 
 , dimensions of, n. 
 
 done by a current, 130. 
 
 , various units of, xvi. 
 
 Working, unit rate of, 25. 
 
 Year, sidereal and tropical, 58. 
 Young's modulus, 47, 53-56. 
 
 
 PRINTED AT THE UNIVERSITY PRESS BY 
 ROBERT MACLEHOSE, GLASGOW.