GIFT OF MICHAEL REESE C.G.S. SYSTEM OF UNITS. ILLUSTRATIONS OF THE C.G.S. SYSTEM OF UNITS WITH TABLES OF PHYSICAL CONSTANTS. BY J. D. EVERETT, M.A., D.C.L., F.RS., F.R.S.E., PROFESSOR OF NATURAL PHILOSOPHY IN QUEEN'S COLLEGE, BELFAST. OF THE UNIVERSITY HJC.critb.cm : MACMILLAN AND CO., AND NEW YORK. 1891. [The right of translation and reproduction is reserved.] First Edition published elsewhere. Second Edition printed for Macmillan & Co., 1879 (Ext. Fcap. 8vo). Third Edition, 1886 (Globe 8vo). Fourth Edition, 1891. PREFACE. THIS work, in its original form, was published by the Physical Society of London, in 1875, as "Illustrations of the C.G.S. System of Units." A greatly enlarged edition was issued in 1879 by the present publishers, and tlie title was changed to " Units and Physical Constants," the letters "C.G.S." being suppressed from an idea that their strangeness would unfavourably influence the sale. This new title was retained in the " second edition " issued in 1886. The C.G.S. units having now become the accepted standards of reference throughout the scientific world, there is no longer any reason for suppressing the name. In the present edition they are accordingly restored to their place in the title, which is thus brought into closer agreement with the contents of the book. The German edition, which was announced in our last preface as about to appear, was published at Leipsic in 1888 by Ambrosius Earth. It is a close translation as regards the portions relating to units, but contains large changes in the experimental data, some years having been devoted to this portion of the undertaking after the original manuscript had been completed. The pre- sent edition has been closely compared with the German vi PREFACE. rendering, and in several instances I have been glad to avail myself of the permission of the editors (Drs. Chappuis and Kreichgauer) to make use of their material. A Eussian translation was published at St. Petersburg in 1888, edited by Mr. P. K Verbitsky and Captain J. Th. Gerabiatieff, and revised by Professor S. A. Ousoff. I am indebted to these gentlemen for pointing out several errors which they discovered in a searching scrutiny not only of the book itself but of the authorities quoted in it. They have, for example, corrected some numbers quoted from P6clet, by the aid of a table of errata in Peclet's book which I had overlooked. The following are the principal items of new matter : 1. A collection of determinations of viscosities of liquids and gases ; in connection with which I have to express my deep obligations to Mr. Carl Barus, who lent me a large bundle of MSS., in- cluding all recent results on this subject. 2. A summary of recent investigations of the mag- netic properties of iron and other substances, with the necessary expositions of theory and terminology. Under this head I have to acknow- ledge very special obligations to Mr. Shelford Bidwell, some six or eight pages of the new matter being substantially from his pen. I have also received valuable aid from Professor Ewing and Dr. Hopkinson. 3. A revolutionising of the introductory sections on Heat, rendered necessary by Rowland's discovery that the specific heat of water decreases by 1 per PREFACE. vii cent, as the temperature rises from 5 to 30 C. The new sections include Eowland's results for the mechanical equivalent of heat, and for the comparison of the thermodynamic scale of tem- perature with the air thermometer and with a Kew standard. 4. Two sections on self-induction, mutual induction, and the theory and terminology of alternating currents. 5. Two sections on the dimensions of electrical and magnetic quantities in terms of K and /z. Under this head I have to acknowledge kind assistance from Professors Rucker and Fitzgerald. 6. A table of values of the magnetic elements at a number of stations in the British Isles, selected from the results of the recent survey of Professors Riicker and Thorpe. Information on several subjects has been brought down to date ; including Wave-lengths of light. Mercury standards of resistance, and their correction for temperature. Standards of light-giving power. Emission of heat. Conduction of heat by liquids. Departure from Boyle's law at very high pressures. Compression of fresh water, sea water, mercury, and glass, at very high pressures. Coefficients of diffusion of liquids and gases, with a simplified exposition. viii PREFACE. Comparison of the mass of unit volume of water with standards of mass. Besides the names above mentioned, I desire to return my thanks to several well-known men of science who have aided me by corrections or suggestions, all of which have been carefully considered, though in some instances I have not seen my way to carry them out. PREFACE TO "ILLUSTRATIONS OF THE C.G.S. SYSTEM OF UNITS," 1875. 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 ex- plained ; 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. PREFACE. ix EXTRACT FROM PREFACE TO FIRST EDITION OF "UNITS AND PHYSICAL CONSTANTS," 1879. 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 multiplication and division, in connection with equations which express not numerical but physical equality. The advantages 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. 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. EXTRACT FROM PREFACE TO THE EDITION OF 1886. IN collecting materials for this edition, I have gone carefully through the Transactions and Proceedings of the Royal Society, the Royal Society of Edinburgh, and the Physical Society of London, from 1879 onwards, besides consulting numerous papers, both English and foreign, which have been sent to me by their authors. I have also had the advantage of the co-operation of Dr. Pierre Chappuis (of the Bureau International des Folds et Mesures), who has for some years been engaged in preparing x PREFACE. a German edition. Several items have been extracted from the very elaborate and valuable Physikalisch-Chemische Tabellen of Landolt and Bo'rnstein (Julius Springer, Berlin, 1883). A Supplemental Section has been added on physical deductions from the dimensions of units ; a simplification has been introduced in the discussion of adiabatic compression ; and the account of thermoelectricity has been rewritten and enlarged. The name "thermoelectric height" has been introduced to denote the element usually represented by the ordinates of a thermoelectric diagram'. The adoption of the Centimetre, Gramme, and Second, as the fundamental units by the International Congress of Electricians at Paris in 1881, led to the immediate execution of a French translation of this work, which was published at Paris by Gauthier-Villars in 1883. The German translation was com- menced about the same time, but the desire to perfect its collection of physical data has caused much delay. It will be brought out by Ambrosius Barth, the publisher of Wiedemann's Annalen. A Polish edition, by Prof. J. J. Boguski, was published at Warsaw in 1885 ; and permission has been asked and granted for the publication of an Italian edition. CONTENTS. PAGE* Tables for Reducing to and from C.G.S. Measures, xiii-xvi Chapter I. General Theory of Units, - 1-14 Chapter II. Choice of Three Fundamental Units, 15-20 Chapter III. Mechanical Units, - 21-30 Supplemental Section, on Physical Deductions from Dimensions, ... - 30-33 Chapter IV. Hydrostatics, - 34-49 Chapter V. Stress, Strain, Elasticity, and Viscosity, - 50-69 Chapter VI. Astronomy, - 70-75 Chapter VII. Velocity of Sound, 76-81 Chapter VIII. Light, - - 82-95 Chapter IX. Heat, - 96-143 Chapter X. Magnetism, 144-160 Chapter XI. Electricity, 161-210 Appendix. Reports of Units Committee of British Association, and Resolutions of Paris Congress, - 211-216 Index, - 17-220 REDUCTION TO AND FROM C.G.S. MEASURES. ACCORDING to Col. Clarke's comparisons of standards of length (printed in 1866), the metre is equal to 1-09362311 yard, or 3-2808693 feet, or 39-370432 inches, the standard metre being taken as correct at C., and the standard yard as correct at 16f C. Hence the inch is 2-5399772 centims., the foot 30-479726 centims.,the square inch 6-4514842 square centims., and the cubic inch 16-3866227 cubic centims. According to the U.S. Coast Survey Bulletin, No. 9, 1889, a more probable value of the metre is 39-36980 inches. 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 would be 2-2046212 pounds, and the pound 453-59265 grammes. Three standard pounds, one of platinum-iridium and the other two of gilded bronze, belonging to the Standards . Department, were compared, in 1883, at the Bureau In- ternational des Poids et Mesures, with standards belonging XIV C.G.S. UNITS AND CONSTANTS. to the Bureau, and their values in grammes were found to be respectively 453-59135, 453-58924, 453-58738. Travaux et Mtmoires, tome IV. In the following tables, cm. denotes centimetre or centimetres, gin. denotes gramme or grammes. The numbers headed " reciprocals " are the factors for reducing from C.G.S. measures. 1 inch, - Ifoot, - 1 yard, - 1 mile, - 1 sea mile, Length. cm. 2-5400 30-4797 91-4392 = 160933- - 185230- Reciprocals. 39370 032809 010936 6-2138 xlO 5-398 xlO 1 sq. inch, 1 sq. foot, 1 sq. yard, 1 sq. mile, Area. sq. cm. 6-45.15 929-01 8361-13 2-59 x 10 10 Reciprocals. 1550 001076 0001196 3-861xlO- n 1 cub. inch, 1 cubic yard, 1 pint, 1 gallon, Volume. cub. cm. 16-387 = 28316- . = 764535- 567-63 4541- Reciprocals. 06102 3 '532 x 10 1-308 xlO- 001762 0002202 1 grain, 1 ounce avoir. 1 pound .,, 1 ton, - Mass. gm. 0647990 = 28-3495 = 453-59 1-01605 xlO 6 Reciprocals. 15-432 035274 0022046 9 -84206 x 10 - TABLES. xv 1 knot, - 1 foot per sec., - 1 mile per hour, - 1 kilometre per hour, - The knot, according to velocity, not a length, per hour." Velocity. cm. per sec. Reciprocals. 51-453 -019435 30-4797 -032809 44-704 -022369 27*777 -036 the best nautical authorities, is a 'Knots" is equivalent to "sea miles Density. gm. per cub. cm. 00395438 016019 Reciprocals. 252-88 62-426 1 grain per cub. inch, 1 Ib. per cub. foot, - The value of g assumed in the following tables is 981 : Force. 1 poundal, 1 pound, - 1 grain, - 1 kilogramme, gm. 453-59 064799 10 3 Dynes. 13825- 4-45 x 10 5 63-6 9-81xl0 5 The ratio of the poundal to the dyne is independent of g. 1 Ib. per sq. foot,. 1 Ib. per sq. inch, 1 ton per sq. inch, - 1 inch mercury at 0, 30 1 centim. ,, ,, 76 ,, 1 grain per linear inch, 1 Ib. ,, foot, Stress. gm. per sq. cm. 48826 70-31 1 -575 x 10 5 34-534 1036- 13-596 1033-3 Dynes per sq. cm. 479- 69000- 1-545 xlO 8 33880- l-0163x!0 6 Surface Tension. gm. per cm. 02551 14-88 1 foot-pound, 1 foot-pound al, 1 foot-grain,- 1 foot4on, - 1 kilogrammetre, - 1 joule, Work and Energy. gm.-centims. 13825- 1-975 3-097 x 10 7 10 5 l-0136x!0 6 Dynes per cm. 25- 14600- Ergs. 1-3562 xlO 7 4-2139 x 10 5 1 -9375 x 10 s 3'0380xl0 lc 9-81 xlO 7 10 7 The value of the foot-poundal in ergs is independent of g. xvi C.G.S. UNITS AND CONSTANTS. Hate of Working. gm.-cm. per sec. Ergs per sec. 1 horse-power, - = 7'604xl0 6 7'46xl0 9 1 force-de-cheval, = 7'5 x 10 6 7'36xl0 9 1 kilowatt, 10 10 1 watt, - 10' Mechanical Equivalents of Heat. (For water at 10 C. or 50 F.) gm. centims. Ergs 1 gramme through 1 C., 4'281 x 10 4 4*2 x 10 7 1 pound through 1 C. , 1 '942 x 10 7 1 '905 x 10 10 1 pound through 1 F. , 1 '079 x 10 7 1 '058 x 10 10 Various Measures of Length, in centimetres. French foot, 32'484 (= 12 inches = 144 lines). Toise, 194-904 (-6 feet). Rhenish or Prussian foot, 31*385; Austrian foot, 31 '611 ; Bavarian, 29'186 ; Hanoverian, 29'209 ; Saxon, 28'319 ; Hessian, 28-770; Wurtemburg, 28 '649 ; Baden, 30 '000; Russian foot, 30-47945. The Russian sagene or sashen is 7 feet. Verst, 106678 (=500 sashen). Prussian mile, 753250 (= 24000 feet). Austrian mile, 758666 ( = 24000 feet). Geographical mile as understood in Germany, 742040. Various Measures of Mass, in grammes. Zollverein pound, 500 ; Prussian pound, 467*711 ; Austrian, 560-012 ; Russian, 409 '52. Each of these pounds is divided into 32 loth or lot, and 100 pounds make one centner. 'SE OF THE :VERSITY CHAPTER I. GENERAL THEORY OF UNITS. Units and Derived Units. 1. 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 Euclidean 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 inde- pendent units would be. For example, when the above 2 C.G.S. UNITS AND CONSTANTS. [CHAP. definition of the unit of area is employed, we can assert 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 acceleration 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 introduction of additional factors, which would involve needless labour in calculating and difficulty in remem- bering.* 5. The correlative term to derived is fundamental. * 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 i.] GENERAL THEORY OF UNITS. 3 Thus, when the units of area, volume, velocity, and acceleration are defined 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, I, 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 T. But V L T these numerical values are , - , ; v I t hence we must have V Lt (1) 7 IT This equation shows that, when the units are changed (a change which does not affect V, L, and T), v must vary directly as I 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 (1) also shows that the numerical value of a given velocity varies inversely as the unit of length, and directly as the unit of time. 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. 4 C.G.S. UNITS AND CONSTANTS. [CHAP. 7. Again, let A denote a concrete acceleration such that the velocity V is gained in the time T, and let o denote the unit of acceleration. Then, since the 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 V t a~ v T' L t V But by equation (1) we may write- -=, for . We thus obtain This equation shows that when the units a, I, t are changed (a change which will not affect A, L, T or T'), a must vary directly as I, and inversely in the duplicate ratio of t ; and the numerical value will vary inversely as I, and directly in the duplicate ratio of t. 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 i.] GENERAL THEORY OF UNITS. 5 saying that the dimensions of acceleration* are e . n ^ , or that the dimensions of the unit of acceleration* are 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 = -r^ 3 J time velocity length acceleration = 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, t 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 * Professor James Thomson (' Brit. Assoc. Report,' 1878, p. 452) objects to these expressions, and proposes to substitute the following : " Change-ratio of unit of acceleration = *a n ge.ratiooju^e ng th ,, (chaiige-ratio of unit of time)~ This is very clear and satisfactory as a full statement of the meaning intended ; but it is necessary to tolerate some abridg- ment of it for practical working. OF THE TT "NT T \r -c i- 6 C.G.S. UNITS AND CONSTANTS. [CHAP. acceleration based on the yard and minute. Equation (2) becomes A = 3 x /1\ 2 _ 1 , ,gv a T X \60/ ~I200' that is to say, an acceleration in which a yard per minute of velocity is gained per minute, is of an acceleration in which a foot per second is gained per second. Meaning of "per" 10. The word per, which we have several times em- ployed 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 dx 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 pro- portional to the time of describing it. Extended Sense of the terms " Multiplication " and "Division" 11. In ordinary multiplication the multiplier is always * It is not correct to speak of interest at the rate of Five Pounds per cent. It should be simply Five jyer 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 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 ; 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 un- changed when the time and distance are both changed in the same ratio. 12. The three quotients 1 mile 5280 ft. 22 ft. 1 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 time 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 denominator. S C.G.S. UNITS AND CONSTANTS. [CHAP. 99 ft 99 Further, since the velocity ' is "-- of the velo- 15 sec. lo 1 ft. ^, , . . 22 ft. 22 ft. city , we are entitled to write = -- . , 1 sec. 15 sec. 15 sec. thus separating the numerical part of the expression from the units part. In like manner we may express the result of 9 by writing yard 1 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 I to denote the unit length, m the unit mass, and t the unit time. Ex. 1. 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 i.] GENERAL THEORY OF UNITS. 9 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 ounce at a place where g (in foot-second units) is 32, what is the unit length ? We have * = sec., =1 and = 32 . ^l or ml - 32 oz. ft. = 2 Ib. ft. sec. 2 sec. J Hence by division / 4 =sY(ft.) 4 , / = ^ft. = 4in. 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 1 = 22 yd.; whence P = -sec. 2 = 121 sec. 2 , t = 11 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 8 = 32- t sec. t 2 secX whence by division t = \ sec., and substituting this value of i in the first equation, we have 4 I = 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. * For the dimensions of density and force, see 14. 10 C.G.S. UNITS AND CONSTANTS. [CHAP. We have I = 2 ft., t = \ sec., .. == , sec. 2 t 2 -j- 1 ^- sec. 2 that is 100 x 32 Ib. = 32 ra, m = 100 Ib. 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] contains 13500 oz. Find the unit of time. Let t = x sec., then / = x ft. ; also let m = y Ib. Then we have ml wlb. aft. y Ib. ft. - KA 00 Ib. ft. -M - 9 - 9- = - 9- = ou x 3^ ------- 3 being the product of moment of inertia by angular velo- city, or the product of momentum by length. Intensity of pressure, or intensity of stress generally, being force per unit of area, is of dimensions - ; that area is M ' EF 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 - e or --- It is numerically mass 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 denned 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 T ^ distance, and its dimensions will be = 9 - Curvature (of a curve) = r , 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. H C.G.S. UNITS AND CONSTANTS. [CHAP. T. 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 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 . 15 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. 16. The following are the most important considera- tions which ought to guide the selection of fundamental units : (1) They should be quantities admitting of very accurate comparison with other quantities of the same kind. 16 C.G.S. UNITS AND 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 comparisons 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 measured 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 magnitude in travelling from one place to another. A spring-balance, it is true, gives a direct measure of n.] THREE FUNDAMENTAL UNITS. 17 force ; but its indications are too rough for purposes of accuracy. IS. 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 defining 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 (0 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 distance between two scratches, and the comparison is made by optical means. The scratches are usually at the bottom of holes sunk half-way 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 UNIVERSITY 18 C.G.S. UNITS AND CONSTANTS. [CHAP. earth's rotation on its axis ; and it is upon the uniformity of this rotation that the preservation of our 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 ; some- times the metre, gramme, and second ; while sometimes 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, ii.] THREE FUNDAMENTAL UNITS. 19 there is great increase of difficulty and of liability to mistake. 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 reference 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 1 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 20 C.G.S. UNITS AND CONSTANTS. [CHAP. n. small numbers. Such numbers are most conveniently written by expressing them as the product of two factors, 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 3^40,000,000 will be written 3-24 x 10 9 , and -00000^24 will be written 3-24 x 10-8 21 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. = 980-6056 -2-5028 cos 2A--000003^, A, clenoting 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 I of the seconds' pendulum being connected with the value of g by the formula g = ifil. Dividing the above equation by ?r 2 we have, for the length of the seconds' pendulum, in centimetres, I = 99-3563 - -2536 cos 2X - -000 OOOSfc. 22 C.G.S. UNITS AND CONSTANTS. [CHAP. At sea-level these formulae give the following values for the places specified : Latitude. Value of y. Value of I. Equator, - 6 6 978-10 99-103 Latitude 45, 45 980-61 99-356 Munich, - 48 9 980-88 99-384 Paris, 48 50 980-94 99-390 Greenwich, 51 29 981-17 99-413 Gottingen, 51 32 981-17 99-414 Berlin, 52 30 981 -25 99-422 Dublin, - 53 21 981 -32 99-429 Manchester, 53 29 981-34 99-430 Belfast, - 54 36 981-43 99-440 Edinburgh, 55 57 981-54 99-451 Aberdeen, - 57 9 981-64 99-461 Pole, - 90 983-11 99-610 The difference between the greatest and least values (in the case of both g and Z) is about T fg of the mean value. 24. The Standards Department of the Board of Trade, being concerned only with relative determinations, has adopted the formula A denoting the latitude, h the height above sea-level, K the earth's radius, g Q the value of g in latitude 45 at sea- level, which may be treated as an unknown constant multiplier. Putting for R its value in centimetres, the formula gives g = g Q (l- -00257 cos 2A- 1-967* x 10~ y ), where h denotes the height in centimetres. in.] MECHANICAL UNITS. 23 The formula which we employed in the preceding section gives - -00255 cos 2 As regards the factor dependent on height, theory indi- cates 1 - as its correct value for such a case as that of a balloon in mid-air over a low-lying country ; the value 1 - - may be accepted as more correct for an elevated 4 JbC plateau on the earth's surface. Fwce. 25. 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 1 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 i , . force acceleration = ; mass ip that is, if a denote acceleration in C.G.S. units, a = - ; 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 /. 24 C.G.S. UNITS AND CONSTANTS. [CHAP. Hence we have F = ma = . Therefore, if mv = 1 and t=I t we have F = l. As a particular case, if m = 1, v=l, t=l, we have 26. 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 1 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, that is, to g. The weight (when weight means force) of 1 gramme is therefore g dynes ; and the weight of m grammes is mg dynes. 27. 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, etc., are, strictly speaking, units of mass. Such an expression as " a force of 100 tons" must be understood as an abbre- viation for " a force equal to the weight [at the locality in question] of 100 tons." 28. The name pounded 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 HI.] MECHANICAL UNITS. 25 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. ~s^c7~ = ~s^cT 2 ' a?= . = 453-59x30-4797 = 13825. gm. cm. Work ami Energy. 29. 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. 30. 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 the velocity required is such that tf = 2gh. Hence we have gmh = ^mv 2 . The kinetic energy of a mass of m grammes moving with a velocity of v centimetres per second is therefore Jwtfl 2 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 26 C.G.S. UNITS AND CONSTANTS. [CHAP. would do against opposing forces before it would come to rest. 31. 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 (30-4797) 2 = 421390 ergs. One foot-pound is 13,825 gramme-centims., which, if g be taken as 981, is 1-356 x 10 7 ergs. -32. The C.G.S. unit rate of working is 1 erg per second. Watt's V horse-power " is defined as 550 foot-pounds per second. This is 7*46 x 10 9 ergs per second. The "force de cheval " is denned as 75 kilogrammetres per second. This is 7 '36 x 10 9 ergs per second. We here assume g = 981. A new unit of rate of working has been lately intro- duced for convenience in certain electrical calculations. It is called the Watt, and is defined as 10 7 ergs per second. A thousand watts make a kilowatt. The following tabular statement will be useful for reference. 1 Watt = 10 7 ergs per second = -00134 horse-power = 737 foot-pounds per second =-1019 kilogram- metres per second. 1 Kilowatt = 1 '34 horse-power. 1 Horse-power = 550 foot-pounds per second = 76-0 kilogrammetres per second = 746 Watts = 1-01385 force de cheval. 1 Force de cheval = 75 kilogrammetres per second = 542-48 foot-pounds per second = 736 watts = -9863 horse-power. in.] MECHANICAL UNITS. 27 In connection with the Watt, a new unit of work has been introduced, called the Joule. It is 10 7 ergs. Examples. 1. 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 ? Ans. Its indications will be too high by about - of the total weight. 2. A cannon-ball, of 10,000 grammes, is discharged with a velocity of 45,000 centims. per second. Find its kinetic energy. Ans. x 10000 x (45000) 2 = 1-0125 x 10 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 10 10 dynes. 4. Given that 42 million ergs are equivalent to 1 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 con- verted into heat and to be taken up by the bullet. We have iv 2 = 15'6xj, where J = 42xl0 6 . Hence 2 = 1310 millions; v-36'2 thousand centims. per second. 5. With what velocity must a stone be thrown verti- cally upwards at a place where g is 981 that it may rise to a height of 3000 centims. ? and to what height would 28 C.G.S. UNITS AND CONSTANTS. [CHAP. 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. 33. A body moving in a curve must be regarded as continually falling away from a tangent. The accelera- /ji2 tion with which it falls away is , v denoting its velocity r 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 m grammes moving in a curve of radius r centimetres, with velocity v centimetres per second, is dynes. This f 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 ^ITY of revolution being T seconds, we have v = T ; hence = f ^ j r, and the force acting on the body is If n revolutions are made per minute, the value of T is 60 -, ,, f /W7rV > , , and the force is mr{ ) dynes. Examples. 1. A body of m grammes moves uniformly in a circle of radius 80 centims., the time of revolution being J of a in.] MECHANICAL UNITS. 29 second. Find the centrifugal force, and compare it with the weight of the body. Ans. The centrifugal force is m x f ~ J 2 x 80 = m x 647r 2 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. 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 OI9J! = 3-176 dynes. If g be 981 the slope will O.-l Ij O 1 be = ; that is, the surface will slope upwards from the concave side at a gradient of 1 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 =s where / denotes the intensity of centrifugal force that is, the centrifugal force per unit mass. We have therefore 30 C.G.S. UNITS AND CONSTANTS. [CHAP. 981 tan SO^IO^V, n denotin g the number of \30/ revolutions per minute, 90 Hence n = 91 -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 10 8 , T = 86164, being the number of seconds in a sidereal day. This is about - of the value of g. 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. SUPPLEMENTAL SECTION. On the help to be derived from Dimensions in investigating Physical Formulas. When one physical quantity is known to vary as some power of another physical quantity, it is often possible to find the exponent of this power by reasoning based on dimensions, and thus to anticipate the results or some of the results of a dynamical investigation. Examples. 1. The time of vibration of a simple pendulum in a small arc depends on the length of the pendulum and the intensity of gravity. If we assume it to vary as the m tJi in.] MECHANICAL UNITS. 31 power of the length, and as the n th power of g, and to be independent of everything else, the dimensions of a time must equal the m th power of a length, multiplied by the n th power of an acceleration, that is _ T rn. + i r P 2 Since the dimensions of both members are to be identi- cal, we have, by equating the exponents of T, 1 = - 2?i, whence n = - J, and by equating the exponents of L, m + n = 0, whence m = \\ that is, the time of vibration varies directly as the square root of the length, and inversely as the square root of g. 2. The velocity of sound in a gas depends only on the density D of the gas and its coefficient of elasticity E, and we shall assume it to vary as D m E n . The dimensions of velocity are LT' 1 . The dimensions of density, or , are ML" 3 . volume The dimensions of E, which will be explained in the chapter on stress and strain, are - -. or (MLT~ 2 )L~ 2 , area or ML- 1 T- 2 . The equation of dimensions is whence, by equating coefficients, we have the three equations 1 = - 3m -n, -I = 2n, m* n = 0, to determine the two unknowns m and n. 32 C.G.S. UNITS AND CONSTANTS. [CHAP. The second equation gives at once n=4. The third then gives m = -i, and these values will be found to satisfy the first equation also. The velocity, then, varies directly as the square root of E, and inversely as the square root of D. 3. The frequency of vibration / for a musical string (that is, the number of vibrations per unit time) depends on its length /, its mass m, and the force with which it is stretched F. The dimensions of/ are T~ J . F MLT-?. Assume that / varies as I* m* P. Then we have giving - 1 = - 2z, x + z = 0, y + z = Q- whence z = }, x = - J, y = - i . VF ,- . im 4. The angular acceleration of a uniform disc round its axis depends on the applied couple G, the mass of the disc M, and its radius E. Assume it to vary as G*M*R*. The dimensions of angular acceleration are T~ 2 . G B Hence we have in.] MECHANICAL UNITS. 33 giving -2=-2./:, whence x = 1, y = - 1, z = - 2. Hence the angular acceleration varies as In the following example the information obtained is less complete. 5. The range of a projectile on a horizontal plane through the point of projection depends on the initial velocity V, the intensity of gravity g, and the angle of elevation a. The dimensions of range are L. V LT->. 9 LT-*. a LT, and the dimensions of all powers of a are LT. Hence we can draw no inferences as to the manner in which a enters the expres- sion for the range. The dimensions of this expression will depend upon V and g alone. Assume that the range varies as Vy. Then L = giving m + =l, whence m = 2, n= - 1. V 2 Hence the range varies as when a is given. CHAPTER IV. HYDROSTATICS. 34. IN the C.G.S. system, density is expressed in grammes per cubic centim. Denning the specific gravity of a substance as the ratio of its density in its actual condition to the density of pure water at 4 C., specific gravity is so nearly identical with density in the C.G.S. system that it is uncertain which is the greater. According to the observations of Kupffer, reduced by Prof. W. H. Miller, the density of pure water at 4 C. is 1 '0000 13. According to the observations of Tralles, reduced by Broch, it is -99988. A fresh determination is in progress at the Bureau International des Poids et Mesures. As regards the density of pure water at 62 F. (16| C.) in British measures, there is still wider divergence of authorities. Rankine, at p. 99 of Rules and Tables, says 277'123 cubic inches "is the correct volume of 10 Ibs. of pure water at 62 Fahr., and is therefore the true value of a gallon in cubic inches. By a former Act of Parliament, since repealed, a gallon was declared to be 277-274 cubic inches." CHAP, iv.] HYDROSTATICS. 35 To find the density in grains per cubic inch, we must divide these numbers into 70,000. We thus obtain 252-595 from the value adopted by Eankine, and 252*458 from the erroneous value in the repealed Act. Mr. H. J. Chaney, Warden of the Standards, in the Proceedings of the Eoyal Society for 1890, No. 294, p. 230, says the hitherto accepted value is 252*458, and gives a new determination from his own measurements, which is 252-286. This value of Mr. Chaney's is equivalent to '997643 gm. per cub. centim. ; and as Mr. Chaney adopts -998881 for the ratio of the volumes at 4 C. and 62 F., the density at 4 C. resulting from this determination is 998752 gm. per cub. centim., which differs from the theoretical value unity by '00125 a departure ten times as great as that found by Tralles and Broch, or 100 times as great as that found in the opposite direction by Kupffer and Miller. 35. The table on next page gives the volume of pure water at temperatures from to 100 C. in terms of the volume at C. To compare with the volume at 4 C. it is necessary to add -00017. The values from to 30 are taken from Broch's table, and those from 35 to 100 from a comparison of Rossetti's and Volkmann's. Herr's formula for the volume at t C., t being between and 30, in terms of the volume at C. is 1 - -000 0603 t+ -000 007 93 t 2 - '000 000 0426 P. The ratio of the density at 4 C. to the density at 62 F. (16f C.) is 1-001118 according to the above table, and this is also the value adopted by Rankine(Rules and OF THE 36 C.G.S. UNITS AND CONSTANTS. [CHAP. Tables, p. 146). Chaney adopts the reciprocal of -998881 , which is 1-001120. Temp. Cent. Volume. Temp. Cent. Volume. 1-000000 15 1 -000735 1 -999948 16 890 2 911 17 1-001057 3 889 18 235 4 883 19 424 5 891 20 624 6 914 21 835 7 952 22 1 -002057 8 1-000003 23 289 9 068 24 530 10 147 25 78 11 239 26 1-00304 12 344 27 31 13 462 28 59 14 593 29 88 Temp. Cent. 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 Volume. 1-00418 1-0057 76 97 1-0118 142 168 195 1 -0225 256 288 1-0321 356 392 1 -0430 36. TABLE OF DENSITIES. Solids. (For the most part rough averages.) Aluminium, Antimony 2-6 6'7 Carbon (diamond), ... (graphite) 3-5 2-3 Bismuth, 9-8 ,, (g&s carbon) 1-9 Brass, 8-4 ,, (wood charcoal), 1-6 Copper, Gold 8-9 19-3 Phosphorus (ordi- nary) 1-83 Iron, 7-8 ,, (red), . 2-2 Lead, 11-3 Sulphur (roll), 2-0 Nickel, 8-9 Quartz (rock cry- Platinum, Silver 21-5 10'5 stal), Sand (dry) 2-6512 1-42 Sodium 98 Clay, . . 1-9 Tin 7-3 Brick 2-1 Zinc 7'1 Basalt 3-0 Cork, 24 Chalk, 1-8 to 2-8 Oak, Ebony -7 to 1-0 1-1 to 1-2 Glass (crown), (flint). 2-5 to 2-7 3-0 to 3-5 Ice. .. 917 Porcelain, . . 2-4 HYDROSTATICS. 37 Liquids at C. Sea water, Alcohol, . 1-026 806 Sulphuric Acid, Nitric Acid, 1-85 1-56 Chloroform, 1-527 Hydrochloric Acid,.. 1-27 Ether. 736 Milk T03 Bisulphide of Carbon, Glycerine, . . Mercury,... 1-293 1-27 13-596 Oil of Turpentine, ... ,, Linseed, ,, Mineral,.. 87 94 76 to -83 More exactly, the density of mercury at C., as com- pared with water at the temperature of maximum density, under atmospheric pressure, is 13*5956. 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 displayed is m - m' grammes, and each gramme of this water occupies a cubic centimetre. Examples. 1. A glass cylinder, / centims. long, weighs m grammes in vacuo and m' grammes in water of unit density. Find its radius. Solution. Its section is Trr 2 , and is also rrl ~ m hence I 2. Find the capacity at C. of a bulb which holds m grammes of mercury at that temperature. Solution. The specific gravity of mercury at being 13-596 as compared with water at the temperature of maximum density, it follows that the mass of 1 cubic centim. of mercury is 13-596. Hence the required ... m , . capacity is ^T-^W cubic centims. * 38 C.G.S. UNITS AND CONSTANTS. [CHAP. 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 unity density, atmos- pheric pressure being neglected. Ans. Ah grammes weight ; that is, gAh dynes. 4. If mercury of specific gravity 13*596 is substituted for water in the preceding question, find the pressure. Ans. 13-596 Ah grammes weight; that is, 13-596 gAJi 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 l-0136x!0 6 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. 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 C., 76 centims. high, is found by putting ft = 76, d= 13-596, and is 1033-3#. It is therefore different at different localities. At Paris, where g is 980-94, it is 1-0136 x 10 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. * The prefix mega denotes multiplication by a million. A megadyne is a force of a million dynes. iv.] HYDROSTATICS. 39 At Greenwich, where g is 981*17, the pressure due to 76 centims. of mercury at C. is 1*0138 x 10 6 ; and the height which would give a pressure of 10 is 74*964 centims., or 29*513 inches. Convenience of calculation would be promoted by adopting the pressure of a megadyne per square centim., or 10 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- portional to the value of g. We shall adopt the megadyne per square centim. as our standard atmosphere in the present work. Examples. 1. What must be the height of a column of water of unit density to exert a pressure of a megadyne per square centim. at a place where g is 981 1 Ans. 1Q QQ QQ = 1019-4 centims. This is 33*445 feet. 981 2. What is the pressure due to an inch of mercury at C. at a place where g is 981 1 (An inch is 2*54 centims.) Ans. 981x2*54x13-596 = 33878 dynes per square centim. 3. What is the pressure due to a centim. of mercury at 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 1 40 C.G.S. UNITS AND CONSTANTS. [CHAP. 981 x 10 5 x 1-027 = 1-0075 x 10 8 dynes per square centim., or 1-0075 x 10 2 megadynes per square centim. ; that is, about 100 atmospheres. 5. What is the pressure due to a mile of the same water ? Ans. 1-6214 x 10 8 C.G.S. units, or 162-14 atmospheres [of a megadyne per square centim.]. Density of Air. 39. Regnault found that at Paris, under the pressure of a column of mercury at 0, 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 10 6 dynes per square centim. Hence by Boyle's law, we can compute the density of dry air at C. at any given pressure. At a pressure of a megadyne (10 6 dynes) per square centim. the density will be - -0012759. 1 *013o The density of dry air at C. at any pressure p (dynes per square centim.) is px 1-2759 x 10~ 9 . ... (4) Example. Find the density of dry air at C., at Edinburgh, under the pressure of a column of mercury at C., of the height of 76 centims. Here we have ^ = 981-54 x 76 x 13-596 - 1-0142 x 10 6 . Ans. Required density = 1-2940 x 10~ 3 = -0012940 gramme per cubic centim. iv.] HYDROSTATICS. 41 40. Absolute Densities of Gases, in grammes per cubic centim.) at C., and a pressure of 10 6 dynes per square centim. Mass of a cubic Volume of a gramme centim. in grammes. in cubic 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 MarshGas, '0007173 1394-1- Chlorine, '0030909 323'5 - Protoxide of Nitrogen, '0019433 514'6 Binoxide '0013254 754'5 Sulphurous Acid, '0026990 370'5 Cyanogen, '0022990 435'0 Olefiant Gas '0012529 798'1 Ammonia, '0007594 1316'8 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 given pressure and temperature are directly as their atomic weights, we have for any gas at zero pvfj.= 1-1316 x!0 10 m; v denoting its volume in cubic centims., m its mass in grammes, p its pressure in dynes per square centim., and /x its atomic weight referred to that of hydrogen as unity. 41. DEPARTURE FROM BOYLE'S LAW. Regnault found that for most gases the product of the volume and pressure decreased as the pressure was in- creased from about 1 atm. to about 30 atm. When the 42 C.G.8. UNITS AND CONSTANTS. [CHAP. initial pressure was about 70 and the final pressure about 140 cm. of mercury, the ratio of the initial to the final value of YP had the following values, which are arranged in order of magnitude Air, 1-00215 j Hydrochloric acid,... 1*00925 Binoxide of nitrogen, 1 -00285 Sulphuretted hydrogen, 1 '01083 Carbonic oxide, 1 -00293 Marsh gas, 1-00634 Protoxide of nitrogen, 1 '00651 Ammonia, 1'01881 Sulphurous acid, 1 '02088 Cyanogen, T02353 Carbonic acid, 1 '00722 For hydrogen, the deviation from Boyle's law was in the opposite direction and smaller in amount. He summed up his results in the empirical formula ^ = 1 + A(m - 1 ) + B(m - 1 ) 2 , m denoting ^?. For air and for nitrogen A was negative and B positive. For carbonic acid both A and B were negative. For hydrogen both A and B were positive. The logarithms of their arithmetical values were Log A. Log B. Air, 3-0435120 5-2873751 Nitrogen, 4*8389375 6'8476020 Carbonic acid, ... 3'9310399 6 '8624721 Hydrogen, 4'7381736 6*9250787 42. The researches of Natterer and the more recent researches of Cailletet and of Amagat have shown that when much higher pressures are employed the value of VP (for gases other than hydrogen) continues to decrease up to a certain point, which is different for different gases, and then continually increases more and more rapidly. Amagat places the minimum value of VP at 50 m. of mercury for nitrogen, 100 m. for oxygen, 65 m. for air, and 50 m. for carbonic oxide. The iv.] HYDROSTATICS. 43 following is a sample of Cailletet's results for nitrogen at 15. They fix the minimum at about 60 m. P in metres of mercury. VP. P. VP. 39-359 8184 89-231 8323 49-271 8022 124-122 8857 59-462 7900 174-100 9191 64-366 7951 181-985 9330 For references, see Jamin et Bouty, torn. L, pp. 213-217. Height of Homogeneous Atmosphere. 43. We have seen that the intensity of pressure at depth h, 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 p - #HD, where p 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 13-596 and height h, we have p = ghx 13-596; .-. HD = A x 13-596, H^* 1 ^' 596 . If it were possible for the whole body of air above the point to be reduced by vertical compression to the 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. 44 C.G.S. UNITS AND CONSTANTS. [CHAP. 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 H-; ... (5) 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 p. If, instead of p 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 H-g. (6) 44. 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 only. For dry air at zero we have, by formula (4), p. 40, H _ 7-8376 x 1Q8 g At Paris, where g is 980*94, we find H = 7-990 x 10 5 . At Greenwich, where g is 981 -17, H- 7-988 x 10 5 . Examples. 1. Find the height of the homogeneous atmosphere at Paris for dry air at 10 C:, and also at 100 C. iv.] HYDROSTATICS. 45 Ans. For given density, p varies as 1 + *00366 t, t de- noting the temperature on the Centigrade scale. Hence we have, at 10 C., H = 1-0366 x 7-99 x 10 5 = S'2824 x 10 5 ; and at 100 C., H = 1-366 x 7-99 x 10 5 = 1-0914 x 10 6 . 2. Find the height of the homogeneous atmosphere for hydrogen at 0, at a place where g is 981. Here we have Diminution of Density with increase of Height in the Atmosphere. 45. 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 1 centim. will involve a diminution of the pressure (and therefore of the density) by of itself, since the layer of air which H has been traversed is - of the whole mass of superincum- H bent air. The density therefore diminishes by the same fraction of itself for every centim. that we ascend ; in other words, the density and pressure diminish in geo- metrical progression as the height increases in arithmetical progression. Denote height above a fixed level by x, and pressure by p. Then, in the notation of the differential calculus, T dx dp we have = -JL H p 46 C.G.S. UNITS AND CONSTANTS. [CHAP. and if p v p 2 are the pressures at the height x^ x 2 , we deduce Hx 2-3026 Iog 10 l. ... (7) P* Pz 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 2 -x l = (l + -00366 1) 7-988 x 10 5 x 2-3026 log^l = (1 + -003660 1,839,300 log-?!, . . . (8) 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 ll in these formulae. The required height will be H log e 2, or, in the latitude of Greenwich, for temperature C., will be 1-8393 x 10 6 x -30103 = 553700. The value of log e 2, or 2-3026 Iog 10 2, is 2-3026 x -30103 =-69315. Hence, by (7), for an atmosphere of any gas at uniform temperature, 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. HYDROSTATICS. 47 2. At what height, in an atmosphere of hydrogen at C., would the density be halved, g being 981 ? Ans. 7-9955 x 10 6 . Capillarity. 46. The phenomena of capillarity, soap-bubbles, etc., 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 in- creases. Superficial tensions at 20 C., in dynes per linear centim., deduced from Quincke's results. Tension of Surface separating the Liquid from Density. Air. Water. Mercury. Water, 0-9982 81 418 Mercury, - 13-5432 540 418 Bisulphide of Carbon 1-2687 32-1 41-75 372-5 Chloroform, 1 -4878 30-6 29-5 399 Alcohol, 7906 25-5 399 Olive Oil, - 9136 36-9 20-56 335 Turpentine, 8867 29-7 11-55 250-5 Petroleum, - 7977 31-7 27-8 284 Hydrochloric Acid, 1-1 70-1 377 Solution of Hyposul ) phite of Soda,- ( 1-1248 77-5 ... 442-5 The values here given for water and mercury are only applicable when special precautions are taken to ensure cleanliness and purity. Without such precautions smaller values will be obtained. (Quincke in Wied. Ann., 1886, page 219.) 48 C.G.S. UNITS AND CONSTANTS. [CHAI-. The following values are from the observations of A. M. Worthington (Proc. Eoy. Soc., June 16, 1881), at tempera- tures from 15 to 18 C., for surfaces exposed to air : Surface Tension. In gm. per cm. In dynes per cm. Water, -072 to '080 70 '6 to 78 '5 Alcohol, -02586 ' 25'3 Turpentine, '02818 27 '6 Olive Oil, -03373 33'1 Chloroform, '03025 29 '6 47. Very elaborate measurements of the thicknesses of soap films have been made by Reinold and Riicker (Phil. Trans., 1881, p. 456 ; and 1883, p. 651). When so thin as to appear black, the thickness varied from 7*2 to 14 -5 millionths of a millimetre, the mean being 11-7. This is 1*17 x 10~ 6 centimetre. The following thicknesses were observed for the colours of the successive orders : FIRST ORDER Red Thickness, cm. 2'84x 10- 5 Yellow-Green,. Red, ] Thickness. cm. 9'64xlO- 5 LO'52 , SECOND ORDER Violet .. .. 3-05 FIFTH ORDER Green, .. 1-119 xlO- 4 Blue, Green 3-53 , 4'09 Red 1-188 1-260 Yellow, . . 4-54 , 1-335 ,, Orange, 4-91 " Red, 5-22 , SIXTH ORDER Green, 1-410 ,, THIRD ORDER Purple 5'59 , j > Red, 1-479 1-548 ,, Blue, 5-77 1-627 ,, Green, 6-03 6-56 SEVENTH ORDER 1 TO'i Yellow, 7-10 , 1 -787 Red, 7'65 Bluish Red, 8-15 , Keel, ooy ,, 1-936 ., FOURTH ORDER Green, 8-41 EIGHTH ORDER 2'OOd. 8-93 , Red, .. 2-115 IV.] HYDROSTATICS. 49 48. Depression of the barometrical column due to capillarity, according to Pouillet : Internal Internal Internal Diameter of tube. Depression. Diameter of tube. Depression. Diameter of tube. mm. mm. mm. mm. mm. 2 4-579 8-5 604 15 2-5 3-595 9 534 15-5 3 2-902 9-5 473 16 3-5 2-415 10 419 16-5 4 2-053 10-5 372 17 4-5 1-752 11 330 17-5 5 1-507 11-5 293 18 5-5 1-306 12 260 18'5 6 1-136 12-5 230 19 6-5 995 13 204 19'5 7 877 13-5 181 20 7'5 775 14 161 20-5 8 684 14-5 143 21 Depression, mm. 127 112 099 087 077 068 060 053 047 041 036 032 028 50 CHAPTER V. STRESS, STRAIN, ELASTICITY, AND VISCOSITY. 49. 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 sup- pose the ratio of the initial to the final length of every line in the strained body to be nearly a ratio of equality. 50. 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 equal and 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 continu- ously 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. CHAP, v.] STRESS, STRAIN, ELASTICITY, VISCOSITY. 51 51. The axes of a strain are the three directions in the body, 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 un- strained 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 an oblique parallele- piped. 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. 52. 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 sec- tions. We shall suppose stress to be homogeneous, in what follows, unless the contrary is expressed. 53. When the force-action across a section consists of 52 C.G.S. UNITS AND CONSTANTS. [CHAP. a simple pull or push normal to the section, the direction 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 p TV/T unit area," or - - are ^=. , which we shall therefore call area LT 2 the dimensions of stress. 54. The relation between the stress acting upon a body and the strain produced depends upon the elasticity* 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 called coefficients of elasticity. A coefficient of elasticity expresses the quotient of a stress (of a given kind) by the strain (of a given kind) which it produces. Steel is an example of a body with large, and cork of a body with small, coefficients of elasticity. In all cases (for solid bodies) equal and opposite strains (supposed small) require for their production equal and opposite stresses. * The word resilience, which was employed in previous editions of this Work, in place of elasticity, has strong recommendations ; but engineers had already appropriated it in a different sense, and hence the proposed change has not found favour. v.] STRESS, STRAIN, ELASTICITY, VISCOSITY. 53 55. The coefficients of elasticity most frequently re- ferred to are the three following : (1) Elasticity 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 elasticity of volume. This is the only kind of elasticity possessed by liquids and gases. (2) Young's modulus, or the longitudinal elasticity 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 I, then - is taken as the measure 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 + Z, the value p T of Young's modulus for the wire is -r- -y- (3) " Simple rigidity " or resistance to shearing. This requires a more detailed explanation. 56. A shear may be denned as a strain by which a sphere of radius unity is converted into an ellipsoid of semiaxes 1, 1 +e, 1 -e- } in other words, it consists of an 54 C.G.S. UNITS AND CONSTANTS. [CHAP. extension in one direction combined with an equal com- pression in a perpendicular direction. 57. A unit square (Fig. 1) whose diagonals coincide with these directions is altered by the strain into a rhombus whose diagonals are (1 + e) ^/2 and (10) J2, and whose area, being half the product of the diagonals, is 1 - e 2 , or, to the first order of small quantities, is 1 , 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 N/l + e 2 or 1 4- 10 2 , and is therefore, to the first order of small quantities, equal to a side of the original square. 58. 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. 1. Then we have ab = J, cb = \e ^/2 = ^, angle cba = -r. Hence the length of the perpendicular cd since ad is ultimately is cb sin- = = - ; and 2 2 equal to ab, we have, to the first order of small quan- tities, i i cd &e angle cab = \- e. ad v.] STRESS, STRAIN, ELASTICITY, VISCOSITY. 55 The semi-angles of the rhombus are therefore - e, and the angles of the rhombus are ^ 2e ; in other Z words, each angle of the square has been altered by the amount 2 Other specimens of copper in abnormal states gave results ranging from 3-86 x 10 11 to 4-64 x 10 11 . The following are reduced from Wertheim's results (Ann. de Chim., ser. 3, torn, xxiii.) g being taken as 981 : Different Specimens of Glass (Crystal). Young's Modulus, 3 '41 to 4 -34, mean 3 '96 } Simple Rigidity, 1-26 to 1'66, T48 V x 10 U Volume Elasticity, .... 3 '50 to 4 '39, 3'89j Different Specimens of Brass. Young's Modulus, 9*48 to 10'44, mean 9'86) Simple Rigidity, 3 "53 to 3'90, ,, 3'67 [ x 10 11 Volume Elasticity, ... 10 '02 to 10 '85, 10'43j 68. Savart's experiments on the torsion of brass wire (Ann. de Chim., 1829) lead to the value 3-61 x 10 11 for simple rigidity. Kupffer's values of Young's modulus for nine different specimens of brass range from 7*96 x 10 n to 11-4 x 10 11 , the value generally increasing with the density. For a specimen, of density 8*4465, the value was 10-58 x 10 11 . v.] STRESS, STRAIN, ELASTICITY, VISCOSITY. 61 For a specimen, of density 8*4930, the value was 11-2x10" His values of Young's modulus for steel range from 2O2xl0 11 to 21-4 x 10". 69. The following are reduced from Rankine's Rules and Tables, pp. 195 and 196, the mean value being adopted where different values are given : Steel Bars Tenacity. , 7 -93 x 10 9 Young's Modulus. 2-45 x 10 12 Iron W^ire 5 -86 , 1-745 Copper Wire, ... 4-14 ,, 1-172 ,, Brass Wire, ... 3-38 ,, 9-81 x 10 11 Lead, Sheet, ... 2-28 xlO 8 5-0 xlO 10 Tin, Cast, Zinc. ... 3-17 5-17 . Ash, M72xl0 9 Spruce, 8-55 x 10 8 Oak, l-026x!0 9 1-10 xlO 11 MO 1-02 , Glass, Brick and Cement,.. 6-48x108 5-52 x 10 11 2-0 xlO 7 The tenacity of a substance may be defined as the greatest longitudinal stress that does not produce rupture. It would be equal to the product of Young's modulus by the ultimate extension if the law of simple proportion- ality held up to the breaking point. In most cases it is very much less "than this product. 70. Professor Tait's results for the compressibilities of water, mercury, and glass (results of Challenger Expedition), are stated with reference to two different units of stress. The pressures actually applied are ex- pressed in tons per square inch, andilw^teaults are UNIVERSITY 62 C.G.S. UNITS AND CONSTANTS. [CHAP. stated in the form of mean compression per atmosphere. Taking g at Edinburgh as 981-5, we have 1 ton per sq. inch = 154'6 megadynes per sq. centim. 1 atmosphere =1-0142 Starting from atmospheric pressure, the mean com- pression of fresh water per atmosphere at t C. (t being between and 15), for the first ton per sq. inch, and for the first 3 tons per sq. inch, had the values (504 - 3-6 t + -04 F) 10~ 7 for first ton per in., (478 - 3-7 1 + -06 P) 10~ 7 for first 3 tons per in., and for sea water the values were (462 -3-2* + -04* 2 )10- 7 for first ton per in., (437-5 - 2-95 t + -05 t 2 ) 10~ 7 for first 3 tons per in. At C., the mean compression per atmosphere for the first p tons per sq. inch, was (520 - 17 p +p 2 ) 10~ 7 for fresh water, (481 - 21-25^ + 2-25^ 2 ) 10~ 7 for sea water. For solutions of common salt at C., containing s parts of salt to 100 of water, the mean compression per atmosphere for the first p tons per sq. inch was 00186 SQ+p+s The temperature of maximum density of water was found to be lowered about 3 C. by 150 atmospheres of additional pressure. The temperature of minimum compressibility was lowered by increase of pressure. At atmospheric pres- sure it was inferred to be about 60 C. for fresh water and about 56 C. for sea water. The compression of mercury per atmosphere was found to be -0000036, and that of glass -000 0026. Owing to v.] STRESS, STRAIN, ELASTICITY, VISCOSITY. 63 slipping of the recording indices during relief of pressure all the above values of compressibility may be a little too 'small. 71. Amagat (Com. Ren. 1889, I., p. 228) found the following values of compressibility per atmosphere : Compressibility. Poisson's Ratio. Mercury, ......... '000003918 Glass (common), ...... 2181 '2451 Do. (crystal), ...... 2405 '2499 He quotes Cautone of Palermo as finding, for four cylinders of glass, the values of Poisson's ratio -246, 261, -264, -256, giving a mean -257. 72. The following are the results of Cailietet's experi- ments (Com. Ren. LXXV. p. 77) which were carried to 600 and 700 atmospheres. We have calculated the column headed "Real" by adding -0000039 to the column headed "Apparent." Density Temp. Compression per atmosphere. Apparent. Real. Bisulphide of carbon, ... 8 . 980 -0001019 ... 8 676 -0000715 Alcohol, ............... -858 9 701 740 ... 9 727 766 Petroleum oil, ........ -865 11 828 '0000867 Essence of petroleum, '720 10 "5 '0000981 '0001020 Ether, ..................... 10 '0001440 1479 Sulphurous acid, ...... -14 '0003014 3053 73. The following values for compression of liquids are reduced from Grassi results as given by Jamin and Bouty, vol. 1, part 2, p. 131. The value for mercury is probably erroneous. 64 C.G.S. UNITS AND CONSTANTS. [CHAP. - Temp. Cent. Volume- Elasticity. Compression for one Atmosphere (megadyne per square centim.) Mercury, Water, o-o o-o [3-436X10 11 ] 2-02 xlO 10 [2-91 x 10- 6 ] 4-96 xlO- 5 . 1-5 1-97 5-08 . 4-1 2-03 4-92 . 10-8 2-11 4-73 . 13-4 2-13 4-70 . 18-0 2-20 4-55 . 25-0 2-22 4-50 . 34-5 2-24 4-47 1 43-0 2-29 4-36 53-0 2-30 4-35 Ether, - f 0-0] 1 'r 1 14-01 9-2 xlO" 7'8 7'2 1-09 xlO- 4 1-29 1-38 Alcohol, f rs\ 1 13-1 / 1-22 xlO 10 1-12 8-17 xlO- 5 8-91 According to experiments by Quincke (Berlin Trans- actions, April 5, 1885) the following are the compressions due to the pressure of one atmosphere. They are ex- pressed in millionths of the original volume : Compression in millionths. at C. Glycerine, 25'24 Rape oil (riibol), 48'02 Almond oil, 48*21 Olive oil, 48-59 Water, 50'30 Bisulphide of carbon, 53 '92 Oil of turpentine, 58'17 Benzol from benzoic acid, Benzol, Petroleum, 64'99 Alcohol, 82-82 Ether, 115'57 at f C. t. 25-10 19-00 58-18 17-80 56-30 19-68 61-74 18-3 45-63 22-93 63-78 17-00 77-93 18-56 66-10 16-78 62.84 16-08 74-50 19-23 95-95 17-51 147-72 21-36 v.] STRESS, STRAIN, ELASTICITY, VISCOSITY. 65 The coefficient of volume elasticity in C.G.S. units can be calculated from the compression per atmosphere by dividing this compression into 1-014 x 10 6 . Thus we have approximately : Compression Elasticity per of atmosphere. volume. Water, 5xlQ- 5 2'03xl0 10 Mercury, 3'9xlO- 2 "60x10" Glass, 2-6xlO~ 6 3'90xlO u Fiscosity. 74. The frictional resistance which a fluid offers to change of shape in other words to "shearing" is found to be directly proportional to the rapidity with which the change is effected in other words to the " shear per unit time." Shearing may be regarded as the relative sliding of parallel planes without change of their mutual distance ; and the tangential force per unit area of one of these planes (or the intensity of the shear- ing stress) is the proper measure of the frictional resist- ance of the fluid at the actual rate of shearing. The quotient tangential force per unit area shear per unit time is accordingly the proper measure of the quality of the fluid in virtue of which it resists distortion. It is called the coefficient of viscosity, or simply the viscosity, of the fluid. The omission of the words " per unit time " from the denominator would convert the definition into that of " simple rigidity " ( 63). The dimensions of " tangential force per unit area " are - , and of " shear per unit time," ^ ; hence the 66 C.G.S. UNITS AND CONSTANTS. [CHAP. dimensions of viscosity are . In the C.G.S. sys- tem the coefficient of viscosity denotes the number of dynes per square centim. necessary to produce unit shear per second. 75. When a fluid is forced through a long and very narrow horizontal tube, calculation shows that, if the velocity at the circumference is zero, the volume of fluid which enters or leaves the tube per unit time is I denoting the length of the tube, a its radius, p l and p 2 the pressures at its ends, and ^ the viscosity. In the case of a gas the " volume " is to be reckoned at the pressure ^(p 1 +p<^- If the velocity at the circumference (called the velocity of slipping) is not zero, but is propor- tional to the tangential force per unit area at the circum- ference, the expression for the volume per unit time will be TTO? p l p 2 /a l\ ^"~r~(^j + pj i where ft denotes tangential force per unit area^ velocity oi slipping This reduces to the previous formula when _ is put equal to zero. Experiment confirms the three laws expressed by the first formula ; that is to say, the flow is directly as the difference of pressures at the ends, inversely as the length, and directly as the fourth power of the radius. Some investigators, by experiments of a different kind, have arrived at the conclusion that - has v.] STRESS, STRAIN, ELASTICITY, VISCOSITY. 67 a finite and sensible value, but their results are disputed. 76. Elevation of temperature increases viscosity in the case of gases and diminishes it in the case of liquids. The pressure of a gas has no influence on its viscosity (within ordinary ranges of pressure). The following determinations of viscosity have been mainly selected from a voluminous collection kindly placed at my disposal by Mr. Carl Barus, of the U.S. Geological Survey. Viscosity of Liquids. 77. The following values for water are based on the results obtained by Sprung (1876), O. E. Meyer (1877), and Slotte (1883): Temp. Viscosity. Temp. Viscosity. Temp. Viscosity. 0181 25 0091 50 0056 5 154 30 81 60 47 10 133 35 73 80 36 15 116 40 67 90 32 20 102 45 61 Meyer adopts the formula 0183 formula 1 + -0369 5212 and Slotte the 26 -00131, as approximately representing their results, denoting temperature Cent. Koch (1881) obtained the following values for mercury (omitting superfluous decimals) : Temp. Viscosity. Temp. Viscosity. Temp. Viscosity. 6 0169 99 0123 249 00964 10 162 154 109 315 918 18 156 197 102 340 897 68 C.G.S. UNITS AND CONSTANTS. [CHAP. 78. Wijkander (1879) gives the following values for other liquids : Alcohol. Ether. Benzine. 10 00156 10 -00283 10 00746 15 141 15 271 20 641 20 127 20 258 30 555 25 114 25 245 40 488 30 104 30 233 50 433 40 86 50 72 Carbon Bisulphide. Chloroform. 12 00393 12 00617 20 370 20 567 25 357 25 539 30 344 30 513 35 332 35 489 40 467 79. Schottner found for glycerine the values 42 at 3, and 8 at 20. Obermayer (1877) found for pitch the values 2-1 x 10 9 between 6 and 7, 5-3 x 10 8 at 10-1, 2-6 x 10 8 at 12-2 ; and for storax the value 13 x 10 10 between 15 and 16. Carl Barus (1890) has found for marine glue at about 25 the value 2 x 10 8 ; and for paraffin at about 25 a value exceeding 2 x 10 11 . Viscosity of Gases. 80. Schumann (1884) found for air (omitting super- fluous decimals), by vibration experiments : emp. Viscosity. Temp. Viscosity. Temp. Viscosity. 000168 40 000190 80 000215 10 172 50 196 DO 222 20 178 60 202 100 229 30 184 70 209 Tomlinson (Phil. Trans. 1886) found -000177 as the value at 12 from a great variety of vibration experiments. v.] STRESS, STRAIN, ELASTICITY, VISCOSITY. 69 Obermayer (1876) gives as the result of his trans- piration experiments the approximate formula 0001683(1 + -00274(9), or the nearly equivalent one 001683(l+a6>y 76 , denoting temperature, and a the coefficient of expansion of air. 81. Carl Barus (1889) from experiments ranging from 418 to 1216 infers, for air and for hydrogen, that the viscosity varies as (1 + aO) 3 . Obermayer (1876) gives the following values at for various gases. Air, Oxygen, . Nitrogen, Hydrogen, 000163 000187 000163 0000822 Nitric oxide, Nitrous oxide, Carbonic oxide Carbonic acid, 000164 141 163 141 0. E. Meyer (1873) deduces the following values at 20 from Graham's transpiration experiments. Air, . . -000190 Sulphurous acid, '000138 Oxygen, 212 Nitric oxide, . 186 Nitrogen, 184 Nitrous oxide, . 160 Hydrogen, 093 Carbonic oxide, 184 Steam, 097 Carbonic acid, . 160 Chlorine, 147 Marsh gas, . 120 Sulphuretted hy-\ JOQ drogen, . / Olefiant gas, . 109 Ammonia, . 108 70 CHAPTER VI. ASTRONOMY. Size and Figure of the Earth. 82. ACCORDING to the latest determination (Geodesy, by Col. A. R. Clarke, 1880), the semiaxes of the ellipsoid which most nearly agrees with the actual earth are, in feet, a = 20926629, 6 = 20925105, c = 20854477, which, reduced to centimetres, are a = 6-37836 x 10 8 , b = 6-37790 x 10 8 , c = 6-35639 x 10 8 , giving a mean radius of 6-37090 x 10 8 , and a volume of 1-0832 x 10 27 cubic centims. The ellipticities of the two principal meridians (defined as difference divided by half-sum of axes) are 28^5 and 295*' The longitude of the greatest axis is 8 15' W. The ellipsoid of revolution which most nearly re- presents the actual earth has for its major and minor semiaxes, in feet, a = 20926202, c = 20854895, or, in centimetres, a = 6-37825 x 10 8 , c = 6-35651 xlO 8 ; the ratio of c to a being 292-465.: 293-465. CHAP, vi.] ASTRONOMY. 71 The average length of one ten-millionth of a quadrant of a meridian is 39 '377 786 inches, whereas a legal metre is 39-370432 inches. The difference is 1 part in 5354. Hence the mean length of a quadrant of the meridian is 1-0001 9 x 10 9 centims. The lengths of a degree of latitude and longitude, in centims., in latitude <, are respectively (1111-317 -5-688 cos <) 10 4 , and (1114-164 cos - -950 cos 3 <) 10 4 . 83. The mass of the earth, assuming Baily's value 5 -6 7 for the mean density, is 6-14 x 10 27 grammes. With the value 5-56 obtained by Bailie and Cornu (Com. Ken., 1878) for the mean density, the mass is 6 - 02 x 10 27 . Day and Year. Sidereal day, 86164 mean solar seconds. Sidereal year, 31,558,150 Tropical year, 31,556,929 2?r 1 Angular velocity of earth's rotation, -^ . Velocity of points on the equator j 4651Q centim> gecond due to earth 's rotation, J Velocity of earth in orbit, about 2973000 Centrifugal force at equator due j 3 . 391g d gra mme. to earth's rotation, J Attraction in Astronomy. 84. The mass of the moon is about -^ of the earth's mass. The mean distance of the centres of gravity of the earth and moon is 60-2734 equatorial radii of the earth that is, 3-8444 x 10 10 centims. The mean horizontal parallax of the sun is about 8" -8; 72 C.G.S. UNITS AND CONSTANTS. [CHAP. hence his mean distance is about 1*493 x 10 13 centims., or 92*8 million miles. The intensity of centrifugal force due to the earth's motion in its orbit (regarded as circular) is f >7 5 ) r, r de- noting the mean distance, and T the length of the sidereal year expressed in seconds. This is equal to the accelera- tion due to the sun's attraction at this distance. Putting for r and T their values, 1-493 x 10 13 and 31558 x 10 7 , we have Yr = -5918. This is about of the value of g at the earth's looO surface. The intensity of the earth's attraction at the mean dis- tance of the moon is about or This is less than the intensity of the sun's attraction upon the earth and moon, which is -59 1 8 as just found. Hence the moon's path is always concave towards the sun. 85. The mutual attractive force F between two masses m and ra', at distance I, is P p mm' F = C Tir 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 = 1, m' = 614 x 10 27 , I = 6-37 x 10 8 ; whence we find c = 6 ^? - 10 8 ~ 1-543 x 10 7 ' vi.] ASTRONOMY. 73 To find the mass m which, at the distance of 1 centim. from an equal mass, would attract it with a force of 1 dyne, we have 1 = Cm 2 ; whence m = -= = 3928 grammes. \C 86. To find the acceleration a produced at the distance of I centims. by the attraction of a mass of m grammes, we have a - f = C m' P where C has the value 648 x 10~ 8 as above. 72 To find the dimensions of C we have C = , where the m dimensions of a are LT~ 2 . The dimensions of C are therefore L2M-1LT- 2 ; that is, LSM^T- 2 . 87. The equation a = C ^ shows that when a = 1 and 1=1, m must equal ^ ; that is to say, the mass which u produces unit acceleration at the distance of 1 centimetre is 1-543 x 10 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 . 74 C.G.S. UNITS AND CONSTANTS. [CHAP. Such a system would be eminently convenient in astro- nomy, 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. 88. 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 981 x (6-37 x!0 8 ) 2 = 3-98xl0 2 o, 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 10 7 dynes ; and its dimensions will be mass x acceleration = (acceleration) 2 x (distance) 2 89. If we adopt a new unit of length equal to / centims., and a new unit of time equal to t seconds, while we define the unit mass as that which produces unit acceleration at unit distance, the unit mass will be / 3 r 2 x 1-543 x 10 7 grammes. If we make / the wave-length of the line F in vacuo, say, 4-86 x lO' 5 , and t the period of vibration of the same ray, so that - v is the velocity of light in vacuo, say, 3 x l()i, (I \ 2 - J is 4-374 x 10 16 , and the unit mass will be the product of this quantity vi.j ASTRONOMY. 75 into 1-543 x 10 7 grammes. This product is 6*75 x 10 23 grammes. The mass of the earth in terms of this unit is 3-98 x 10 20 -r (4-374 x 10 16 ) = 9100, and is independent of determinations of the earth's density. 76 CHAPTER VII. VELOCITY OF SOUND. 90. THE propagation of sound through any medium is due to the elasticity of the medium; and the general formula for the velocity of propagation s is fl D' where D denotes the density of the medium, and E the coefficient of elasticity. 91. 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, Y v the initial and final volumes,^* and v being small in comparison with P and V, we have E = -4=*I 0/V v If the compression took place at constant temperature, we should have |=* andE = P. But in the propagation of sound, the compression is CHAP, vii.] VELOCITY OF SOUND. 77 effected so rapidly that there is not time for any sensible part of the heat of compression to escape, and we have where y = 1*41 for dry air, oxygen, nitrogen, or hydrogen. p The value of jj for dry air at t Cent, (see p. 44) is (1 + -003660 x 7-838 x 10 8 . Hence the velocity of sound through dry air is s = 10 4 x/1-41 x (1 + -003662) x 7-838 = 33240^1 + -003662; or approximately, for atmospheric temperatures, s = 33240 + 602. 92. The following are the principal experimental deter- minations of the velocity of sound in air, reduced to C. :- Metres per sec. Academic des Sciences (1738), 332 Benzenberg (1811), j j|f| Goldingham (1821), 331 '1 Bureau des Longitudes (1822), 330'6 Moll, van Beek, and Kuytenbrouwer (1822), ... 332'2 Stampfer and Myrback, 332'4 Bravais and Martins (1844), 332'4 Wertheim, 331'6 Stone (1871), 332-4 Le Roux, 330-7 Regnault, 330'7 93. In the case of any liquid, E denotes the elasticity of volume.* * See foot note in following page. 78 C.G.S. UNITS AND CONSTANTS. [CHAP. For water at 8-l C. (the temperature of the Lake of Geneva in Colladon's experiment) we have E = 2-08 x 10 10 , D = 1 sensibly; = ^E = 144000: the velocity as determined by Colladon was 143500. 94. 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 different for different specimens of the same material. Employing the values given in the Table ( 64), we have ;E Values of E. Values of Values of *J ^> or velocity. Glass, first specimen, - 6-03 xlO 11 2-942 4-53xl0 5 ,, second specimen, Brass, l-075xl0 12 2-935 8-471 4-42 3-56 Steel, - 2-139 7-849 5-22 Iron, wrought, 1-963 7-677 5-06 ,, cast, - 1-349 7-235 4-32 Copper, 1-234 8-843 3-74 95. If the density of a specimen of red pine be -5, and its modulus of longitudinal elasticity be T6 x 10 6 pounds per square inch at a place where g is 981, compute the velocity of sound in the longitudinal direction. * Strictly speaking, E should be taken as denoting the elasticity for sudden applications of stress so sudden that there is not time for changes of temperature produced by the stress to be sensibly diminished by conduction. This remark applies to both 93 and 94. For the amount of these changes of temperature, see a later section under Heat. VII.] VELOCITY OF SOUND. 79 By the table of stress, at the beginning of this volume, a pound per square inch (g being 981) is 6-9 x 10 4 dynes per square centim. Hence we have, for the required velocity, q-6xlQ6x6-9xl0 4 _ centims. per second. 96. The following numbers, multiplied by 10 5 , are the velocities of sound through the principal metals, as determined by Wertheim : At 20 C. At 100 C. At 200 C. Lead, - 1-23 1-20 Gold, - 1-74 1-72 1-73 Silver, - 2-61 2-64 2-48 Copper, - Platinum, 3-56 2-69 3-29 2-57 2-95 2-46 Iron, 5-13 5-30 4-72 Iron Wire (ordinary), Cast Steel, - 4-92 4-99 5-10 4-92 4-79 Steel Wire (English), 4-71 5-24 5'00 j> " 4-88 5-01 The following velocities in wood are from the observa- tions of Wertheim and Chevandier, Comptes Rendus, 1846, pp. 667 and 668: Along Fibres. Radial Direction. Tangential Direction. Pine, - 3'32xl0 5 2-83X10 5 1-59 xlO 6 Beech, - 3-34 3-67 2-83 Hornbeam, 3-92 3-41 2-39 Birch, - 4-42 2-14 3-03 ,, Fir, 4-64 2-67 1-57 Acacia, - 4-71 Aspen, - 5-08 80 C.G.S. UNITS AND CONSTANTS. [CHAP. Musical Strings. 97. 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 Example. For the four 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, 2931 195; 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 M# 2 , in dynes, are 7-89 x 10 6 , 5-64 x 10 6 , 3-97 x 10 6 , 4-43 x 10 6 . Faintest Audible Sound. 98. Lord Eayleigh (Proc. E. S., 1877, vol. xxvi. p. 248), from observing the greatest distance at which a whistle giving about 2730 vibrations per second, and vii.] VELOCITY OF SOUND. 81 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 8-1 x 10~ 8 centims., the calculation being made on the supposition that the sound spreads uniformly in hemispherical waves, and no deduc- tion being made for dissipation, nor for waste energy in blowing. 82 CHAPTER VIII LIGHT. 99. ALL kinds of light are believed to have the same velocity in vacuo. The absolute index of refraction for light of given refrangibility in a given medium is equal to the quotient obtained by dividing the velocity of that kind of light in the medium into the velocity of light in vacuo. The frequency of vibration (that is the number of vibrations per second) is unchanged when a ray passes out of one medium into another; but the wave-length changes in the inverse ratio of the index of refraction. The product of the wave-length in any medium by the index of refraction for that medium is equal to the wave- length in vacuo. 100. The best determinations of the velocity of light are those of Michelson and Newcomb, by the method of the revolving mirror, and of Cornu, by the method of the toothed wheel. The resulting velocity in vacuo is about 2-999 x 10 10 centims. per sec. Professor Newcomb remarks that the value 299860 km. per sec. for the velocity of light, combined with Clarke's value 6378 -2 km. for the earth's equatorial radius, CHAP, viii.] LIGHT. 83 and Nyren's value 20"-492 for the constant of aberration, gives for the solar parallax the value 8" '7 9 4. The following summary of results for velocity in vacuo is given by Professor Newcomb : Km. per sec. Michelson, at Naval Academy, in 1879, 299910 Michelson, at Cleveland, 1882, 299853 Newcomb, at Washington, 1882. using only \ results supposed to be nearly free from V- 299860 constant errors, J Newcomb, including all determinations, 299810 Foucault, at Paris, in 1862, 298000 Cornu, at Paris, in 1874, 298500 Cornu, at Paris, in 1878, 300400 This last result as discussed by Helmert 299990 Young and Forbes, 1880-81, 301382 101. Wave-lengths are usually stated in tenth metres, a tenth-metre being 10~ 10 of a metre, or 10~ 8 of a centim. Thus 5893 tenth-metres = 5-893 x 10~ 5 centims. The following wave-lengths are adopted by Rowland and Bell (May, 1888), from their own observations, for air at 20 C. and 760 mm. (at Baltimore), and are probably the best determinations yet made : A (line between head and tail of group), 7621 '31 B ,, 6884-11 C 6563-07 D! 5896-18 D 5890-22 EJ 5270-52 b x ................................................... 5183-82 F ................................................... 4861-51 Their result for D l reduced to vacuum is 5897-90. UNIVERSITY OF 84 C.G.S. UNITS AND CONSTANTS. [CHAP. They give the following list of results obtained by various observers for D : : Mascart, 5894 '3 Angstrom corr. by Thai en,. 5895 '89 Van der Willigen,..5898'6 Miiller and Kempf , 5896 '25 Angstrom, 5895 -13 Mace de Le"pinay, 5896 '04 Ditscheiner, 5897 '4 Kurlbaum, 5895-90 Peirce, 5896 -27 Bell, ,-. 5896-18 102. The following list of adopted values for the chief lines in the solar spectrum, including the ultra- violet, is given in the new edition of Watts' Index of Spectra (1889). Designation and Origin. Wave-length in Air. Refractive Index of Air. A 7604-0 1-00029286 B 6867-0 29350 (H) 6562-1 29383 D (Na) 5892 ' 12 {5889-12J 2947 E (CaandFe) 5269-13 29584 k (Mg) 5183-10 60 (Mg) 5172-16 63 (NiandFe) 5168'48 6 4 (Mgand Fe) 5166'88 F (H) 4860-72 29685 G (Fe) 4307-25 29873 H (Ca) 3968-l\ o n noQ K (Ca) 3933-OJ L (Fe) 3819-8 300955 M (Fe) 3727-0 301475 N (Fe) 3580-5 30212 (Fe, double) .. 3439'8 30336 P (FeandTi) 3359'2 30397 Q (Fe) 3284-9 30459 E, (FeandCa) 3179'0 30555 r (Fe, double) 3144'3 3073? 51 (Ni, double) 3100>6 Uinn-n 5 2 (Fe, triple ) 3099-5/ dl s (Fe) 3046-4 T (Fe, double) 3019-7 t (Fe) 2994-3 U (Fe) 2947-8 LIGHT. 85 For the principal bright line of lithium the wave- length in air is 6706 ; and for the thallium line 5349. 103. The following wave-lengths were adopted by Angstrom for air at 16C. and 760 mm. pressure (at Upsal), and were long regarded as the standard values. We give in the third column the corresponding fre- quencies deduced by taking the velocity as 3 x 10 10 centims. per sec. Fraunhofer Wave-length, Vibrations Line. tenth-metres. per sec. A 7604 3-945xl0 14 B 6867 4-369 C 6562-01 4-572 D (mean) 5892-12 5'092 E 5269-13 5-693 P 4860-72 6-172 G 4307-25 6-965 H! 3968-01 7'560 H 2 3933-00 7-628 According to Langley (Com. Ren., Jan., 1886), the solar spectrum extends beyond the red as far as wave- length 27000, and the radiation from terrestrial bodies at temperatures below 100 C. extends as far as wave- length 150000 tenth-metres. The corresponding fre- quencies are 1-1 x 10 14 and 2 x 10 13 . On the other hand the frequency corresponding to the wave-length 2948 in the extreme ultra-violet is 1-017 x 10 15 . INDICES OF REFRACTION OF SOLIDS. 104. Dr. Hopkinson (Proc. E. S., June 14, 1877) has determined the indices of refraction of the principal varieties of optical glass made by Messrs. Chance, for 86 C.G.S. UNITS AND CONSTANTS. [CHAP. the fixed lines A, B, 0, D, E, b, F, (G), G, h, H v 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 p-l = a{l+bx(l + cx)}, where is a numerical name for the definite ray of which /x is the refractive index. In assigning the value of x, four glasses hard crown, soft crown, 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 j, the value assigned to x for this ray is ju- - /^ F where /Z F denotes the value of ju. for the line F. The value of JH as a function of A, the wave-length in 10~ 4 centimetres, was found to be approximately P = 1-538414 + 0-0067669-1 -0-00017341 + 0-0000231. The following were the results obtained for the differ- ent specimens of glass examined : Hard Crown, 1st specimen, density 2*48575. a = 0-523145, b = 1-3077, c = -2-33. Means of observed values of /*. 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 1-528348; h 1 '530904; H x 1-532789. viii. ] LIGHT. 87 Soft Crown, density 2 '55035. a = 0-5209904, b = 1-4034, c = -1-58. Means of observed values of /A. A. 1-508956; B 1-510918; C 1 '511910; D 1-514580; E 1-518017; 61-518678; Fl '520994; (G) 1-526208 ; G 1-526592; h 1 "529360; H x 1-531415. Extra Light Flint Glass, density 2'86636. a = 0-549123, b = 1-7064, c = -0-198. Means of observed values of /*. A 1-534067; B 1-536450; C 1-537682; D 1-541022; E 1-545295 ; 61 '546169; F 1-549125; (G) 1-555870; G 1 '556375; h 1-559992; Hj 1 '562760. Light Flint Glass, density 3 '20609. a = 0-583887, 6=1*9605, c = 0'53. Means of observed values of p. B 1-568558; C 1 '570007; D 1 '57401 3; E 1-579227; 61-580273; F 1-583881 ; (G) 1-592184; G 1'592825; h 1*597332; H! 1-600717. Dense Flint, density 3 '65865. a = 0-634744, 6 = 2-2694, c = l'48. Means of observed values of p.. B 1-615704; C 1-617477; D 1-622411; E 1-628882; 6 1-630208; F 1 '634748; (G) 1-645268; G 1-646071; h 1-651830; Hj 1-656229. Extra Dense Flint, density 3 '88947. a = 0-664226, 6 = 2'4446, c = 1'87. Means of observed values of /A. A 1-639143; B 1-642894; C 1 "644871; D 1-650374; E 1 '657631 ; 61*659108; F 1-664246; (G) 1 '676090; G 1-677020; h 1-683575; H x 1 '688590. C.G.S. UNITS AND CONSTANTS. [CHAP. A 1-696531; D 1-710224; F 1 -727257 ; h 1-751485. Double Extra Dense Flint, density 4-42162. a = 0-727237, 6 = 2-7690, c = 2 '70. Means of observed values of ft. B 1-701080; C 1-703485; E 1-719081; 6 1-720908; (G) 1-742058; G 1-743210; 105. The following indices of rock salt, sylvin, and alum for the chief Fraunhofer lines are from Stefan's observations : Rock Salt at 17 C. 1-53663 1-48377 53918 -48597 54050 -48713 54418 -49031 Sylvin at 20 C. 54901 55324 56129 56823 49455 49830 50542 51061 Alum at 21 C. 1 -45057 45262 45359 45601 45892 46140 46563 46907 106. Indices of other singly refracting solids : Index of Kind of Refraction. Light. Diamond, 2'470 D Fluorspar, 1-4339 D Amber, 1'532 D Rosin, 1-545 Red Copal, 1-528 Red Gum Arabic , 1 '480 Red Peru Balsam, T593 D Canada Balsam, ... 1-528 Red Observer. Schrauf. Stefan. Kohlrausch. Jamin. Baden Powell. Wollaston. Effect of Temperature. According to Stefan, the index of refraction of glass increases by about '000 002 for each degree Cent, of increase of temperature, and the index of rock salt VIII.] LIGHT. 89 diminishes by about '000 037 for each degree of increase of temperature. 107. Doubly refracting crystals : Uniaxal Crystals. : Observer. Heusser. Reusch. v. d. Willigen. F.Kohlrausch. Heusser. if de Senarmont. G. Miiller gives the following values for quartz. (The subscript under the figure denotes the number of zeros after the decimal point.) Extraordinary Index. 1-54784 - -0 5 457 875 - Apatite, Ordinary Index. 1-64607 Extraordi- nary Index. 1-64172 Kind of Temp. Light. D 21 Ice,.. 1-3060 1 -3073 Red Iceland-spar, Nitrate of Soda, Tourmaline, Tourmaline, Zircon.. . 1-65844 1-5854 1-6366 1-6479 1-92 1-48639 1-3369 1-6193 1-6262 1-97 D 24 D 23 D 22 Green 22 Red Ordinary Index. B 1-54108 - -0 5 432 C 197 - 402 1 D 432 - 432 1 F 976 - 426 1 G 1-55404 - -0 5 459 1-55116 - 674 - 454 485 462 1 1-56114 - -0 5 467 Biaxal Crystals. Least. Arragonite, 1 '53013 Borax, 1*4463 Mica, 1-5609 Nitre 1-3346 Nitrate of) , .,/. Potash, / L Selenite, 1-52082 Sulphur \ (prismatic)/ Topaz, 1-9505 1-61161 ES OF REFRACTION FOR SODIUM LIGHT. .termediate 1-68157 1-4682 . Greatest. 1-68589 1-4712 Temp. 23 Observer. Rudberg. Kohlrausch. 1-5941 1-5997 23 tt 1-5056 1-5064 16 Schrauf. 1-5056 1-5064 16 1-52287 1-53048 17 v. Lang. 2-0383 2-2405 16 Schrauf. 1-61375 1-62109 Rudberg. 90 C.G.S. UNITS AND CONSTANTS. [CHAP. INDICES OF REFRACTION FOR LIQUIDS. . 108. The following indices of refraction are taken from van der Willigen's results. d t denotes density at temp. Jt-A Carbon Bisulphide. Water. Benzol. d, 5 = d l8 = Density, ... ....1-2709 0-8775 Temp., ... 17 20 -2 18 A 1-61136 1-32889 1-4879 B 1756 - 3038 910 C 2086 3113 926 D 3034 3298 1-4972 E 4320 3522 1-5033 -~F 5529 3713 089 G 7975 4057 197 H 1-70277 1-34343 1-5295 The following are from Kundt's results : Alcohol. Ether. Chloroform. Density, 0-800 0-713 1-501 TemD 15 15 15 A , \.0 B 1-3596 609 1-3550 565 1-4440 458 C 615 573 467 D K E F 633 656 675 594 618 641 492 525 554 G 713 681 1-4611 H 1-3745 1-3713 INDICES FOR GASES. 109. Indices of refraction of air at C. and 760 mm. for the principal Fraunhofer lines. viii.] LIGHT. 91 According to Ketteler. According to Lorenz. A 1-00029286 1-00028935 B 29350 28993 C 29383 29024 D 29470 29108 E 29584 29217 F 29685 29312 G 29873 29486 H 30026 29631 110. The formula established by the experiments of Biot and Arago for the index of refraction of air at various pressures and temperatures was = -0002943 Jh_ l+o* * 760' a denoting 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 10 6 dynes per sq. cm., the general formula applicable to all localities alike will be ' _ l _ -0002943 P 9 1 + -00366* ' 1-0136 x 10 6 ' where P denotes the pressure in dynes per sq. cm. This can be reduced to the form _ 1 _ -0002903 P 1 + -00366 r io 6 ' 111. According to Mascart, /x 1 for any gas is pro- portional not to h but to ^/^ 2 . 1 + at l+at where /? and a are coefficients which vary from one gas to another. In the following table, the column headed ju, contains the indices for and 760 mm. at Paris. The 92 C.G.S. UNITS AND CONSTANTS. [CHAP. next column contains the value of ft multiplied by 10 7 (it being understood that h is expressed in millimetres), and the next column the value of a. All these data are for the light of a sodium flame : Air, 1 Nitrogen, Mo 0002927 2977 /3xl0 7 7'2 8-5 a' 00382 382 Oxygen,.. 2706 in Hydrogen, . 1387 -8'6 378 Nitrous Oxide, Nitrous Gas, Carbonic Oxide, Carbonic Acid, Sulphurous Acid, .... Cyanogen, 5159 2975 3350 4544 7036 8216 88 7 8-9 72 25 27-7 388 367 367 406 460 More recent observations by Benoit with Fizeau's dilatometer give 1-0002923 as the index of refraction of air for the line D at C. and 760 mm., and 003667 as the temperature coefficient. of Dispersive Power. 112. Assuming Cauchy's formula b (where A is the wave-length), which is known to be approximately true for air within the limits of the visible spectrum, the constant b may be called the coefficient of power. Employing as the unit of length for viii.] LIGHT. 93 A the 10~ 4 of a centimetre, Mascart (Ann. de 1'Ecole Normale, 1877, p. 62) has obtained the following values for 1) : Coefficient of Dispersion. Air, -0058 Nitrogen, '0067 Oxygen, '0064 Hydrogen, '0043 Carbonic Oxide, '0075 Carbonic Acid, -0052 Nitrous Oxide, '0125 Cyanogen, '0100 According to Mascart, the ratio of dispersion to devia- tion for the two lines B and H is *024 for air, '032 for the ordinary ray in quartz, '038 for light crown glass, 040 for water, and -046 for the ordinary ray in Iceland- spar. Eolation of Plane of Polarization. 113. The rotation produced by 1 millim. of thickness of quartz cut perpendicular to the axis has the following values for different portions of the spectrum, according to the observations of Soret and Sarasin (Com. Ren. t. xcv. p. 635, 1882), the temperature of the quartz being 20 C. : A 12-66S B 15-746 C 17'31S D : 2l-684 D, . 21-727 E 27'543 F 32773 G 42-604 H 51-193 According to the same observers, the rotation at f C. is equal to the rotation at C. multiplied by 1 + -0001 791 94 C.G.S. UNITS AND CONSTANTS. [CHAP. The rotation (for sodium light) produced by a length of 10 centims. of solution of cane sugar, at 20 C., con- taining n parts by weight of sugar in 100 of the solution, is 0-665 x n. For milk sugar it is 0-5253 x n. Units of Illuminating Power. 114. The British "Candle" is a spermaceti candle, j inch in diameter (6 to the lb.), burning 120 grains per hour. The " Pentane Standard," introduced by Mr. Vernon Harcourt, is intended to represent an average " candle," with the advantage of greater uniformity and a whiter light. It is obtained by burning a mixture of pentane vapour and air from a burner of specified construction. The pentane is a volatile liquid prepared by a process of purification and rectification from American petroleum. When 3 cub. ft. of air and 9 cub. in. of liquid pentane are brought together in a gasholder, the pentane vapor- izes and mixes with the air, yielding altogether 4*05 cub ft. of standard air-gas. This gas is burnt from a J in. circular orifice so as to form a steady conical flame 2| in. high. The rate at which the gas burns is also measured, and must approximate closely to half a cubic foot of air- gas per hour. The French " Carcel " is a lamp of specified construc- tion, burning 42 grammes of pure Colza oil per hour. One " carcel " is equal to about 9 \ " candles." The unit adopted by the International Congress at Paris, April, 1884, is a square centimetre of molten platinum at the temperature of solidification. The surface illuminated by it in photometric tests is to be vin.] LIGHT. 95 normally opposite to the surface of the molten platinum. According to the experiments of M. Violle, the author of this unit, it is equal to 2-08 carcels. It is therefore about 19| candles. At the International Congress of 1890 it was agreed to adopt one twentieth part of this unit as the interna- tional standard candle, and to call it the "Decimal Candle " (bougie decimale). The commonest unit in Germany is the Hefner- Alteneck lamp, which, with a cylindrical wick 8 mm. in diameter, burns amyl-acetate, with a flame 40 mm. high. It is about 1J candle, 15-9 Hefner units being equal to 2-08 carcels, or 19f British candles, or 20 decimal- candles. CHAPTER IX. HEAT. 115. THE primary unit of heat is the quantity of heat which is equivalent to the unit quantity of work. Thus in the C.G.S. system the heat which is equivalent to 1 erg is the primary unit of heat ; or, more briefly (since heat is a form of energy), the erg is the primary unit of heat in the C.G.S system. It is more usual to employ a secondary unit of heat, namely, the quantity of heat required to raise unit mass of cold water through 1 of the thermometric scale employed. The dimensions of the secondary unit of heat are MA, A denoting length of a degree. Mechanical Equivalent of Heat. 116. The ratio of the secondary to the primary unit of heat is commonly called the " Mechanical Equivalent of Heat," or the " Mechanical Equivalent of the Unit of Heat," or, more briefly, "Joule's Equivalent," and is denoted by the symbol J. It is the number of units of work required to raise the temperature of unit mass of water 1. For a more precise definition see 126. CHAP, ix.] HEAT. 97 117. Since the secondary unit of heat varies jointly as the unit of mass and the length of a degree of tempera- ture, the dependence of the numerical value J upon the units employed can be calculated as follows, the length of a degree being denoted by A : Secondary unit of heat varies as MA. Absolute unit of work ML 2 T~ 2 . Gravitation unit of work ,, ML. Hence the numerical value J varies as T , if gravitation Lt AT 2 units are employed, and as ^- if absolute units are em- ployed ; and the dimensions of the unit in terms of which T T 2 J is the numerical value are and in the two cases respectively. Ex. 1. In gravitation units, 772 . 1U " U , = 1390 , x "" u - 424 metre deg. Fahr. deg. Cent. deg. Cent, centim. delTC? the numerical values, 772, 1390, etc., being the values of J in the several systems of reckoning. Ex. 2. In absolute units, if 25187 be the value of J in the foot-pound-second-Fahrenheit system, what is its value in the C.G.S. system ? We have OKI Q 7 (lOOu)'" (ccriuiiii. ) 2i 1 O I -s ^r == X -= pr \ deg. F. deg. C. whence x = 25187 x (30-48) 2 x I = 4-212 x 10 7 . To reduce from gravitational to absolute units, the 98 C.G.S. UNITS AND CONSTANTS. [CHAP. value of J must be multiplied by the value of g. For example, in the C.G.S. system, if 42400 be taken as the value of J in gravitation units, and 981 as the value of g, the value of J in absolute units will be 42400 x 981 = 4-159 x 10 7 nearly. 118. The mean thermal capacity of a body between two stated temperatures is defined as the quantity of heat required to raise it from the lower temperature to the higher, divided by the difference of the temperatures. The thermal capacity at a stated temperature is the limit to which the mean thermal capacity tends when the two temperatures approach the stated temperature ; that is, it is the value of the differential coefficient -S, Q CLv denoting the quantity of heat given, and t the tempera- ture produced. The thermal capacity of unit mass of any substance is called the specific heat of that substance. 119. Rowland has directly determined the specific heat of water at temperatures from 5 to 36 by a Temp. C. Temp. C. Temp. C. 5 4-212 xlO 7 16 4-187xl0 7 27 4-171 x 10 7 6 4-209 17 185 28 171 7 4-207 18 183 29 170 8 4-204 19 181 30 171 9 4-202 20 179 31 171 10 4-200 21 177 32 171 11 4-198 22 176 33 172 12 4-196 23 175 34 172 13 4-194 24 174 35 173 14 4-192 25 173 36 173 15 4-189 26 172 IX.] HEAT. 99 modification of Joule's method of raising the temperature of water by stirring it; and his final results for the number of ergs required to raise the temperature of a gramme of water through 1 C. of the absolute thermo- dynamic scale are given in the annexed table. 120. The following are the corresponding gravitation values, (1) for Baltimore (g = 980-0), where the ex- periments were made, (2) for Manchester (# = 981 '3), where Joule's experiments were made. The values for Baltimore are given in metres per degree Cent. Those for Manchester are given both in met. per deg. Cent, and in feet per degree Fahr., so as to be comparable with Joule's well-known result, 772. Temp. Baltimore. Manchester. Temp. C. Baltimore. Manchester. 5 429-8 429-2 782-3 21 426-2 4257 775-9 6 429-5 428-9 781-8 22 426-1 425-6 775-7 7 429-3 428-7 781-4 23 426-0 425-5 775-5 8 429-0 428-4 780-9 24 425-9 425-4 775-3 9 428-8 428-2 780-5 25 425-8 425-3 775-1 10 428-5 428-0 780-1 26 425-7 425-2 774-9 11 428-3 427-8 779-7 27 425-6 425-0 774-7 12 428-1 427-6 779-4 28 425-6 425-0 774-7 13 427-9 427-4 779-0 29 425-5 424-9 774-5 14 427-7 427-2 778-6 30 425-6 425-0 774-7 15 427-4 426-9 778-1 31 425-6 425-0 774-7 16 427-2 426-7 7777 32 425-6 425-0 774-7 17 427-0 426-5 777-3 33 425-7 425-2 774-9 18 426-8 426-3 777-0 34 425-7 425-2 774-9 19 426-6 426-1 776-6 35 425-8 425-3 775-1 20 426-4 425-9 776-2 36 425-8 425.3 775-1 It thus appears from Rowland's experiments that the round number 42 millions is the true value of J in ergs per gm. deg., at the temperature 10 C. or 50 F., and 100 C.G.S, UNITS AND CONSTANTS. [CHAP. that, at this temperature, the gravitation values of J at Manchester are (to the nearest integer) 428 kgm. metres per kgm. deg. C., 780 ft. Ibs. per Ib. deg. F. 1404 ft. Ibs. per Ib. deg. C., 121. An inspection of the Table in 119 shows that the specific heat of water does not regularly increase with the temperature, as was formerly believed, but first decreases and then increases, attaining its minimum at about 29 C. When temperatures are expressed in degrees of a mercurial thermometer with its tube divided into parts of equal volume, the results will vary according to the kind of glass of which the thermometer is made. Kow- land gives results for four different standard mercurial thermometers, and they place the minimum at points ranging from 18 or 19 (Geissler thermometer) to 29 (Kew thermometer). On the other hand Regnault's determinations make the specific heat of water at t proportional to 1 + -000 04 t+'OOO 000 9 2 and the mean specific heat between and t propor- tional to 1 + -000 02 1+ -000 000 3ft Rowland remarks, " The principal experiments on the subject were published by Eegnault in 1850, and these have been accepted to the present time. It is un- fortunate that these experiments were all made by mixing water above 100 with water at ordinary tem- peratures, it being assumed that water at ordinary tempera- tures changed little, if any. An interpolation formula was ix.] HEAT. 101 then found to represent the results ; and it was assumed that the same formula held at ordinary temperature, or even as low as C. It is true that Regnault experi- mented on the subject at points around 4 C. by deter- mining the specific heat of lead in water at various temperatures ; but the results were not of sufficient accuracy to warrant any conclusions except that the variation was not great." 122. The relation between heat and work 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 1 Fahr. This is 1389-6 foot-pounds for a pound of water raised 1 C., or 1389-6 foot-grammes for a gramme of water raised 1 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 98T3, that is 4-156 x 10 7 ergs, for 1 gramme of water raised 1 C. A later experiment on the heat produced by the friction of water is described by Joule in Phil. Trans., 1878, Part II., leading to the result that the work re- quired to raise a pound of water from 60 to 61 Fahr. is 772-55 foot-pounds at sea-level and the latitude of Greenwich. The number of ergs required to raise a gramme of water through 1 C. at the temperature of 60-5 Fahr. or 15 -8 C. would accordingly be 772-55 xfx 30-48x981-17 = 4-159 x 10 7 . Rowland's result for this temperature (15f C.) is 4-187 x 10 7 . In the British Association Report for 1867 (and " Reprint of Reports on Electrical Standards," p. 186) 102 C.G.S. UNITS AND CONSTANTS. [CHAP. is given a determination by Joule based on the heating of a conductor by an electric current. The result given is 25187 foot-grain-second (absolute) units of work for a grain of water raised 1 Fahr. The corresponding num- ber of ergs for a gramme of water raised 1 0. is 25187 x I x (30-48) 2 - 4-212 x 101 In view of the fact that the electrical resistance was ex- pressed in terms of the B.A. standard, which is now known to be only -98656 of a true ohm, the above result must be multiplied by -98656, giving 4-155 x 10 7 . The average temperature of the water was 18 -63 C. The principal part of the difference between Joule's results and Rowland's has been traced to want of agree- ment between the mercurial thermometers employed by Joule and the perfect air-thermometer. Comparison of Mercurial with Air Thermometer. 123. Eowland, at p. 89 of his paper on the Mechanical Equivalent of Heat, gives for the excess of the reading T of a mercurial thermometer above the reading t of an air thermometer, the formula T-t = at(WO-t] (b-t); and gives as a specimen of actual values a =-00000033, b = 290. The observed excess T-t had in this instance the following values between and 100 : Temp., 10 20 30 40 50 60 70 80 90 100 Excess, -08 -14 -18 '20 -20 '18 '14 -10 "05 P. Chappuis gives, in vol. VI. of "Travaux et Memoires du Bureau International," p. 116, the follow- IX.] HEAT. 103 ing differences between a hydrogen thermometer and a mercury thermometer of hard glass by Tonnelot : Mercury Thermometer. 20 10 10 20 30 40 Mercury minus Mercur. Hydrogen. TnGrmoni' -172 50 -073 60 70 + 052 80 085 90 102 100 107 Mercury minus Hydrogen. 103 090 072 050 026 Comparison with the Absolute Thermo-dynamic Scale. 124. The absolute thermodynamic scale of tempera- ture is defined by the statement that, when a perfect thermodynamic engine takes in heat at one temperature and gives out heat at another, the quantities of heat which it takes in and gives out are proportional to these temperatures. In the following comparisons, the con- stant which expresses the difference between the zero of the absolute scale and that of the ordinary centigrade scale is neglected. Eowland gives, at p. 114 of his paper on the Mechani- cal Equivalent of Heat, the following table of corrections for reducing temperatures by air thermometer (at con- stant volume) to the absolute scale : Air Ther. 10 20 30 40 50 60 70 Correction. Air Ther. 80 - -0028 90 - -0048 100 - -0061 200 -0067 SOO - -0068 400 - -0063 500 -0054 Correction. - -0044 - -0022 + -037 + 092 + 157 + 228 104 C.G.S. UNITS AND CONSTANTS. [CHAP. These corrections were calculated by the formula -00088^, t denoting temperature by air thermometer. To reduce his Kew standard thermometer to the absolute scale, he employs the following corrections : Temp. Correction. 10 - -03 20 - '05 30 - '06 40 -'07 50 - -07 Temp. Correction. 60 - "06 70 - '04 80 -'02 90 - -01 100 Relations between various Units of Heat. 125. The secondary unit of heat employed in the C.G.S. system is the quantity of heat required to raise a gramme of cold water 1 C. It is called for brevity the gramme-degree. No definite temperature has been agreed upon for the cold water. We shall adopt 4-2 x 10 7 , or 42 millions, as the number of ergs equivalent to the gramme- degree. This, according to Eowland, is the true value at 10 C. The gramme-degree is often called the lesser calorie, and the kilogramme-degree, which is 1000 times as large, is called the greater calorie. The following is a list of the principal units of heat which are in use. 1 joule = 10 7 ergs. 1 gramme degree or minor calorie = 4-2 joules nearly. 1 kilogramme-degree or major calorie = 4200 ,, 1 pound-degree Centigrade = 1905 ,, 1 pound-degree Fahrenheit = 1058 ix.] HEAT. 105 More precise Definition of the Secondary Unit. 126. Rowland's results exhibit a difference of some- thing like one per cent, between the specific heats of cold water at different temperatures. To render the definition of the secondary unit of heat exact, a standard temperature must be selected, the specific heat of water at this temperature must be called unity, and the second- ary unit of heat will be n times the heat required to raise unit mass of water from this initial temperature through l/n of a degree when n is indefinitely great. 127. When a secondary unit of heat is employed, the specific heat of water at some standard temperature is by definition unity ; and the specific heat of any substance is a mere numerical ratio, namely, the ratio of the capacity of any quantity of that substance to the capacity of an equal mass of water (the water being at the standard temperature). 128. The thermal capacity of unit volume of a substance is another important element ; we shall denote it by e. Let s denote the specific heat, and d the density of the substance ; then c is the capacity of d units of mass, and therefore c = sd. The dimensions of c are the same as those of d, namely, . Its numerical value will not be altered by any change in the units of length, mass, and time, which leaves the density of water unchanged. In the C.G.S. system the value of c for water at ordinary temperatures is approximately unity ; hence the value of c for any substance is approximately identical with the ratio heat given to the substance heat given to or THE UNIVERSITY OF 106 C.G.S. UNITS AND CONSTANTS. [CHAP. for the same rise of temperature, when the comparison is between equal volumes. 129. Mr. Herbert Tomlinson (Proc. Roy. Soc., June 19, 1885) has obtained the following determinations of specific heat from observations conducted in a uniform manner with metallic wires well annealed. The wires were heated sometimes to 60 C. and sometimes to 100 C., and were plunged in water at 20. The formulae are for the true specific heat at f C. : Aluminium, -20700+ '000 2304 t Copper, -09008+ -000 0648 t German Silver, -09413+ '0000106* Iron, -10601 + '000 140 t Lead, -02998 + -000031* Platinum, -03198 + '000 013 1 Platinum Silver, -04726 + '000 028 t Silver, -05466+ '000044* Tin, -05231 + '000072* Zinc, -09009+ "000075* The formulae for the mean specific heat between and t are obtained from these by leaving the first term un- changed and halving the second term. Violle has made the following determinations of specific heat at t : Platinum, -0317 + 000012* Iridium, -0317 + 000012* Palladium, -0582 + 000020* H. F. Weber has determined the specific heat of diamond to be 0947 + -000 994 *--000 000 36 1\ and consequently the mean specific heat of diamond from to t to be 0947 + -000 497 2 -'000 000 12 t 2 . IX.] HEAT. 107 The mean specific heat of ice according to Regnault is 504 between 20 and 0, and '474 between 78 and 130. The following list of specific heats of elementary substances is condensed from that given in Landolt and Bernstein's tables : Substance. Temperature. Sp. Heat. Observer. Aluminium, 15 to 97 2122 Regnault. Antimony, 13 106 0486 Bede. Arsenic (crystalline), 21 68 0830 ( Bettendorff & j Wullner. ,, (amorphous), 21 ,, 65 0758 55 55 Bismuth, 9 ,, 102 0298 Bede. Boron (crystalline), ,, 100 2518 Mixter&Dana ,, (amorphous), 18 48 254 Kopp. Bromine, solid, -78 ,,-20 0843 Regnault. ,, liquid, 13 45 1071 Andrews. Cadmium, 100 0548 Bunsen. Calcium, 100 1804 55 Carbon, diamond, 11 1128 H. F. Weber graphite, 11 1604 55 ,, wood charcoal,.. to 99 1935 55 Cobalt, 9 97 1067 Regnault. Copper, 15 100 0933 Bede. Gold, 100 0316 Violle. Iodine, 9 98 0541 Regnault. Iridium, 100 0323 Violle. Iron, 50 1124 Bystrom. Lead, 19 to 48 0315 Kopp. Lithium, 27 ,, 99 9408 Regnault. Magnesium, 20 51 245 Kopp. Manganese, 14 97 1217 Regnault. AT 1 " rl 78 4-0 .ftOlQ /o ,, tv 55 liquid, 17 48 0335 Kopp. Molybdenum, 5 15 0659 ( De la Rive and | Marcet. Nickel, 14 97 1092 Regnault. Osmium, 19 98 0311 55 108 C.G.S. UNITS AND CONSTANTS. Substance. Temperature. Sp. Heat. Observer. Palladium, to,100 0592 Violle. Phosphorus (yellow, solid) - 78 ,, 10 1699 Regnault. ,, ( liquid) 49 98 2045 Person. (red), 15 ,, 98 1698 Regnault. Platinum, 100 0323 Violle. Potassium, -78 1655 Regnault. Rhodium, 10 97 0580 Regnault. Selenium, crystalline, 22 ,, 62 0840 ( Bettendorff & j Wullner. Silicon, crystalline, 22 1697 H. F. Weber. Silver, to 100 0559 Bunsen. Sodium, -28 6 2934 Regnault. Sulphur (rhomb, cryst.), 17 45 163 Kopp. ,, (newly melted), 15 97 1844 Regnault. Tellurium, crystalline,... 21 51 0475 Kopp. Thallium, 17 ,, 100 0335 Regnault. Tin, cast, 100 0559 Bunsen. Zinc, ,, 100 0935 tt Substances not Elementary. Brass (4 copper 1 tin), hard, 15 to 98 '0858 Regnault. soft, 14 ,, 98 -0862 Ice, -20,, '504 131. The following determinations of specific heat of liquids are by Regnault. We have omitted decimal figures after the fourth, as even the second figure is different with different observers : Alcohol. Oil of Tur- pentine. Ether. Chloro- form. Bisulphide of Carbon. -20 5053 3842 -30 5113 2293 2303 5475 4106 5290 2324 2352 40 6479 4538 30 5468 2354 2401 80 7694 4842 60 2384 120 5019 160 5068 IX.] HEAT. 109 Schiiller has found the specific heat of liquid benzine at r tobe -37980 + -00144 rf. 132. The following table (from Miller's Chemical Physics, p. 308) contains the results of Regnault'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 denned 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 at Constant Pressure. Gas or Vapour. Equal Gas or Vapour. Equal Vote, Weights. Vols. Weights Air, Oxygen, - 2375 2405 2375 2175 Hydrochloric ) Acid, - j 2352 1842 Nitrogen, Hydrogen, 2368 2359 2438 3-4090 Sulphuretted ) Hydrogen, i 2857 2432 Chlorine, 2964 1210 Steam, - 2989 4805 Bromine, 3040 0555 Alcohol, - 7171 4534 Nitrous Oxide, 3447 2262 Wood Spirit, - 5063 4580 Nitric Oxide, - 2406 2317 Ether, - 1-2266 4796 Carbonic Oxide 2370 2450 Ethyl Chloride 6096 2738 Carbonic Anhydride, Carbonic Di- ) sulphide, \ 3307 4122 2163 1569 ,, Bromide Disul- ) phide, \ Cyanide, 7026 1-2466 8293 1896 4008 4261 Ammonia, 2996 5084 Chloroform, - 6461 1566 Marsh Gas, - 3277 5929 Dutch Liquid, 7911 2293 Olefiant Gas, - 4106 4040 Acetic Ether, - 1-2184 4008 Arsenious / Chloride, \ 7013 1122 Benzol, - Acetone, 1-0114 8341 3754 4125 Silicic Chloride Titanic 7778 8564 1322 1290 Oil of Tur- ) pentine, - \ 2-3776 5061 Stannic ,, Sulphurous ) 8639 041 0939 1 ^4 Phosphorus ) Chloride, - \ 6386 1347 Anhydride, \ O^rJ. lUrr 110 C.G.S. UNITS AND CONSTANTS. [CHAP. 133. E. Wiedemann (Pogg. Ann., 1876, No. 1, p. 39) has made the following determinations of the specific heats of gases : Specific Heat. AtO. At 100. At 200. Air, ...................... 0-2389 ... ... 1 Hydrogen, .............. 3'410 ... ... 0'0692 Carbonic Oxide, ...... 0'2426 ... ... 0'967 Carbonic Acid, ....... 0'1952 0-2169 0'2387 1'529 Ethyl, .................... 0-3364 0'4189 0'5015 0-9677 Nitrous Oxide, ........ 0'1983 0'2212 0'2442 1-5241 Ammonia, ............... 0'5009 0'5317 0'5629 0'5894 Multiplying the specific heat by the relative density, he obtains the following values of Thermal Capacity of Equal Volumes. At 0. 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 Nitrous Oxide, ... 0'3014 0'3362 0'3712 Ammonia, ......... 0'2952 0*3134 0'3318 The same author (Pogg. Ann., 1877, New Series, vol. ii. p. 195) has made the following determinations of specific heats of vapours at temperature f : Va P ur - in a Sfper f iment P , Specific Heat. Chloroform, ...... 26 '9 to 189'8 '1341 + '000 1354 t Bromic Ethyl, ... 27 '9 to 189 '5 '1354 + "000 3560 1 Benzine, ........... 34-l to 179'5 -2237+ '001 0228 1 Acetone, ........... 26'2 to 179'3 '2984 + -000 7738 t Acetic Ether,. ... 32 "9 to 188 '8 '2738+ '000 8700 1 Ether, .............. 25 '4 to 188 '8 '3725 + '000 8536 1 IX.] HEAT. Ill Regnault's determinations for the same vapours were as follows : Mean Specific Heat for this Range. Range of Temperature. Vapour. Chloroform,.... Bromic Ethel, . Benzine, Acetone, Acetic Ether, . 117 to 228 77 '7 to 196-5 116 to 218 129 to 233 115 to 219 According to According to Regnault. Wiedemann. 1567 1573 1896 1841 3754 3946 4125 3946 4008 4190 4797 4943 Ether, 70 to 220 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. MELTING POINTS AND HEATS OF LIQUEFACTION. 134. The following table of melting points and latent heats of liquefaction is taken from the German edition of this work. Those which are marked with an asterisk are recent very careful determinations by Violle. Melting Latent Melting Latent Point. Heat. Point. Heat. Iridium, 1950* Cadmium, 315 13-6 Platinum, 1775* 27-2* Bismuth, 260 12-6 Palladium, 1500* 36-3* Tin, 230 14-6 Iron, 1600 Selenium, 217 Nickel, 1450 4-64 Sulphur, 115 9-4 Cast steel, 1370 Sodium, 96 Pig iron, 1075 Potassium, 62 Copper, 1054* Phosphorus, 44 5-0 Gold, 1045* Paraffin, 54 Silver, 954* 24-7 Bromine, - 7 Glass, 1100 Mercury, - 40 2-82 Aluminium, 600 Iodine, 11-7 Antimony, 432 Nitrate of soda, 63-0 Zinc, 412 28-1 Nitrate of potash, 47'4 Lead, 326 5-4 Ice (Bunsen), 80 112 C.G.S. UNITS AND CONSTANTS. [CHAP. Violle states that 1950, the melting point of iridium, is very near to that of the hottest part of the oxy- hydrogen flame. The latent heat of liquefaction of ice was found by Regnault, and by Provostaye and Desains, to be 79. Bunsen (Pogg. Ann. cxli., p. 30) obtained the value 80-025. He found the specific gravity of ice to be '9167. 135. Chandler Roberts and Wrightson have compared the densities of molten and solid metals by weighing a solid metal ball in a bath of molten metal either of the same or a different kind (Phys. Soc., 1881, p. 195, and 1882, p. 102). They find that "iron expands rapidly (as much as 6 per cent.) in cooling from the liquid to the plastic state, and then contracts 7 per cent, to solidity ; whereas bismuth appears to expand in cooling from the liquid to the solid state about 2-35 per cent." The following is their tabular statement of results : Percentage of change in volume from cold solid to liquid. Decrease of vol. 2 '3 Increase of vol. 7'1 9-93 6-76 11-1 11-2 1-02 Metal. Bismuth, Sp. Grav. of Solid. 9-82 Sp. Grav. of Liquid. 10-055 Copper, . . . 8-8 8-217 Lead, 11-4 10-37 Tin 7-5 7-025 Zinc, Silver... 7-2 . 10-57 6-48 9-51 136. Change of volume in melting, from Kopp's ex- periments (Watts' Diet., Art. Heat, p. 78) : Phosphorus. Calling the volume at unity, the volume at the melting point (44) is 1'017 in the solid, and T052 in the liquid, state. Sulphur. Volume at being 1, volume at the melting point (115) is T096 in the solid, and 1-150 in the liquid, state. ix.] HEAT. 113 Wax. Volume at being 1, volume at melting point (64) is 1-161 in solid, and 1*166 in liquid, state. Stearic Acid. Volume at being 1, volume at melting point (70) is 1-079 in solid, and 1-198 in liquid, state. Rose's Fusible Metal (2 parts bismuth, 1 tin, 1 lead). Volume at being 1, 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. 137. The following table (from Miller's Chemical Physics, p. 344) exhibits the change of volume of certain substances in passing from the liquid to the vaporous condition under the pressure of one atmosphere : 1 volume of water yields 1696 volumes of vapour. alcohol 528 ,, ,, ,, ether 298 ,, oil of turpentine 193 ,, ,, 138. The following table of boiling points and heats of vaporization, at atmospheric pressure, is condensed from Landolt and Bornstein, pp. 189, 190 : Boiling Latent Heat of ov^r^r Pointf Vaporization. Alcohol, 77-9 202-4 Andrews. Bisulphide of Carbon, 46 '2 86 '7 ,, Bromine, 58 45'6 Ether, 34'9 90'4 Mercury, 350 62'0 Person. Oil of Turpentine, 159'3 74'0 Brix. Water, 100 535*9 Andrews. A specially careful determination of the boiling point of sulphur has been made by Callendar and Griffiths (Proc. R. S., Dec. 1890, p. 58). The result is, "the tem- perature, by normal air thermometer at constant pressure, of the saturated vapour of sulphur boiling freely under a pressure of 760 mm. of mercury at C. and g = 980-61 C.G.S. (sea level in lat. 45), is 444-53C." H 114 C.G.S. UNITS AND CONSTANTS. [CHAP. 139. Regnault's approximate formula for what he calls "the total heat of steam at t" that is, for the heat required to raise unit mass of water from to t in the liquid state and then convert it into steam at t, is 606 -5 + -305 t. If the specific heat of water were the same at all tempera- tures, this would give 606-5- -695 1 as the heat of evaporation at t. But since, according to Regnault, the heat required to raise the water from to t is t + -000 02 P + -000 000 3 *, the heat of evaporation will be the difference between this and the " total heat," that is, will be 606-5 - -695 t - -000 02 $ - -000 000 3 $, which is accordingly the value adopted by Regnault as the heat of evaporation of water at t. 140. According to Regnault, the increase of pressure at constant volume, and increase of volume at constant pressure, when the temperature increases from to 100, have the following values for the gases named : Q ag At Constant At Constant Volume. Pressure. Hydrogen, -3667 '3661 Air, -3665 '3670 Nitrogen, -3668 Carbonic Oxide, '3667 '3669 Carbonic Acid, '3688 -3710 Nitrous Oxide, '3676 '3719 Sulphurous Acid, '3845 '3903 Cyanogen, '3829 '3877 Jolly has obtained the following values for the co- efficient of increase of pressure at constant volume : IX.] HEAT. 115 Air, -00366957 Oxygen, '00367430 Hydrogen, "00365620 Nitrogen, '0036677 Carbonic Acid, '0037050 Nitrous Oxide, '0037067 Rowland, from his own experiments, has deduced for air at constant volume, the value -0036675 or -0036707, according as Regnault's or Wiillner's determination of the expansion of mercury is employed in the calculation. 141. According to recent researches by P. Chappuis, the mean coefficient of increase of pressure of hydrogen at constant volume from to 100 (the pressure at being that of 100 centims. of mercury) is -00366254; and taking this as the standard of "uniform" increase the true coefficient of increase of pressure for nitrogen, with the same pressure at 0, is 0036770 - 7 '8267 x 10~ 8 1 + 4*780 x lO" 10 2 ; and for carbonic acid, 00373537 - 2-6754 x 10~ 7 + 2-6157 x 10' 10 1 2 + 7-5992 xl 142. Pressure of Aqueous Vapour. Temp. in Lat. 45 j_/yiius per Temp. in Lat. 45 jjynes per at Sea Level. sq. cm. at Sea Level. sq. cm. 99--0 73-316 9-775xl0 5 100-0 76-000 l'0133xl( 1 581 9-810 1 273 1-0169 2 846 9-846 2 547 1 -0206 3 74-113 9-881 3 821 1-0242 4 380 9-917 4 77-097 1-0279 '5 648 9-953 5 373 1-0316 6 917 9-988 6 650 1-0353 7 75-186 1 -0024 x 10 6 7 928 1-0390 8 457 1-0060 i 8 78-207 1-0427 9 728 1-0097 9 486 1-0464 100-0 76-000 1-0133 , 101-0 767 1-0502 116 C.G.S. UNITS AND CONSTANTS. [CHAP. The above table, showing the maximum pressure at temperatures near 100 C., is based on Regnault's deter- minations as revised by Broch. 143. Maximum pressure of aqueous vapour at various temperatures, in dynes per sq. centim. Temp. Pressure. Temp. Pressure. Temp. Pressure. -20 1-26x10- 20 2-315 xlO 4 80 4 -731 xlO 5 -15 1-92 25 3-135 , 90 7-006 ,, -10 2-87 30 4-201 j 100 1-0133 xlO 8 - 5 4-21 40 7-315 120 1-989 6-09 50 1-226x1 b 5 140 3-624 5 8-67 60 1-985 j 160 6-204 10 1-219x10* 70 3-111 200 1 -559 x 10 7 15 1-690 ,, 144. The following are approximate values of the maximum pressure of aqueous vapour at various tempera- tures, in millimetres of mercury. They can be reduced to dynes per sq. cm. by multiplying by 1334: mm. 4-6 92 rnm. 567 112 mm. 1150 132 mm. 2155 10 9-2 94 611 114 1228 134 2286 20 17-4 96 658 116 1311 136 2423 30 31-5 98 707 118 1399 138 2567 40 54-9 100 760 120 1491 140 2718 50 92-0 102 816 122 1588 142 2875 60 149 104 875 124 1691 144 3040 70 233 106 938 126 1798 146 3213 80 355 108 1004 128 1911 148 3393 90 525 110 1075 130 2030 150 3581 145. The density (in gm. per cub. cm.) of aqueous vapour at any temperature t and any pressure^? (dynes per sq. cm.), whether equal to or less than the maximum pressure, is 622 x -001276 1 + -00366 _ 10 6 IX.] HEAT. 117 If q denote the pressure in millimetres of mercury, the approximate formula is 622 x -001293 a 1 + -00366 t 760 146. Temperature of evaporation and dew-point (Glaisher's Tables, second edition, page iv.). The follow- ing are the factors by which it is necessary to multiply the excess of the reading of the dry thermometer over that of the wet, to give the excess of the temperature of the air above that of the dew-point : Reading of Dry Bulb Therm. -10C. = 14F. - 5 23 32 + 5 41 + 10 50 Factor. 8-76 7-28 3-32 2-26 2-06 Reading of Dry Bulb - Therm. 15C. = 59F. 20 68 25 77 30 86 35 95 Factor. 1-89 1-79 1-70 1-65 1-60 For density of moist air see end of this chapter. 147. Maximum Pressure of various Vapours, in dynes per s<2. cm. Alcohol. Ether. Sulphide of Carbon. Chloroform. -20 4455 9-19 xlO 4 6-31 xlO 4 -10 8630 T53 xlO 5 1-058 xlO 5 16940 2-46 1-706 10 32320 3-826 2-648 20 59310 5-772 3-975 2-141 xlO 5 30 l-048x!0 5 8-468 5-799 3-301 40 1-783 l-210x O 6 8-240 4-927 50 2-932 1-687 l-144x O 6 7:14 60 4-671 2-301 1-554 1 -007 x O 6 80 1-084 x O 6 4-031 2-711 1-878 100 2-265 6-608 4-435 324 120 4-31 1-029 xlO 7 6-87 5-24 118 C.G.S. UNITS AND CONSTANTS. [CHAP. Critical Temperatures and Pressures. 148. The following table of critical temperatures of gases (above which they cannot be liquefied), and the corresponding maximum pressures, is given by van der Waals (p. 464 of English translation, Phys. Soc., Lond.): Critical. Carbonic Acid, Ether, Carbon Bisulphide, 5? 5 Sulphurous Acid, . . . Alcohol, . , Monochlorethane, Benzol, Acetone, Chloroform, Ethylene, Hydrochloric Acid, ... Acetylene, Carbon Tetrachloride, Nitrous Oxide,.. Temp. Pressure. o atm. 30-92 73 190 36-9 271-8 74-7 273 77-9 155-4 78-9 234-3 62-1 234-6 65 256 119 182-5 52-6 280-6 49-5 232-8 52-2 237-5 60 260 54-9 9-2 58 51-25 86 37 68 277-9 58-1 36-4 73-1 Andrews. Sajotschewsky. 3 J Hannay. Sajotschewsky. Hannay. Cagniard. Sajotschewsky. van der Waals. Ansdell. Hannay. Janssen. Conductivity. 149. By the thermal conductivity of a substance at a given temperature is meant the value of k in the expression Q kA^^t, (1) where Q denotes the quantity of .heat that flows, in time t, through a plate of the substance of thickness x, the area of each of the two opposite faces of the plate being A, ix.] HEAT. 119 and the temperatures of these faces being respectively #! and v 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, conduc- tivity may be defined as the quantity of heat that passes in unit time, through unit area of a plate whose thickness is unity, ichen its opposite faces differ in temperature ly one degree. 150. Dimensions of Conductivity. From equation (1) we have 7- - Q x (9\ - ~v~-v\ ' At' The dimensions of Q and v 2 v^ are respectively MA and A, giving M as the dimensions of the first factor. The dimensions of the factor are ^ : hence the dimen- At Li M sions of 7v are =-=. This is on the supposition that the Jul unit of heat is the heat required to raise unit mass of water one degree. In calculations relating to conduc- tivity 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 ^ will then be L 3 , and L 2 the dimensions of k will be . In the C.G.S. system these two units of heat are practically identical. 120 C.G.S. UNITS AND CONSTANTS. [CHAP. 151. 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), ' 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 proposed to call the thermometric conductivity, as dis- C tinguished from k the thermal or calorimetric conductivity. We prefer, in accordance with Sir Wm. Thomson's article, " Heat," in the Encyclopaedia Britannica to call the c diffusivity of the substance for heat, a name which is 7 based on the analogy of to a coefficient of diffusion. Coefficient of Diffusion. 152. By the coefficient of diffusion of a given sub- stance in a given solution is meant the value of K in the expression Z^^, ...... (3) where m denotes the mass of the substance that passes in time t through a layer of the solution of thickness x, the area of each face of the layer being A, and the con- ix.] HEAT. 121 centrations at these faces s 1 and s 2 ; concentration being so defined that in a solution of uniform concentration the mass of the substance is equal to the volume of the solution multiplied by the concentration. Since m divided by s 2 - s 1 is a mass divided by a con- centration, and is therefore a volume, the dimensions of L 2 K are found to be , . Let y denote the thickness of a layer of concentration s v which would be raised to concentration s 2 by the mass of the substance transmitted in time t. The mass re- quired is (s 2 - sj y A, which when put for m in equation (3) gives K=?? ....... (4) In the analogous problem for heat, denoting by y the thickness of a stratum at temperature v v which would be raised to v. 2 by the heat transmitted in time t, we must put c, (v. 2 -Vi) y A for Q in equation (1) or (2), and we deduce * = !? ..... (5) Hence - is the analogue of K. 153. The following table gives the coefficients of diffusion between certain inorganic liquids and water according to Scheffer's observations. His units are the centimetre and the day ; hence to reduce these coefficients to C.G.S. values they must be divided by the number of seconds in a day, that is by 86400. The first column gives the concentration in grammes per cub. cm., and the other columns the values of m in grammes. 122 .C.G.S. UNITS AND CONSTANTS. [CHAP. Substance in 100 cub. cm. 3} C. 5* C. 2$ C. 1\ C. 10$ C. .of solution. HC1. NaCl. NaN0 3 . AgNO 3 . Xa 2 So0 3 4-55 gm. 1-622 4-96 ... ... ... -899 5-45 ... -756 5-6 ... ... ... ... -635 10-35 ... ... -622 12-53 ... -727 22-7 2-01 26-3 ... -732 26-88 ... ... ... ... -543 35-97 ... -774 49-09 ... ... -565 65-58 ... ... ... -649 Temperature- Coefficient. De Heen finds that the coefficients of diffusion of solu- tions of the undermentioned salts are affected with the following factors depending on the temperature t : MgS0 4 1 - -000 119 t K 2 C0 3 1 - -000 127 t KN0 3 1 - -000 127 t Na 2 HP0 4 1 - '000 128 1 154. The following values of K in terms of the centi- metre and second are given in Professor Clerk Maxwell's 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 ix.] HEAT. 123 These may be compared with the value of - for air, c 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 10~ 5 , and is independent of the pressure. Pro- fessor Maxwell, by a different method, calculates its value at 5-4 x 10- 5 . Results of Experiments on Conductivity of Solids. 155. 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 /QQ./j. Q\2 C.G.S., is - n~^' or 15'48; and his results must be 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 1 J inch, and of the other an inch. The results when reduced to C.G.S. units are as follows : Cterit'' 1 J-inch bar. 1-inch bar. -207 '1536 25 -1912 -1460 50 -1771 '1399 75 -1656 -1339 100 -1567 -1293 125 -1496 -1259 150 -1446 -1231 175 -1399 -1206 200 -1356 -1183 225 -1317 '1160 250 -1279 -1140 275 -1240 -1121 156. Neumann's results (Ann. cle Chim., vol. Ixvi. p. 124 C.G.S. UNITS AND CONSTANTS. [CHAP. 185) must be multiplied by -000848 to reduce them to our scale. They then become as follows : Copper, TIGS Brass, -302 Zinc, -307 Iron, -164 German Silver, -109 Ice, -0057 In the same paper he gives for the following substances the values of the diifusivity - . These require the same C reducing factor as the values of k, and when reduced to our scale are as follows : lc Values of - Coal, -00116 Melted Sulphur, -00142 Ice, -0114 Snow, -00356 Frozen Mould, -00916 Sandstone, -0136 Granite (coarse), -0109 Serpentine, '00594 157. Sir W. Thomson's results, deduced from observa- tions of underground thermometers at three stations at Edinburgh (Trans. K. 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 (30-48) 2 or 929 to reduce them to our scale. The following are the reduced results : *, or *. Conductivity. c Trap-rock of Calton Hill, -00415 '00786 Sand of experimental garden, '00262 '00872 Sandstone of Craigleith Quarry, '01068 '02311 ix.] HEAT. 125 The values of - were directly derived from the observa- L/ tions ; and the values of k were deduced from them by the help of determinations of c made by Regnault. k My own result for the value of - from the Greenwich C underground thermometers (Greenwich Observations, 1860) is in terms of the French foot and the year. As a French foot is 32'5 centims., and a year is 31557000 seconds, the reducing factor is (32'5) 2 - 31557000; that is, 3-347 x 10~ 5 . The result is * c Gravel of Greenwich Observatory Hill, '01249 Professors Ayrton and Perry (Phil. Mag., April, 1878) determined the conductivity of a Japanese build- ing stone (porphyritic trachyte) to be '0059. 158. Angstrom, in Pogg. Ann., vols. cxiv. (1861) 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 : Value of * c Copper, first specimen, 1-216(1- '00214 t) second specimen, 1 '163 (1 - '001519 t) Iron, ...' '224 (1 - '002874 t) He adopts for c the values 84476 for copper , '88620 for iron, and thus deduces the following values of k : Conductivity. Copper, first specimen, 1'027 (1 - '00214 t) second specimen, '983 (1 - '001519 t) Iron, '199 (1 - '002874 t) 126 C.G.S. UNITS AND CONSTANTS. [CHAP. 159. A Committee, consisting of Professors Herschel and Lebour, and Mr. J. F. Dunn, appointed by the British Association to determine the thermal conduc- tivities of certain rocks, have obtained results from which the following selection has been made by Pro- fessor Herschel : -. Calcareous sandstone (firestone), '00211 '0049 Plate-glass, German and English, -00198 to '00234 -00395 to German glass toughened, ......... "00185 '00395 Heavy spar, opaque rough crystal, '00177 ...... Fire-brick, ............................ '00174 '0053 Fine red brick, ...................... '00147 '0044 Fine plaster of Paris, dry plate, '00120 '0060) Do., thoroughly wet, '00160 '0025 / White sand, dry, ................... '00093 '0032 Do., saturated with water, about ....................... '00700 '0120 about House coal and cannel coal, ...... '00057 to '001 13 '0012 to '0027 Pumice stone, ....................... '00055 ...... 160. Peclet in Annales de Chimie, se"r. 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), 150, shows that the value of this conductivity in the C.G.S. system, is His results must accordingly be divided by 100. The same author published in 1853 a greatly extended series of observations, in a work entitled " Nouveaux documents relatifs au 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 1 . , , . l_ 1 '10000*3600' '3bO' 128 C.G.S. UNITS AND CONSTANTS. The results must therefore be divided by 360. Those of, them which refer to metals appear to be much too small. The following are for badly conducting substances : Density. Conductivity. Fir, across fibres, -48 '00026 along fibres, -48 -00047 Walnut, across fibres, '00029 along fibres, -00048 Oak, across fibres, -00059 Cork, -22 -00039 Caoutchouc, -00041 Gutta Fercha, -00048 Starch Paste, 1'017 -00118 Glass, 2-44 -0021 Glass, 2-55 -0024 Sand, quartz, 1'47 '00075 Brick, pounded, coarse-graiued, 1-0 -00039 p Si", ough } " - 00046 Fine brick dust, obtained by de-) ^.g- -00039 Chalk, powdered, slightly damp, '92 -00030 washed and dried, '85 '00024 washed, dried, and) ,. 02 -OOO^Q compressed, J Potato-starch,. -71 '00027 Wood-ashes, -45 -00018 Mahogany sawdust -31 -00018 Wood charcoal, ordinary, powdered, -49 -00022 Bakers' breeze, in powder, passed ) .- through silk sieve, / Ordinary wood charcoal in pow-\ . 41 der, passed through silk sieve, J Coke, powdered, '77 -00044 Iron filings, 2'05 -00044 Binoxide of Manganese, 1-46 '00045 Woolly Substances. Cotton Wool, of all densities, .... Cotton swansdown (molleton de) coton), of all densities, J Calico, new, of all densities, Wool, carded, of all densities, ... 000111 000111 000139 000122 IX.] HEAT. 129 Density. Woollen swansdown (molleton de^ laine) of all densities), / Eider-down, Hempen cloth, new, '54 old, -58 Writing-paper, white, '85 Grey paper, unsized, '48 Conductivity. 000067 000108 000144 000119 000119 000094 161. In Professor George Forbes's paper on conduc- tivity (Proc. E.S.E., February, 1873) the units are the centim. and the minute ; hence his results must be divided by 60. In a letter dated March 4, 1884, to the author of this work, Professor Forbes remarks that the mean tempera- ture of the substances in these experiments was - 10, and expresses the opinion that bad conductors (such as most of these substances) conduct worse at low than at high temperatures an opinion which was suggested by the analogy of electrical insulators. His results reduced to C.G.S. are Ice, along axis, -00223 Ice, perpendicular to\ .QAO ilX 1 S, J Black marble, -00177 White marble, -00115 Slate, -00081 Snow, -00072 Cork -000717 Glass, -0005 Pasteboard, -000453 Carbon, -000405 Roofing-felt, -000335 0003 000088 Fir, parallel to fibre, Fir, across fibre, and\ along radius, J Boiler-cement, -000162 Paraffin, -00014 Sand, very fine, '000131 Sawdust, -000123 00011 sedindia- 1 . 000089 ' J Kamptulikon, .. Vulcanized india- rubber, Horn, -000087 Beeswax, '000087 Felt, -000087 Vulcanite, -000 0833 Haircloth, -0000402 Cotton-wool, divided, -000 0433 ,, pressed, -0000335 Flannel, -0000355 000 0298 000922 00124 00057 00083 0040 0044 Coarse linen, Quartz, along axis, Quartz, perpendicu-) lar to axis, / Professor Forbes quotes a paper by M. Lucien De la 130 C.G.S. UNITS AND CONSTANTS. [CHAP. 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 Ann ales de Chimie. ser. 4, torn. i. pp. 504-6. Conductivity of Liquids. 162. H. F. Weber (Sitz. kon. Preuss. Akad., 1885), has made the following determinations of conductivities of liquids at temperatures of from 9 to 15 C. He employs the centimetre, the gramme, and the minute as units : we have accordingly divided the original numbers by 60 to reduce to C.G.S. Water, Conduc- tivity. 00136 Amyl 4.cetate Conduc- tivity. 000302 Aniline, 000408 Glycerine, 000670 Chloro Benzol 000302 Ether, 000303 Chloroform, 000288 Methyl Alcohol, Ethyl Alcohol, 000495 000423 Chloro Carbon, Propyl Chloride, Isobutyl Chloride, 000252 000283 000278 Propyl Alcohol, Butyl Alcohol, Amy 1 Alcohol, 000373 000340 000328 Amyl Chloride, Bromo Benzol, 000284 000265 Formic Acid,... . 000648 Ethyl Bromide, Propyl Bromide, . 000247 000257 Acetic Acid, 000472 Isobutyl Bromide, 000278 Propion Acid, 000390 Amyl Bromide 000237 Butyric Acid, 000360 Isobutyric Acid, Valerian Acid 000340 000325 Ethyi Iodide, Propyl Iodide 000222 000220 Iso valerian Acid, Isocapron Acid, 000312 000298 Isobutyl Iodide, Amyl Iodide, . ... 000208 000203 Methyl Acetate, 000385 Benzol, 000333 Ethyl Formiate, 000378 Toluol, 000307 Ethyl Acetate, 000348 Cymol, 000272 Propyl Formiate, Propyl Acetate, Methyl Butyrate, Ethyl Butyrate, Methyl Valerate, EthvlValerate... 000357 000327 000335 000318 000315 000307 Oil of Turpentine, Sulphuric Acid, Bisulphide of Carbon, Oil of Mustard, Ethvl SulDhide.... 000260 000765 000343 000382 000328 ix.] HEAT. 131 163. C. Chree (Proc. Eoy. Soc., April, 1887) found, by two slightly different methods, at temperatures between 18 and 20, values which, when divided by 60 to reduce them to C.G.S., are : Conductivity. 1st Method. 2nd Method. Water, '001245 '001358 Methylated Spirit, "000590 "000577 Paraffin Oil, '000440 '000455 Turpentine Oil, ... '000315 Bisulphide of Carbon, . . . '000537 More recently, Gratz has found at 13 : Conductivity. I Conductivity. Glycerine, '000637 | Ethyl Alcohol, '000545 Ether, '000378 i Bisulphide of Carbon, '000266 Oil of Turpentine, .... '000325 | Chloride of Sodium Solu- tion, density 1'152, '000112 Assuming k t = k Q (l - at), where k , lc t denote the conduc- tivities at and t, the temperature-coefficient a had, in the neighbourhood of 13, the values : Glycerine, '012 | Oil of Turpentine, '007 Sodium Chloride Solution, -006 Conductivity of Gases. 164. Winkelmann gives the following values for gases at 7'5 : Conductivity. Air, -0000516 Hydrogen, '00033 Nitrous Oxide, . . . '0000312 Conductivity. Carbonic Acid, . . . '0000273 Ethyl, '0000356 Marsh Gas, '000065 For the temperature-coefficient (defined as in the pre- ceding section) he finds : Air -000208 | Carbonic Acid, -0038 Emission and Surface Conduction. 165. Mr. D. M'Farlane has published (Proc. Roy. Soc., 1872, p. 93) the results of experiments on the loss 132 C.G.S. UNITS AND CONSTANTS. [CHAP. of heat from blackened and polished copper balls 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 + 3-06 x 10~ 6 t - 2-6 x 1Q- 8 / 2 for a blackened surface, and x = -000168 + 1-98 x 10~ c t - 1-7 x 10~ 8 t 2 . for polished copper, x denoting emissivity, that is, the quantity of heat lost 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 14C. The air within the enclosure was kept moist by a saucer of water. The greatest difference of temperature employed in the experiments (in other words, the highest value of t) 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 : Difference of Temperature. Emissivity. Ratio. Polished Surface. Blackened Surface. 5 OC0178 000252 707 10 000186 000266 699 15 000193 000279 692 20 000201 000289 695 25 000207 000298 694 30 000212 000306 693 35 000217 000313 693 40 000220 000319 -693 45 000223 000323 690 50 000225 000326 -690 55 000226 000323 -690 60 000226 000328 690 ix.] HEAT. 133 166. Rowland (p. 170 of paper on Mechanical Equiva- lent) found for the coefficient of total emission of heat from a nickel-plated calorimeter to the surrounding jacket, the jacket being at 20 : Difference of Temp. Coefficient of Emission. 6 -000081 5 -000082 10 -000086 15 -000089 20 -000093 25 -000096 and points out that these are rather less than half the values obtained by M'Farlane for polished copper. The jacket was not blackened. Influence of Size. According to Prof. Ayrton, who quotes a table in " Box on Heat," the coefficient of emission increases as the size of the emitting body diminishes, and for a blackened sphere of radius r centims. may be stated as OOP 4928 + - 0003609 r The value of r in M'Farlane's experiments was 2. Radiation in Vacuo. 167. Dr. J. T. Bottomley (Phil. Trans., 1888, and subsequently in 1890, the final results being not hitherto published), experimenting with the same two globes em- ployed by M'Farlane, and keeping the walls of the enclosure at about 14 or 15, found the emissivity in a 134 C.G.S. UNITS AND CONSTANTS. [CHAP. Sprengel vacuum, when the surface of the globe was coa,ted with soot, to range from -000 1 25 for an excess of temperature of 83 0< 6 to '000095 for an excess of 32-7. When the globe was silvered and very highly polished, the emissivity in vacuo ranged from -000 0354 for an excess of 721 to -0000296 for an excess of 43-7. Employing higher temperatures for the globes, but the same temperature of the enclosure, the mean emis- sivity for the sooted globe was -000 223 when the excess was between 230 and 205, and -000122 when it was between 84-l and 78 -8. With dry air at atmospheric pressure in place of vacuum, the emissivity for the sooted globe was multi- plied by about 2J, and that for the silvered and polished globe by about 5. With a platinum wire at 408 as the emitting body, the temperature of the enclosure being about 16, the emission at the highest vacuum being taken as the unit, the emission was 1'92 at a pressure of '034 mm., 3'12 at 094 mm., 3'96 at -14 mm., 7-1 at '444 mm., 9-2 at -88 mm., 11-5 at 1*7 mm., 13-5 at 2-5 mm., 15-9 at 4 mm., 16-7 at 5-7 mm., 19-14 at 17'2 mm., 20-04 at 42 mm., 21-00 at 340 mm., and 21-48 at 740 mm. 168. 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 approximately 8 C. When reduced to C.G.S., Professor Tait's Table of Results will stand as follows : IX.] HEAT. 135 Pressure 1 '014 x 10 6 [760 millims of mercury]. Blackened. Temp. Cent. Loss per sq. cm. per second. 61-2 -01746 50-2 -01360 41-6 -01078 34-4 -00860 27-3 ' -00640 20-5 , -00455 Bright. Temp. Cent. Loss per sq. cm. per second. 63-8 -00987 57-1 -00862 50-5 -00736 44-8 -00628 40-5 -00562 34-2 -00438 29-6 -00378 23-3 -00278 18-6 -00210 Pressure 1 '36 x 10 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 49'7 -00340 43 -00791 44-9 -00302 28-5 -00490 40-8 . -00268 Pressure 1'33 x 10 4 [10 millims. of mercury]. 62-5 -01182 57-5 -01074 54-2 -01003 41-7 -00726 37-5 -00639 34 -00569 27-5 -00446 24-2 . -00391 65 60 50 40 30 23-5 00388 00355 00286 00219 00157 00124 169. Heat and Energy of Combination with Oxygen. 1 gramme of Compound formed. Gramme- degrees of heat ! produced. Equivalent Energy, in Ergs. Hydrogen, H 2 O 34000 A F l-43x!0 12 Carbon, ..1 CO 2 8000 A F i 3-36xlO n Sulphur, SO 2 2300 A F 1 9'66xl0 10 Phosphorus, P 2 5 5747 A 2-41 x 10 11 Zinc, ZnO 1301 A 5-46xl0 10 Iron, Fe 3 4 1576 A 6'62x 10 10 Tin, SnO 2 1233 A 5-18 ,, Copper CuO 602 A 2-53 ,, Carbonic Oxide,.. Marsh-gas CO 2 CO 2 and H 2 2420 A 13100 A F l-02x!0 n 5 '50 Olefiant gas, Alcohol, 11900 A F 6900 A F 5-00 2'90 ,, " 136 C.G.S. UNITS AND CONSTANTS. [CHAP. Combustion in Chlorine. Hydrogen, HC1 23000 F T 9-66 x 10 11 Potassium, Zinc, . KC1 ZnCP 2655 A 1529 A 1-12 6'42x 10 10 Iron,.... Fe 2 Cl 6 1745 A 7-33 Tin SnCl 4 1079 A 4-53 Copper, CuCP 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. 170. Difference between the two specific heats of a gas. Let s l denote the specific heat of a given gas at con- stant pressure, s 2 the specific heat at constant volume, a the coefficient of expansion per degree, v the volume of 1 gramme of the gas in cubic centim. at pressure p dynes per square centim. When a gramme of the gas is raised from to 1 at the constant pressure p, the heat taken in is s 1? the increase of volume is av, and the work done against external resist- ance is avp (ergs). This work is the equivalent of the difference between s : and s 2 : that is, we have Sl -s 2 = ~f, where J = 4-2 x 10?. For dry air at the value of vp is 7 "838 is -003 665. Hence we find s - s = -0684. x 10 8 , and a 2 . The value of s p according to Regnault, is -2375. Hence the value ofs 2 is -1691. ix.] HEAT. 137 The value of - 1 ^, or !f , for dry air at and a v ' J megadyne per square centim., is .9, - s 9 '0684 v 783-S = 8-727 and this is also the value of-^ - for any other gas (at the same temperature and pressure) which has the same coefficient of expansion. 171. Change of freezing point due to change of pressure. Let the volume of the substance in the liquid state be to its volume in the solid state as 1 to 1 + e. 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 c, 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 + 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 +p, I 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 138 C.G.8. UNITS AND CONSTANTS. [CHAP. But the mechanical effect is pe. Hence we have _ P e *_404 - JC 1 - m 174. Expansions of Volume per degree Cent, (abridged from Watts' 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, .......................... -000087 '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. 175. Very exact determinations of the coefficients of linear expansion of certain substances have been made by Benoit at the Bureau International des Poids et Mesures, and are described in torn. VI. of Travaux et Memoires, where the chief results are collected at p. 190 In terms 142 C.G.S. UNITS AND CONSTANTS. [CHAP. of temperature by hydrogen thermometer, some of them are as follows : Quartz, along axis, 10~ 9 ( 7161 '4+ 8'01 1). ,, perpendicular to axis, 10" 9 ( 1 3254 '6 4-1 1*63 ). Beryl, along axis, lO' 9 ( - 1347'8 + 4-12U). ,, perpendicular to axis, 10~ 9 ( + 1002'5 + 4-570- Iceland-spar, along axis, 10- 9 ( 25135 '3 + 11 '80 1). perpendicular to axis, 10 - 9 ( - 5578-2 + 1 '38 t). Platinum, pure, 10~ 9 ( 8901 + 1'2U). Iridium, ,, 10' 9 ( 6358 + 3'2U). Platinum-iridium, containing 10 per ) , A _ q / cfiKa cent, of iridium, about, / U Steel, tempered, 10~ 9 ( 10354 + 5'23 1). ,, ,, another specimen, 10' 9 ( 10457 + 5'20). Bronze, about, lQ- 9 ( 17545 + 5 -25*). Where we have added the word " about," the original gives results for various specimens, and we have adopted a mean. 176. Expansion of Mercury, according to Regnault (Watts' Dictionary, p. 56). Temp.., .Voluine.tr. "g^SS?? 1-000000 -00017905 10 1-001792 -00017950 20 1-003590 -00018001 30 1-005393 -00018051 50 1-009013 -00018152 70 1-012655 -00018253 100 1-018153 -00018405 The temperatures are by air thermometer. The formula adopted by the Bureau International des Folds et Mesures for the volume at t C. (derived from Regnault's results) is 1 + -000 181 792 1 + -000 000 000 175 t z + -000 000 000 035 116* 3 . ix.] HEAT. H3 177. Expansion of certain Liquids, as determined by Kopp and Pierre. Volume of Temp. Alcohol. Ether. Bisulphide of Carbon. Oil of Turpentine. ! 1- r 1- 10 1-01050 1-01518 1-01156 1-00919 20 1-02128 1-03122 1-02350 1-01875 30 1-03242 1-04829 1 -03594 1-02865 40 1 -04404 1 -06654 1-04901 1-03886 178. Collected Data for Dry Air. Expansion from to 100 at const, pressure, as 1 to 1 *367 or as 273 to 373 Specific heat at constant pressure, ......................... '2375 , , , , at constant volume, ........................... '1691 Pressure-height at C., about 7 "99 x 10 5 cm., or about 26210 ft. Standard barometric column, ........ 76 cm. = 29 '922 inches. Standard pressure, ...................... 1033'3 gm. per sq. cm. or 14 "7 Ibs. per sq. inch. or21171bs. ,, foot. or 1*0136 x 10 G dynes per sq. cm. Standard density, at C. , ......... -001293 gm. per cub. cm. or -0807 Ibs. per cub. foot. Standard bulkiness, .................. 773 -3 cub. cm. per gm. or 12-39 cub. ft. per Ib. Dry and Moist Air. Mass of 1 Cubic Metre in Grammes. Temp. Dry Air. Saturated Air. ......... 1293-1 ............ 1290-2 ............ 4-9 10 ......... 1247-3 ............ 1241-7 ............ 9'4 20 ......... 1204-6 ............ 1194-3 ............ 17'1 30 ......... 1164-8 ............ 1146-8 ............ 30'0 40 ......... 1127-6 ............ 1097-2 ............ 50'7 If A denote the density of dry air and W that of vapour at saturation, the density of saturated air is A - f W, or more exactly A - -608 W. 144 CHAPTER X. MAGNETISM. 179. THE unit magnetic pole, 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 1 centimetre, with a force of 1 dyne. If P denote the strength of a pole, it will repel an equal P 2 pole at the distance L with the force y^. Hence we have the dimensional equations P 2 L~ 2 * force = MLT" 2 , P 2 = ML S T- 2 , P = M^T' 1 ; that is, the dimensions of a pole (or the dimensions of strength of pole) are M^L^T 1 . 180. 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 _L 181. The intensity or strength of a, magnetic field is the rce which a unit pole will experience when placed in it. CHAP, x.] MAGNETISM. 145 Denoting this intensity by H, the force on a pole P will be HP. Hence HP - force = MLT-' 2 , H = MLT^. M^IT^T - that is, the dimensions of field intensity are 182. The moment of a magnet is the product of the strength of either of its poles by the distance between them. Its dimensions are therefore LP; that is, M 5 L*T . 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 of force 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^T' 1 as before. 183. If different portions be cut from a uniformly magnetised substance, their moments will be simply as their volumes. Hence the intensity of magnetisation of a uniformly magnetised body is defined as the quotient of its moment by its volume. But we have moment _! > L - 3 , MLT~\ volume Hence intensity of magnetisation (often called simply mag- netisation) has the same dimensions as intensity of field. When a magnetic substance (whether paramagnetic or diamagnetic) is placed in a magnetic field, it is magnet- ised by induction. From this point of view the intensity of the field to which the magnetisation is due is called the magnetising force. 184. If we suppose a narrow crevasse to be excavated in the magnetised substance, there will be no free mag- 146 C.G.S. UNITS AND CONSTANTS. [CHAP. netism on its sides if their direction be longitudinal, that is, parallel to the direction of magnetisation ; and when we speak of the magnetising force at a point in a lody we mean the field which would exist in such a crevasse excavated about the point. Magnetising force is now usually denoted by H. It is called, indifferently, mag- netising force, magnetic force, or strength of field. On the other hand, if the narrow crevasse be transverse, that is, if it cut the lines of magnetisation at right angles, there will be free magnetism of opposite signs on the two faces of the crevasse, the surface-density of this magnet- ism on either face being numerically equal to the intensity of magnetisation, which is denoted by I. These two surface-layers produce a field of intensity 4:r I in the narrow space between them, and this field must be compounded with the field H in order to obtain the resultant field within the transverse crevasse. This resultant is called the intensity of induction, or more briefly the induction, at the point in question, and is denoted by B. Accordingly, whether the body is iso- tropic or not, B is the resultant of H and 4^- 1. 185. If the substance is isotropic, and has no mag- netism except what is called out by the existing field, H and I have parallel directions, which are the same or opposite according as the substance is paramagnetic (like iron) or diamagnetic (like bismuth). In the former case H and I must be regarded as having the same sign, and in the latter case opposite signs. In both cases we have the algebraic equation 5 B-H + 47r|, (1), which is often written JU=1+47TK, (2), x.] MAGNETISM. 147 // denoting the ratio of B to H, and K the ratio of I to H. /u is called the permeability and K the susceptibility of the substance. These ratios are by no means constant for a given substance, but largely depend on the value of H. As H increases from zero, their values (in most cases at least) first increase to a maximum and then decrease. The value of K when negative is always small : so that ^ is always positive, being greater than unity for paramagnetic and less than unity for diamag- netic substances. The dimensions of B, H, and I are 1VH L~* T" 1 ; /x and K are mere numerical quantities independent of the units of mass, length, and time. 186. In air I is sensibly zero even in strong fields, and B is therefore sensibly equal to H. It can be shown that the component of B normal to the surface of a magnetised body (whether the mag- netisation be temporary or permanent) has the same value just outside as just inside the body ; whence it can be deduced that if tubes be drawn following the direction of B, the value of the product B x section of tube will be the same at all parts of one and the same tube. Every such tube returns into itself. One portion of it may be within a magnetised body, and the other portion in the external air. It is convenient to make the tubes of such sizes that the value 'of the constant product B x section is unity for each. Then the number of these " tubes of induction" that cut any area is called the " flux of induction " across the area. If the area is bounded by a conducting 143 C.G.S. UNITS AND CONSTANTS. [CHAP. circuit, any change in the flux of induction will produce in the circuit an electromotive force equal at each instant to the rate of change of this flux. 187. It was formerly thought that the ratios K and // were nearly independent of the intensity of the field so long as the magnetisation was far below saturation. This however is very far from being the fact. The following table based upon observations by Ewing* and Bidwellf indicates approximately the changes which occur in the susceptibility and permeability of wrought iron as the strength of the field is increased from a small to a high value. The results obtained with different specimens of iron may vary considerably in details, but the general character of the phenomenon is always the same. H 1 K B /* Magnetising force. Magnetisation. Susceptibility. Induction. Permeability. 0-3 3 10 41 128 1-4 32 23 413 299 2-2 117 53 1460 670 3-5 574 164 7230 2070 4-9 917 187 11540 2350 67 1078 161 13520 2020 10-2 1173 115 14840 1450 22-3 1249 56 15710 705 78 1337 17 16900 215 208 1452 7 18500 89 585 1530 2-6 19800 34 24500 1660 0-067 45300 1-9 * Phil. Trans. 1885, II., p. 541, and 1889, A. p. 226. f Proc. Roy. Soc., No. 245 (1886), p. 493. x.] MAGNETISM. 149 188. When I ceases to increase sensibly for further increase of H, "saturation" is said to be attained. Ewing (Phil. Trans. 1889, A. p. 242) finds that it is attained with magnetising forces of less than 2000 C.G.S. for wrought iron and nickel, and less than 4000 for cast iron and cobalt ; the following being the saturation values of I : Wrought iron, 1700 Cast iron, 1240 Nickel (annealed), 515 ,, (hard drawn),.... 400 Cobalt, 1300 189. Magnetic susceptibility and permeability are also affected by such causes as mechanical stress, vibration, and changes of temperature. In weak magnetic fields the susceptibility of an iron wire is increased by longitudinal tension, whereas in strong fields it may be diminished. The strength of field at which the reversal of the effect of tension takes place is called by Sir W. Thomson the " Villari critical point." It occurs earlier the greater the stress is. The following table, due to Ewing (Phil. Trans., 1885, II., p. 623), gives the critical strength of field and the corre- sponding magnetisation for a certain iron wire under various loads. Load Grms. per Cm. i Critical Field. Magnetisation. 215000 7'3 1220 430000 4-3 1040 860000 3-4 840 1290000 305 690 150 C.G.S. UNITS AND CONSTANTS. 190. Vibration or jarring appears generally to increase the permeability and susceptibility. With a small mag- netising force Ewing has found an apparent permeability of no less than 20,000 in a rod of soft annealed iron kept in a state of vibration (Journ. Inst. Elec. Eng., xix. 41). 191. The permeability of iron for small magnetising forces increases with rise of temperature, slowly at first, and afterwards more rapidly, until the iron is red hot, when it suddenly falls to 1, and the iron becomes non- magnetic. In an experiment by Hopkinson with a magnetising force of - 3 the permeability of an iron ring was 11,000 at a temperature of 770,, and only 1*14 at a temperature of 785 (Journ. Inst. E. E., xix. 26). Nickel becomes non-magnetisable at a temperature of about 300, and cobalt at a bright yellow heat. Critical Temperature for Iron. 192. In the case of iron, certain other remarkable changes of property occur at the same temperature at which it ceases to be magnetic. If an iron or steel wire is gradually raised to a red heat by an electric current, a spontaneous fall in its temperature and contraction of its length occur momentarily when it first reaches this temperature. When it has been heated higher and is allowed to cool, a still more marked spontaneous rise of temperature called " recalescence " and momentary elon- gation occur on arriving at the same temperature. The thermoelectric properties of iron also exhibit a striking irregularity at and near this temperature. Various Substances. 193. For nickel and cobalt Eowland found that as the MAGNETISM. 151 magnetising force was increased, ^ increased to a maxi- mum and then diminished, the following being the approximate values of the maxima and of the magnetis- ing forces with which they were obtained : Nickel at 15 C., 220, Cobalt at 5 C., -5, .. ,, 230, .. Max. M 220 315 142 144 236 Field. 9 5 or 6 18 above 18 11 or 12 194. The following results are selected from those contained in Prof. Chrystal's article " Magnetism " in the Encyc. Brit. For bismuth, Yon Ettingshausen obtained - 1 4 as the value of 10 K under magnetising forces ranging from 26 to 110 C.Gr.S., with some evidence of diminishing value for the larger forces. For ferric chloride, Silow found a maximum value of 10 6 K amounting to about 170, corresponding to a mag- netising force -4 C.G.S. The value was only 34 when the magnetising force was '08. The following values of 10 6 K for liquids and gases were obtained by Schuhmeister. Liquids. Magnetising force 61-5 130-8 252-7 Water, ... - -55 - -45 - '44 Alcohol, Bisulphide of carbon, Ether, -45 -46 -40 - -42 39 -29 -38 -37 152 C.G.S. UNITS AND CONSTANTS. Gases. [CHAP. Magnetising force, 66-8 141-8 272-2 Oxygen, 05 06 12 Nitrogen, . 025 038 047 For concentrated solutions of salts of iron Pliicker obtained the following values of susceptibility divided by density, in terms of an arbitrary unit. Ferrous chloride, ... 84 Ferric sulphate, 58 the scale being such that iron was 100,000, magnetic iron ore 40,000, specular iron ore 533, and hematite 134. Ferrous sulphate,... Ferric chloride, 126 98 Changes of Length. 195. The length of a bar of magnetisable metal is in general altered by longitudinal magnetisation. In a continually increasing field, the length of an iron bar is at first increased and afterwards diminished ; that of a Magnetic Elongations in Ten-millionths of original lengths. Iron. Cobalt. Nickel. 65 13 -104 125 19* -10 -167 237 7 -31 -218 293 -37 -233 343 - 6 -44t -240 500 -35 -30 745 -50 1120 -65 45 1400 -66 75 - 245 Maximum increment. t Maximum decrement. x.] MAGNETISM. 153 cobalt bar is at first diminished and afterwards increased, while that of a nickel bar is always diminished. The annexed table shows the nature of the positive and negative elongations of certain rods of iron, cobalt and nickel under magnetisation as observed by Bidwell (Phil. Trans., 1888, A. pp. 226-8) : ELECTROMAGNETISM. 19G. A circular current C, of radius ?, exerts upon a unit pole placed at its centre a force - r , the strength of the current being expressed in electromagnetic units. A north pole is urged along the axis of the circuit in a direction which is related to that of the current as the longitudinal to the rotational motion of a right-handed screw. The intensity of force inside a solenoid or coil, the length of which is great compared with its transverse dimensions, is AirCn/l, n being the number of turns of wire, and I the length of the solenoid. The direction of the force may be specified as before. Inside a solenoid forming a hollow circular ring of circular section (like string wound upon an anchor ring) the strength of field 9

; + -I5y) minutes. For Dip, -(1-6- -O9.r+-04y) minutes. For Horizontal Force, + (-00019 - -000 006# + '000 003y). See table, page 325 of Bakerian Lecture. 201. The following mean values of the magnetic ele- ments at Greenwich have been kindly furnished by the Astronomer Eoyal : West Declination,... 11" 41' -5 - 6'-75 x (t- 1888) Dip, 6725'-3-l'-29x(f-1888) Horizontal Force, ... 0-18196 + -000 187 x (t- 1888) Vertical Force, ... 0-43760 - -000 013 x (t - 1888) = Horizontal Force x tan. Dip. Each of these formulae gives the mean of the entire year t. 202. According to Lord Rayleigh's determination (Phil. Trans., 1885, II. p. 343) the rotation of the plane of polarisation between two points, 1 centim. apart, in bisul- phide of carbon at 18 C., whose magnetic potentials (in C.G.S. measure) differ by unity, is -04202 minute, or 1-22231 x 10~ 5 radian, for sodium light. At C. the rotation is according to H. Becquerel, -04341 min. J. E. H. Gordon, -0433 L. Arons, -0439 161 CHAPTER XI. ELECTRICITY. Electrostatics. 203. 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 I will be |j. 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 1 centim. with a force of 1 dyne. Since the dimensions of force are ^-, we have, as regards dimensions, C IP whence q z = ~, q = m^-i. 204. 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 1 erg. 162 C.G.S. UNITS AND CONSTANTS. [CHAP. Or, we may define potential as the quotient of quantity of .electricity by distance. This gives v = mfir l . l~ l = mtyty- 1 , as before. In the C.G.S. system the unit of potential is the potential due to unit quantity at the distance of 1 centim. 205. 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 = ? = mfrl*t~ l . m~%-fy = L v The same conclusion might have been deduced from the fact that the capacity of an isolated spherical con- ductor is equal (in numerical value) to its radius. The C.G.S. unit of capacity is the capacity of an isolated sphere of 1 centim. radius. 206. 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 |j that is, mfyty~ z . t The C.G.S. unit of current is that current which con- veys the above defined unit of quantity in 1 second. 207. 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^lk~ l . wr^H/ 2 = l~ l t. Or, the resistance of a conductor is equal to the time required for the passage of a unit of electricity through it, XL] ELECTRICITY. 163 when unit difference of potential is maintained between its ends. Hence resistance = time x potential = f ^j^ m _ 1H/ = ^ quantity 208. As the force upon a quantity q of electricity, in a field of electrical force of intensity i, is iq, we have '1 The quantity here denoted by i is commonly called the " electrical force at a point." Electromagnetics. 209. A current (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 = lAV^T- 1 . 210. 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^. 211. 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~i, or 164 C.G.S. UNITS AND CONSTANTS. [CHAP. 212. 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 . Tjl 213. Eesistance is ^; its dimensions are therefore \j M WT- 2 . M - *L - *T ; that is, LT' 1 . 214. 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 i L t T -i M*L* LT- 1 Current, M*Lh:- M*L*T-i LT- 1 Capacity, L L -1 T 2 L 2 T -2 Potential ancU electronic- J- tive force, j MiA. MWT-. L-.T Resistance,.... L-'T LT- 1 L- 2 T 2 215. The heat generated in time T by the passage of a current C through a wire of resistance R (when there are no Peltier or Thomson effects in the wire) is J gramme degrees, J denoting 4-2 x 10 7 ; and this is true whether C and R are expressed in electromagnetic or in electrostatic units. XL] ELECTRICITY. 165 Ratios of the two sets of Electric Units. 2 1C. 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 M*L*T- ] C.G.S. electrostatic units of quantity, and the electromagnetic unit of quantity equal to M^L^ 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^L^) LT' 1 C.G.S. electrostatic units = 1 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. 217. 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 four electrical elements. The last column of the above table shows that M does not enter into any of the 166 C.G.S. UNITS AND CONSTANTS. [CHAP. ratios, and that L and T enter with equal and opposite indices, showing that all the ratios depend only on the L velocity rp- Thus, if the concrete velocity ^, be a velocity of r centims. per second, the following relations will subsist between the C.G.S. units : 1 electromagnetic unit of quantity = r electrostatic units. 1 current = i- 1 capacity = r- v electromagnetic units of potential = 1 electrostatic unit. v 2 resistance = 1 218. The latest determination of the value of v is that of Prof. J. J. Thomson and G. F. C. Searle (Phil. Trans., 1890, p. 620), and is 2-9955 x 10 10 centims. per second. On page 621 of the same paper is given (as a quotation from a paper by E. B. Eosa) the following list of previous determinations : 1856 Weber and Kohlrausch, ... 3'107 x 10 10 1869 W. Thomson and King 2'808 1868 Maxwell, 2'842 1872 M'Kichan, 2'896 1879 Ayrton and Perry, 2'960 ., 1880 Shida, 2"955 1883 J.J.Thomson, 2'963 1884 Klemencic, 3'019 1888 Himstedt, 3'009 1889 W. Thomson, 3'004 ,, 1889 E. B. Rosa, 2'999 ,, All these values agree closely with the velocity of light XL] ELECTRICITY. 167 in vacuo, of which the best determinations are, some of them a little less, and some a little greater than 3 x 10 10 . We shall adopt this round number as the value of r. 219. 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 ( 214), we have the following results: Electrostatic. Electroma gnetic. Quantity, Current, Capacity, L Potential, D^L Resistance, L^T LT J It will be noted that the exponents of L and T in these expressions are free from fractions. Specific Inductive Capacity. 220. 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 conden- ser in other respects equal and similar in which air is the dielectric. It is of zero dimensions, and its value exceeds unity for all solid and liquid insulators. According to Maxwell's electromagnetic theory of light, the square root of the specific inductive capacity is equal to the index of refraction for the rays of longest wave-length. Messrs. Gibson and Barclay, by experiments performed in Sir W. Thomson's laboratory (Phil. Trans., 1871, 168 C.G.S. UNITS AND CONSTANTS. [CHAP. p. 573), determined the specific inductive capacity of solid paraffin to be 1'977. 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, Light, . 2-87 3 '2 6-57 6 '85 2-29 2'14 1-541 1-574 Dense, 3-66 7'4 2'02 1-622 Double extra \ dense, / 4-5 10-1 2-25 1-710 Specific Inductive Capacity. Paraffin, 2 '29 In a later series of experiments (Phil. Trans., 1881, Dec. 16), Dr. Hopkinson obtains the following mean determinations : Specific Glass. Inductive Density. Capacity. Hard crown, 6'96 2-485 Very light flint, 6 '61 2 '87 Light flint, 6'72 3'2 Dense flint, 7'38 3'66 Double extra-dense flint, 9 '90 4 '5 Plate, 8-45 221. For liquids Dr. Hopkinson (Proc. Roy. Soc., Jan. 27, 1881) gives the following value of ^ (computed) and K (observed), K denoting the specific inductive capacity and /^ the index of refraction for very long waves deduced by the formula where b is a constant. XT.] ELECTRICITY. 169 Petroleum spirit (Field's), .................. 1-922 1'92 Petroleum oil (Field's), ...................... 2'075 2*07 (common), .................... 2'078 2'10 Ozokerit lubricating oil (Field's), ......... 2-086 2-13 Turpentine (commercial), ................... 2'128 2'23 Castoroil, ....................................... 2'153 4'78 Sperm oil, ....................................... 2'135 3'02 Oliveoil, ......................................... 2-131 3'16 Neatsfoot oil, ................................... 2-125 3'07 This list shows that the equality of / to K (which Maxwell's theory requires) holds nearly true for hydro- carbons, but not for animal and vegetable oils. 222. Wiillner (Sitzungsber. konigl. bayer. Akad., March, 1877) finds the following values of specific induc- tive capacity : Paraffin, ............ 1'96 Shellac,.... 2 '95 to 3 '73 Ebonite, ............ 2'56 Sulphur,... 2 -88 to 3 '21 Plate glass, ........ 6'10 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, ..... I'85to2'47 Caoutchouc, ......... 2'12 to 2'34 Ebonite, ..... 2 -21 to 276 Do., vulcanized, 2 -69 to 2 -94 Plate glass,. 5 '83 to 6 '34 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 2-071 170 C.G.S. UNITS AND CONSTANTS. [CHAP. Boltzmann (Wien. Akad. Ber. (2), Ixx. 342, 1874) finds for sulphur in directions parallel to the three princi- pal axes, the values 4-773, 3-970, 3-811. 223. Quincke has investigated the specific inductive capacity of certain liquids by two independent methods, namely, by measuring the capacity of a condenser and by measuring the mutual attraction of the two parallel plates which composed the condenser. According to Maxwell's theory, the charging of a condenser produces tension (or diminution of pressure) in the dielectric along the lines of force, and repulsion (or increase of pressure) perpendicular to the lines of force, the tension and repulsion (per unit of cross section) being each equal to K(A-B) 2 87TC 2 K denoting the specific inductive capacity of the dielectric c the distance between the two parallel plates of the con- denser, and A B their difference of potentials. The following are Quincke's latest determinations as given in the R. S. Proc., 1886, Dec. 16, p. 459 : Ether. BeMoL By capacity, ...... 4'211 2'574 2'359 2'025 By attraction, ... 4 "394 2-582 2 '360 2*073 224. Professors Ayrton and Perry have found the following values of the specific inductive capacities of gases, air being taken as the standard : Air, 1-0000 Vacuum, 0-9985 Carbonic acid,... 1*0008 Hydrogen, 0'9998 Coal gas, 1-0004 Sulphurous acid, 1 '0037 XL] ELECTRICITY. 171 Practical Units. 225. The unit of resistance chiefly employed by practical electricians is the Ohm, which is theoretically defined as 10 9 C.G.S. electromagnetic units of resistance. The practical unit of electromotive force is the Volt, which is defined as 10 8 C.G.S. electromagnetic units of potential. The practical unit of current is the Ampere. It is de- fined as -**^ T V of the C.G.S. electromagnetic unit current, or as the current produced by 1 volt through 1 ohm. The practical unit of quantity of electricity is the Coulomb. It is defined as -s of the C.G.S. electromagnetic unit of quantity, or as the quantity conveyed by 1 ampere in 1 second. The practical unit of capacity is the Farad* It is defined as 10~ 9 of the C.G.S. electromagnetic unit of capacity, or as the capacity of a condenser which holds 1 coulomb when charged to 1 volt. The practical unit of work employed in connection with these is the Joule. It is defined as 10 7 ergs, or as the work done in 1 second by a current of 1 ampere in flowing through a resistance of 1 ohm. * As the farad is much too large for practical convenience, its millionth part, called the microfarad, is practically employed, and condensers are in use having capacities of a microfarad, and its decimal subdivisions. The microfarad is 10~ 15 of the C.G.S. electromagnetic unit of capacity. 172 C.G.S. UNITS AND CONSTANTS. [CHAP. The corresponding practical unit of rate of working is the Watt. It is defined as 10 7 ergs per second, or as the rate at which work is done by 1 ampere flowing through 1 ohm. Electric Spark. 226. 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 be- Intensity of Difference of Potential between Surfaces. tween Surfaces. f orco in Electrostatic Units. In Electrostatic Units. Iii Electromagnetic Units. 0086 267-1 2-30 6-90xl0 10 0127 257-0 3-26 9-78 0127 262-2 3-33 9-99 0190 224-2 4-26 12-78 0281 200-6 5-64 16-92 0408 151-5 6-18 18-54 0563 144-1 8-11 24-33 0584 139-6 8-15 24-45 0688 140-8 9-69 29-07 0904 134-9 12-20 36-60 1056 132-1 13-95 41-85 1325 131-0 17-36 52-08 XI.] ELECTRICITY. 173 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 10 10 . 227. Dr. Warren De La Eue and Dr. Hugo W. Miiller (Phil. Trans., 1877) have measured the striking dis- tance between the terminals of a battery of chloride of silver cells, the number of cells being sometimes as great as 11000, 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 Eue 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 Distances. Nn of Pp11 1-10 l-37x!0 10 1-04 xlO 10 845 x 10 10 737 x 10 10 1-20 1-33 926 666 486 a 1-25 1-31 896 ,, 624 434 ,, 1-30 1-36 *94 662 472 ,, 1-40 1'69 1-30 1-05 896 ,, 1-50 2-74 2-13 ,, 1'72 1-52 ,, 1-60 4-82 3-62 2-75 2-21 , } T70 9-41 6'25 4-23 3-07 ,, Resistance of Carbons. 237. The specific resistance of Carre's electric-light carbons at 20 C. is stated to be 3-927 x 10 C.G.S., whence it follows that the resistance of a cyclinder 1 metre long and 1 centimetre in diameter is just half an ohm. The specific resistance of Gaudin's carbons is about 8-5x1 0* ; retort carbon ,, 6'7 x 10 7 graphite from 2-4 x 10 6 to 4-2 x 10 7 The resistance of carbon diminishes as the temperature increases, the diminution from to 100 C. being T \- for Carry's and - for Gaudin's. The resistance of an incan- xi.] ELECTRICITY. 181 descent lamp when heated as in actual use is about half its resistance cold. Resistance of the Electric Arc. 238. The difference of potentials between the two carbons of an arc lamp has been found by Ayrton and Perry (Phil. Mag., May, 1883) to be practically in- dependent of the strength of the current, when the dis- tance between them is kept constant. It was scarcely altered by tripling the strength of the current. The apparent resistance of the arc (including the effect of reverse electromotive force) is therefore inversely as the current. The difference of potentials was about 30 volts when the current was from 6 to 12 amperes. 239. 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 2-240xl0 18 - 6-2 ....; 1-023 ,, - 5-02 9-486xl0 17 - 3-5 6-428 - 3-0 5-693 - 2-46 4-844 ,, - 1-5 3-876 - 0-2 2-840 ,, + 0-75 1-188 about + 2-2 2-48 x 10 16 + 4-0 9-1 xlO 15 + 7-75 5-4 xlO 14 + 11-02 3-4 182 C.G.S. UNITS AND CONSTANTS. [CHAP. The values in the original are given in megohms, and wo have assumed the megohm = 10 15 C.G.S units. According to F. Kohlrausch (Wied. Ann., xxiv. p. 48, 1885) the resistance at 18 C. of water purified by distillation in vacuo is 4 x 10 10 times that of mercury. This makes its specific resistance 3-76 x 10 15 . 240. The specific resistance of glass of various kinds at various temperatures has been determined by Mr. Thomas Gray (Proc. Roy. Soc., Jan. 12, 1882). The following are specimens of the results : Bohemian Glass Tubing, density 2 '43. At 60 6-05 xlO 22 At 160 2'4 x 10 19 100 2 xlO 21 174 87 x 10 18 130 2 x 10 20 Thomson's Electrometer Jar (flint glass), density 3 '172. At 100 2-06 x 10 23 At 160 2'45 x 10 21 120 4-68 xlO 2 " 180 5'6 x 10 20 140 1-06 ,, 200 1-2 The following are all at 60 C. : Bohemian Beaker, 4 '25 x 10~ density 2'427 ,, 7-15 ,, 2-587 Florence Flask, 4'69xl0 20 2 "523 Test Tube, 1'44 2'435 3-50 2-44 Flint Glass Tube, 3 '89 x 10 22 2 753 Thomson's Electro- "\ meter Jar (flint [ I'02xl0 24 3'172 glass), J 241. The following approximate values of the specific resistance of insulators after several minutes' electrifi- cation are given in a paper by Professors Ayrton and XI.] ELECTRICITY. 183 Perry (Proc. Royal Society, March 21, 1878), " On the Viscosity of Dielectrics " : Specific Temperature. Resistance. Centigrade. Mica, 8'4xl0 22 20 Gutta-Percha, 4 '5 x 10 23 24 Shellac, 9'OxlO 24 28 Hooper's Material, To x 10 25 24 Ebonite, 2-8 xlO 25 46 Paraffin, 3'4xl0 25 46 p, /Not yet measured with accuracy, but greater ' ( than any of the above. Air, Practically infinite. 242. Particulars of Board of Trade Standard Gauge of Wires (Imperial Gauge) Nos. 4 to 20. Ayrton and Perry. /Standard adopted by \ Latimer Clark. Ayrton and Perry. Recent cable tests. Ayrton and Perry. No. Diameter. Sectional area. Sq. inches. Resistance in ohms of 1 metre length pure copper at C. Milli- metres. Thou- sandths of inch. Annealed. Hard-drawn. 4 5-89 232 04227 0005929 0006065 5 5-38 212 03530 7107 7269 6 4-88 192 02895 8638 8835 7 4-47 176 02433 001029 001053 8 4-06 160 02011 1248 1276 9 3-66 144 01629 1536 1571 10 3-25 128 01287 1948 1992 11 2-95 116 01057 2364 2418 12 2-64 104 008494 2951 3019 13 2-34 92 006647 3757 3842 14 2-03 80 005026 4992 5106 15 1-83 72 004070 6142 6283 16 1-63 64 003216 7742 7919 17 1-42 56 002463 01020 01043 18 T22 48 001809 01382 01414 19 1-016 40 001256 01993 02038 20 0-914 36 000917 02462 02518 184 C.G.S. UNITS AND CONSTANTS. [CHAP. The heat generated per second in 1 metre length of (P\ 2 j gm. deg., and at 40 C. is -0055 U- j gm. deg., C denoting the current in amperes, and D the diameter in millimetres. 243. Resistance of I metre length of Wires of Imperial Gauge at C. (For copper see preceding table.) No. Iron annealed. German Silver, either annealed or hard-drawn. Platinum annealed. Silver, annealed. 4 003606 007768 003361 0005583 5 4322 9311 4028 6692 6 5253 01132 4896 8184 \ 6261 01349 5836 9694 8 7590 01635 7074 001175 9 9339 02012 8705 1446 10 01184 02552 01104 1834 11 01438 03096 01340 2226 12 01795 03867 01673 2779 13 02285 04922 02129 3538 14 03036 06540 02829 4700 15 03736 '08047 03482 5784 16 04708 1014 04388 7290 17 06204 1336 05782 9606 18 08405 1811 07834 01301 19 1212 2611 1130 01876 20 1498 3226 1396 02319 Electromotive Force. 244. The electromotive force of a Daniell's cell was found by Sir W. Thomson (p. 246 of Papers on Electricity and Magnetism ) to be 00374 electrostatic unit, XL] ELECTRICITY. 185 from observation of the attraction between two parallel discs connected with the opposite poles of a Daniell's battery. As 1 electrostatic unit is 3 x 10 10 electromag- netic units, this is -00374 x 3 x 10 10 = M22 x 1C 8 electro- magnetic units, or 1-122 volt. According to Latimer Clark's experimental determina- tions communicated to the Society of Telegraph Engineers in January, 1873, the electromotive force of a Daniell's cell with pure metals and saturated solutions, at 64 F., is 1*105 volt., and the electromotive force of a Grove's cell 1'97 volt. These must be diminished by 1 per cent, because they were deduced from the assumption that the B.A. unit of resistance was correct. They will thus be reduced to 1*094 and 1*9.5 volts. 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 xlO 8 , and that of a Grove's cell 1-942 x 10 8 . These must be diminished by 3 per cent., because they were deduced from the value -9717 x 10 9 for Siemens' unit which is 3 per cent, too great. They will thus be reduced to 1-104 and 1-884 volts. H. S. Carhart (Amer. Jour. Sci. Art., Nov. 1884) found the following different values -for the electro-motive force of a Daniell's cell according to the strength of the zinc sulphate solution : Electromotive force . 1-118 1-115 1-111 1-111 Per Cent, of ZnS0 4 . Electromotive force in volts. Per Cent. 1 1-125 10 3 1-133 15 5 1-142 20 74 1-120 25 186 C.G.S. UNITS AND CONSTANTS. [CHAP. He finds by the same method the electromotive force of Latimer Clark's standard cell to be 1'434 volt. Lord Eayleigh (Phil. Trans., June, 1884, p. 452) determined the electromotive force of a Clark cell at 15C. to be 1-435 volt. In a supplementary paper (Jan. 21, 1886) he gave the general result for any temperature f, 1-435(1 -0-00077^ -15)}, together with full particulars as to the precautions neces- sary for securing constancy. 245. 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 inductivety in air. The method of experiment- ing is described in the Proceedings of the Eoyal Society for March 21, 1878. The following abstract of their latest results was specially prepared for this work by Professor Ayrton in January, 1879 : XI.] ELECTRICITY. 187 jjnoq-B '0 O gl ' SUTV lauii iodxa }?,.! jo aratq. aq^ ^TJ aan; eaad uia.T oS^jaAy S -5 ^ 5 sstug; * (M $ (N O5 C^ r I** 0* co cp cc o UK i ! 1 1 1 1 * # * O Oi TJ- S ^ ^ ^a^-g .mrfp.nv Tj< l^-l O 00 lip | o o 00 O5 i o 3 S Jfll 11 i f || * | lO CO CO O5 o --( C^ (M os Ci 2 -*-* rH fe 'UIJQ t- ^ CO p co O CJ TJH CO +3 (D gj +=> J- ' ! I ||^| ^ CO 00 O i i co co p > s a s sf .? g^3 g^^'G umin:i'ej x j yH C* CO 5* 9 s T 1 9" 1 a* a |E3 1 1 i , 7 i ^ 1 o 53 tliill ll^lf P^J 00 o 5 OS O l^ * 1 I 's |j2: | 10 CO * ( Oi * * * I? a-2 J*S oo ^ o UOJJ CO co co i^ o -2'5^_ ^ 1 1 i l T5 ^ 0*0^ aaddo ^^ CD 1 1 CD O <* I^> iQ *O O^ 00 Tj< t OO O | P || | 1 III! ' c J ^ 3 a o to t^* oo uoqa'BQ ^ "^ I I 1 * co 1 1 1 ?! ! ? 9ii 103 1'087 at 16'6 C., j Copper sulphate, satu- ) ratedat 15 C., J 070 % Sea salt, specific ) gravity, 1'18 at } -475 605 -267 -856 -334 ^H 20'5C., J O Sal-ammoniac, satu- ) rated at 15 -5 C.,... f -396 -652 -189 057 -364 OQ Zinc sulphate solu-] tion, specific grav- V ity,]-125atl6-9C.,j Zinc Sulphate, satu- \ rated at 15*-3C.,... ) 1 Distilledwater mixed | with 3 zinc sulphate, saturated solution, J O f * ( 20 Distilled water, ] 5 &( 1 strong sulphuric r S 11 acid J I, * 10 Distilled water, " } 1 strong sulphuric abou -035 acid, , CQ o "^ S / 5 Distilled water, 1 < o.'Si I 1 strong sulphuric g M acid, J . ^ 1 Distilled water, "| 5 strong sulphuric 3 to '01 -120 -256 3 11 acid, J Sulphuric Acid, - 85to-5 depeiu in.- o 1-113 720 1 1-252 1-600 to CONCEN- 1 carbon 1*300 TRATED.1 Nitric Acid 672 Mercurous sulphate paste Distilled water, with a trace of sulphuric acid, XL] ELECTRICITY. 189 Solids with Liquids and Liquids with Liquids in Air. 1 N Amalgamated Zinc. , Mercury. Distilled Water. AlumSolution , satu- rated at 16' '5 C. If. 3 S a^ .2 ^ PS Igg Zinc Sulpnate solu- tion, Specific Grav- ity 1-125 at 16-9C. /line euipnaie ooiu- tion, saturated at 15-3 C. 1 Distilled Water, 3 Zinc Sulphate. Strong Nitric Acid. -105 to 100 231 -043 164 + 156 -536 -014 090 -043 095 102 -565 -435 -637 -348 -238 -430 -"284 -200 -095 -444 -102 -344 -358 -429 016 848 1-298 1-456 1-269 1-699 475 -24 078 190 C.G.S. UNITS AND 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 poten- tial to consider, and in a Grove's cell five, viz.: Daniell's Cell Volts. 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, +0*750 1-155 Grove's 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 Thermoelectricity. 246. The electromotive force of a thermoelectric circuit is called Thermoelectric force. It is proportional ccet. par. to the number of couples. The thermoelectric force of a single couple is in the majority of cases equal to the product of two factors, one being the difference of temperature of the two junctions, and the other the difference of the thermoelectric heights of the two metals at a temperature midway between those of the junctions. The current through the hot junction is from the lower XL] ELECTRICITY. 191 to the higher metal when their heights are measured at the mean temperature. Our convention as to sign (that is, as to up and down in speaking of thermoelectric height) is the same as that adopted by Prof. Tait, and is opposite to that adopted in the first edition of this work. We have adopted it because it leads to the rule (for the Peltier and Thomson effects) that a current running down generates heat, and a current running up consumes heat. The following table of thermoelectric heights relative to lead can be employed when the mean temperature of the two junctions does not differ much from 19 or 20 C. It is taken from Jenkin's Electricity and Magnetism, p. 176, where it is described as being compiled from Matthiessen's experiments. We have reversed the signs to suit the above convention, and have multiplied by 100 to reduce from microvolts to C.G.S. units. Thermoelectric Heights at about 20 (J. Bismuth, pressed com- ~\ _ o-nn Antimony, pressed wire + 280 mercial wire, / ' Silver, pure hard, -f 300 Bismuth, pure pressed ) conn i Zinc, pure pressed, + 370 wire, / Copper, galvano-plas-^ , qfift Bismuth, crystal, axial, - 6500 tically precipitated, J + ,, equatorial,.... -4500 ! Antimony, pressed! Cobalt, -2200 commercial wire, .../ + German Silver, -1175 Arsenic, -f 1356 Quicksilver, - 41'8| Iron, pianoforte wire,.. + 1750 Lead, Antimony, axial, + 2260 Tin, + 10 i ,, equatorial, + 2640 Copper of Commerce, . . . + 10 Phosphorus, red, + 2970 Platinum, + 90 Tellurium +50200 Gold, + 120 Selenium, +80700 247. The following table is based upon Professor Tait's thermoelectric diagram (Trans. Roy. Soc., Edin., vol. 192 C.G.S, UNITS AND CONSTANTS. [CHAP, xxvii. 1873) joined with the assumption that a Grove's cell has electromotive force T97 x 10 s : Thermoelectric Heights at t C. in C.G.S. units. Iron, + 1734- 4'87 t Steel, + 1139- 3'28 t Alloy, believed to be Platinum Iridium, + 839 at all temperatures. Alloy, Platinum 95 ; Iridium 5. + 622 - -55 t 90; 10, + 596- 1'34 85; 15, + 709- '63 1 ,, ,, 85; ,, 15, + 577 at all temperatures. Soft Platinum, - 61- I'lQt Alloy, platinum and nickel, + 544- 1'10 Hard Platinum, + 260- '75 Magnesium, + 244- '95 German Silver, -1207- 5'12t Cadmium, + 266+ 4'29 Zinc, + 234+ 2'40 t Silver, + 214+ 1-50 1 Gold, + 283+ 1-02 1 Copper, :...+ 136+ -95t Lead, Tin, - 43+ '55t Aluminium, - 77+ '39 Palladium, - 625- 3'59 Nickel to 175" C., -2204- 5'12 250 to 310 C., -8449 + 24-1 1 ,, from 340 C., - 307- 5'12< The lower limit of temperature for the table is - 18 C. 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. Ex. 1. Required the electromotive force of a copper- iron couple, the temperatures of the junctions being C. and 100 C. XL] ELECTRICITY. 193 We have, for iron, + 1734 - 4-87 t ; copper, + 136+ '95)5; ,, iron above copper, 1598-5-82 t. The electromotive force per degree is 1598-5-82x50=1307, and the electromotive force of the couple is 1307 (100-0) --130,700, tending from copper to iron through the hot junction. By the neutral point of two metals is meant the tem- perature at which their thermoelectric heights are equal. Ex. 2. To find the neutral point of copper and iron we have 1598-5-82* = 0, t = 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. Ex. 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 10 5 millimetre-milligramme-second units for 1 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 10~ 10~^ = 10~ s , giving 2400 as the electromotive force per degree of difference. From the above table we have Iron above German silver, 2941 + -25 t, 194 C.G.S. UNITS AND CONSTANTS. [CHAP. which, for t 100, gives 2966 as the electromotive force per degree of difference. Peltier and Thomson Effects. 248. When a current is sent through a circuit com- posed of different metals, it produces in general three distinct thermal effects. 1. A generation of heat to the amount per second of C 2 R ergs, C denoting the current, and E, the resistance. 2. A generation of heat or cold at the junctions. This is called the Peltier effect, and its amount per second in ergs at any one junction can be computed by multiplying the difference of thermoelectric heights at this junction by t + 273 and by the current, t denoting the centigrade temperature of the junction. If the current flows down (that is from greater to less thermoelectric height) the effect is a warming; if it flows up, the effect is a cooling. Ex. 4. Let a unit current (or a current of 10 amperes) flow through a junction of copper and iron at 100 C. The thermoelectric heights at 100 C. are Iron, 1247 Copper, 231 Iron above copper, 1016 Multiplying 1016 by 373, we have about 379,000 ergs, or of a gramme-degree, as the Peltier effect per second. Heat of this amount will be generated if the current is from iron to copper, and will be destroyed if the current is from copper to iron. 3. A generation of heat or cold in portions of the cir- cuit consisting of a single metal in which the temperature XL] ELECTRICITY. 195 varies from point to point. This is called the Thomson effect. Its amount per second, for any such portion of the circuit, is the difference of the thermoelectric heights of the two ends of the portion, multiplied by 273 + , where t denotes the half sum of the centigrade tempera- tures of the ends, and by the strength of the current. The Thomson effect, like the Peltier effect, is reversed by reversing the current, and follows the same rule that heat is generated when the current is from greater to less thermoelectric height. Experiment shows that the Thomson effect is insensible in the case of lead ; hence the thermoelectric height of lead must be sensibly the same at all temperatures. It is for this reason that lead is adopted, by common con- sent, as the zero from which thermoelectric heights are to be reckoned. Ex. 5. In an iron wire with ends at C. and 100 C., the cold end is the higher (thermoelectrically) by 4'87 x 100 that is, by 487. Multiplying this differ- ence by 273 + 1(0 + 100) or 323, we have 157300 as the Thomson effect per second for unit current. This amount of heat (in ergs) is generated in the iron when the current through it is from the cold to the hot end, and is destroyed when the current is from hot to cold. Ex. 6. In a copper wire with ends at C. and 100 C., the hot end is the higher by -95 x 100 or 95. Multiply- ing this by 323, we have 30700 (ergs) as the Thomson effect per second per unit current. This amount of heat is generated in the copper when the current through it is from hot to cold, and destroyed when the current is from cold to hot. 196 C.G.S. UNITS AND CONSTANTS. [CHAP. , The effect of a current from hot to cold is opposite in these two metals, because the coefficients of t in the expressions for their thermoelectric heights ( 247) have opposite signs. Relation between Thermoelectric Force and the Peltier and Thomson Effects. 249. The algebraic sum of the Peltier and Thomson effects (expressed in ergs) due to unit current for one second in a closed metallic circuit, is equal to the thermo- electric force of the circuit ; and the direction of this thermoelectric force is the direction of a current round the circuit which would give an excess of destruction over generation of heat (so far as these two effects are concerned). Ex. 7. In a copper-iron couple with junctions at C. and 100 C., suppose a unit current to circulate in such a direction as to pass from copper to iron through the hot junction, and from iron to copper through the cold junction. The Peltier effect at the hot junction is a destruction of heat to the amount 1016 x 373 = 379,000 ergs. The Peltier effect at the cold junction is a generation of heat to the amount 1598 x 273 = 436,300 ergs. The Thomson effect in the iron is a destruction of heat to the amount 487 x 323 = 157,300 ergs. The Thomson effect in the copper is a destruction of heat to the amount 95 x 323 = 30,700 ergs. The total amount of destruction is 567,000, and of generation 436,300, giving upon the whole a destruction of 130,700 ergs. The electromotive force of the couple XL] ELECTRICITY. 197 is therefore 130,700, and tends in the direction of the current here supposed. This agrees with the calculation in Example 1. Electrochemical Equivalents. 250. The quantity of a given metal deposited in an electrolytic cell or dissolved in a battery cell (when there is no " local action ") depends on the quantity of elec- tricity that passes, irrespective of the time occupied. Hence we can speak definitely of the quantity of the metal that is " equivalent to " a given quantity of elec- tricity. By the electrochemical equivalent of a metal is meant the quantity of it that is equivalent to the unit quantity of electricity. In the C.G.S. system it is the number of grammes of the metal that are equivalent to the C.G.S. electromagnetic unit of electricity. Special attention has been paid to the electrochemical equivalent of silver, as this metal affords special facilities for an accurate determination. The latest experiments of Lord Rayleigh and Kohlrausch agree in giving 01118 as the C.G.S. electrochemical equivalent of silver.* The number of grammes of silver deposited by 1 ampere in one hour is 01118 x T Vx 3600 = 4-025. 251. The electrochemical equivalents of the most im- portant of the elements are given in the following table. They are calculated from the chemical equivalents in the preceding column by simple proportion, taking as basis the above-named value for silver. Their reciprocals are *Rayleigh's determination is '01 11794; Kohlrausch's. '011183; Mascart's, -011156. See Phil. Trans., 1884, pp. 439, 458. 198 C.G.S. UNITS AND CONSTANTS. [CHAP. the quantities of electricity required for depositing one gramme. The quantity of electricity required for de- positing the number of grammes stated in the column " chemical equivalents " is the same for all the elements, namely, 9634 C.G.S. units. Elements. Atomic Weight. i I i i i 3 1 2 1 2 1 4 2 3 2 2 2 2 3 2 1 1 1 3 Chemi- cal Equiva- lents. Electro- chemical equivalents or grammes per unit of electricity. Recipro- cal or Electri- city per gramme. Electro-positive Hydrogen, 1 39-03 23-00 196-2 107-7 63-18 199-8 117-4 55-88 58-6 64-88 206-4 27-04 15-96 35-37 126-54 79-76 14-01 1 - 39-03 23-00 65-4 107-7 31-59 63-18 99-9 199-8 29-35 587 18-63 27-94 29-3 32-44 103-2 9-01 7'98 35-37 126-54 79-76 4-67 0001038 004051 002387 006789 01118 003279 006558 01037 02074 003046 006093 001934 002900 003042 003367 01071 000935 OC08283 003671 013134 008279 0004847 9634 246-9 418-9 147-3 89-45 305-0 152-5 96-43 48-22 328-3 164-1 517-1 344-8 328-7 297-0 93-37 1070 1207 272-4 76-14 120-8 2063 Potassium, Sodium Gold, Silver Copper (cupric), . . ,, (cuprous) Mercury (mercuric), ... ,, (mercurous), Tin (stannic), ,, (stannous), Iron (ferric) ,, (ferrous), Nickel, Zinc, Lead Aluminium, Electro-negative Oxvsen Chlorine, Iodine Bromine Nitrogen . . . . To find the equivalent of 1 coulomb, divide the above electrochemical equivalents by 10. To find the number of grammes deposited per hour by xi. ELECTRICITY. 199 1 ampere, multiply the above electrochemical equivalents by 360. 252. Let the " chemical equivalents " in the above table be taken as so many grammes : then, if we denote by H the amount of heat due to the whole chemical action which takes place in a battery cell during the consumption of one equivalent of zinc, the chemical energy which runs down, namely JH ergs, must be equal (if there is no wasteful local action) to the energy of the current pro- duced. But this is the product of the quantity of electricity 9634 by the electromotive force of the cell. TTT The electromotive force is therefore equal to 9634 In the tables of heats of combination which are in use among chemists, the equivalent of hydrogen is taken as 2 grammes, and that of zinc as 64-88 or 65 grammes. The equivalent quantity of electricity will accordingly be 9634 x 2, and the formula to be used for calculating the JH electromotive force of a cell will be . In applying this calculation to DanielPs and Grove's cells, we shall employ the following heats of combination, which are given on page 614 of Watts' Dictionary of Chemistry, vol. vii., and are based on Julius Thomson's observations : Zn, 0, SO 3 , Aq., 108,462 Cu, 0, SO 3 , Aq., 54,225 N 2 2 , O 3 , Aq., 72,940 N 2 2 , 0, Aq., 36,340 In Darnell's cell, zinc is dissolved and copper is set free ; we have, accordingly, H = 108,462 - 54,225 = 54,237. 200 C.G.S. UNITS AND CONSTANTS. [CHAP. In Grove's cell, zinc is dissolved and nitric acid is changed into nitrous acid. The thermal value of this latter change can be computed from the third and fourth data in the above list, as follows : 72,940 is the thermal value of the action in which, by the oxidation of one equivalent of N 2 2 and combination with water, two equivalents of NHO 3 (nitric acid) are produced. 36,340 is the thermal value of the action in which, by the oxidation of one equivalent of N 2 2 and combination with water, two equivalents of NHO 2 (nitrous acid) are produced. The difference 36,600 is accordingly the thermal value of the conversion of two equivalents of nitrous into nitric acid, and 18,300 is the value for the conversion of one equivalent. In the present case the reverse changes take place. We have, therefore, H = 108,462 - 18,300 = 90,162. TTT Taking J as 4-2 x 10 7 , the value of- -.. will be lyzbo ] -182 x 10 s for Daniell's cell. 1-965 x 10 s Grove's These are greater by from 2 to 8 per cent, than the direct determinations given in 244. Examples in Electricity. 1. Two conducting spheres, each of 1 centim. radius, are placed at a distance of r centims. from centre to centre, r being a large number j and each of them is 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 uni- formly distributed over their surfaces, and the force will XL] ELECTRICITY. 201 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 1 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 r 2 dyne, r being the distance between their centres in cen- timsJ (See 244.) One sphere must be charged to potential 1 and the other to potential - 1. The number of cells required is 2 =535. 00374 3. How many Daniell's cells would be required to pro- duce 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 226, 244.) 4. Compare the capacity denoted by 1 farad with the capacity of the earth. The capacity of the earth in static measure is equal to its radius, namely 6-37 x 10 s . Dividing by v 2 to reduce to magnetic measure, we have *71 x 10~ 12 , which is 1 farad multiplied by '71 x 10~ 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 20'2 C.G.S. UNITS AND CONSTANTS. [CHAP. plate of zinc, A square centims. in area, and a plate of copper of the same dimensions, separated by an acid solution of specific resistance 10 9 , the distance between the plates being 1 centim. Ans. . , or - - of an ohm. A A 6. Find the heat developed in 10 minutes by the passage of a current from 10 Daniell's cells in series through a wire of resistance of 10 10 (that is, 10 ohms), assuming the electromotive force of each cell to be 1-1 x 10 8 , and the resistance of each cell to be 10 9 . Here we have Total electromotive force = 1*1 x lO-\ Resistance in batter}^ =10 1() . Resistance in wire =10 ]0 . Current = ^ x 1 9 9 = -55 x 10" 1 = -055. Heat developed in ) = (-Q55) 2 x IQio = wire per second j 4-2 x 10^ 'Hence the heat developed in 10 minutes is 4321-4 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 line of the ordinary gauge [4 feet 8J inches] due to cutting the lines of force of terrestrial magnetism, the vertical intensity being '438. The electromotive force will be the product of the velocity of travelling, the distance between the rails, and the vertical intensity, that is, (44-7 x 30) (2-54 x 56-5) (-438) = 84,300 electromagnetic units. This is about TO^ of a volt. XL] ELECTRICITY. 203 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, ^and self-induction being left out of account. 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 ^H?ra 2 sin 0, where H = -1794, rt = 15, and w = 300. The electromotive force at any instant is the rate at which this quantity increases or diminishes ; that is, nHira? cos . w, if w denote the angular velocity. At the instant of passing the meridian, cos 6 is 1, and the electromotive force is MH7ra 2 a>. With 10 revolutions per second the value of w is 27r x 10. Hence the electromotive force is 1794 x (3-142) 2 x 225 x 20 x 300 = 2-39 x 10 (; . This is about of a volt. 42 Electrodynamics. 253. Ampere's formula for the repulsion between two elements of currents, when expressed in electromagnetic units, is cc'ds.ds' 2 (2 sin a sin a cos v - cos a cos a ), i>04 C.G.S. UNITS AND CONSTANTS. [CHAP. 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 ; the angle between the plane of r, ds, and the plane of r, ds'. For two parallel currents, one of which is of infinite length, and the other of length / 5 the formula gives, by integration, an attraction or repulsion 21 D cc ' where D denotes the perpendicular distance between the currents. Example, Find the attraction between two parallel wires a metre long and a centiui. apart when a current of -^ is passing through each. Here the attraction will be sensibly the same as if one of the wires were indefinitely increased in length, and will be 20( Y!Y _ i uo/ 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. Coefficients of Self-induction and of Mutual Induction. 254. When a steady current C flows through a coil of resistance R, the difference of potential between the ends of the coil is OR. When the current, instead of being XL] .ELECTRICITY. 205 steady, is increasing at the rate dG/dt, the difference of potentials between the ends is CR + L-J , L being a constant called the self-induction (or coefficient of self-induction) of the coil. This is on the supposition that no current except that in the coil itself exerts an inductive influence on the coil. LdC/cU may be described as the reverse e. m. f. due to the fact that the current is increasing. When two neighbouring coils influence each other, the reverse e. m. f. in the first coil is and the reverse e. m. f. in the second coil is LJ being the self-induction of the first coil, L 2 that of the second, and M the mutual induction of the two coils. Since the two terms of the expression CR + L dC/dt must be of the same dimensions, L has the dimensions lit. In the electromagnetic system, R is a velocity, hence L has the dimensions of length. In C.G.S. units, R is in centims. per second, and L and M are in centims. In the "practical" system, R is in ohms, t is in seconds, and the unit for L and M has often been called the secohm, a name compounded of ohm and second. The official name recently adopted by an International Con- gress at Paris, and by the Electrical Standards Committee of. the British Association, is the quadrant, an abbre- viation of the name " earth-quadrant," the ohm being a velocity of an earth-quadrant per second. The accurate value is 10 centims. 206 C.G.S. UNITS AND CONSTANTS. [CHAP. 255. The same authorities have introduced some new names in connection with alternating currents. Let E denote the difference of potential between the two ends of the armature coil, and let its value in terms of the time t be E = E sinW, E and to being constants. If R denote the resistance of the armature, the current would be if there were no self-induction. Let L be the self- induction, then the value of C is J3 being a constant such that tan (3 = - -~ . The quantit + L 2 w 2 ) is called the impedance (the accent to be on the second syllable). This name is adopted by the Standards Committee in preference to resistance apparente which was recommended by the Paris Congress. Both authorities agree in giving the name effective current to the quantity whose square is equal to the mean value of the square of C. It must be distinguished from the mean current, which, with the above value of C., would be zero. 256. To investigate the magnitudes of units of length, 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. XT.] P:LECTRICITY. 207 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 (86), acceleration = C .,---., where C = 6-48 x 10~ 8 ; (distance)-* and in the new system we are to have i ' , . mass acceleration = -_, - (distance) 2 Hence, by division, acceleration in C.G.S. units acceleration in new units _ Q mass in C.G.S. units (distance in new units) 2 mass in new units ' (distance in C.G.S. units) 2 ' that is,^ = Cg. This equation expresses the first of the three conditions. The equation _ = c expresses the second, v denoting 3 x 10 10 . The equation M = L 3 expresses the third. Substituting L 3 for M in the first equation, we find T = ^ . Hence, from the second equation, \C and from the third, 208 C.G.S. UNITS AND CONSTANTS. [CHAP. Introducing the actual values of C and v, we have approximately T - 3928, L = 1-178 x 10 14 , M - 1-63 x 10 42 ; that is to say, The new unit of time will be about l h 5J IU ; The new unit of length will be about 118 thousand earth quadrants ; The new unit of mass will be about 2-66 x 10 14 times the earth's mass. Modern Views on Electrical and Magnetic Dimensions. 257. Maxwell has pointed out (Elec. and Mag. 622, 2nd edition) certain relations which must exist between the dimensions of various electrical and magnetic quan- tities in any consistent system, and has shown ( 623) that the dimensions of any electrical or magnetic quantity can be definitely expressed (1) In terms of mass, length, time, and quantity of electricity ; (2) In terms of mass, length, time, and quantity of magnetism ; the dimensions thus obtained being the same for all systems. Riicker (Phil. Mag., Feb. 1889) has pointed out, as an algebraic deduction from equations given by Maxwell, that the dimensions of all electrical and magnetic quan- tities can be definitely expressed (1) In terms of mass, length, time, and specific in- ductive capacity ; (2) In terms of mass, length, time, and magnetic permeability ; XL] ELECTRICITY. 209 and maintains, in common with several of the leading exponents of Maxwell's views, that specific inductive capacity and magnetic permeability, which are usually regarded as mere numerical quantities, ought to be regarded as quantities of unknown dimensions. The numerical ratios usually understood by these terms must then be regarded not as absolute but merely as relative values, all of which are to be multiplied by the values for vacuum, which are at present unknown, and which depend, in some way not at present known, on the units of length, mass, and time employed. The following are examples of the dimensions thus obtained, K denoting specific inductive capacity, and p magnetic permeability. In terms ef K. In terms of /*. J Quantity of electricity, ..... M*L*T"*K* M*L*>"* | Strength of pole, .............. M*L*K'* M*L*T'V* Electric force at a point, ... M^L'^T^K'* *~* Magnetic ,, ... J Electric potential, ............ \Magnetic ............ Intensity of magnetization, M^L'^K"^ Electrical resistance, ........ L " 1 TK " ' LT " * fi ,, capacity, .......... LK L *T%" a 258. In every case the ratio of the dimensions in terms of K to the dimensions in terms of //. is a power of If electrical and magnetic actions are to be regarded as manifestations of the ordinary laws of d 210 C.G.S. UNITS AND CONSTANTS. [CHAP. xi. to motions which our present knowledge does not enable us to specify, then every electrical or magnetic quantity has definite dimensions in terms of mass, length, and time ; hence the expression K^/x^LT" 1 must be of dimen- sions M LT , in other words K^/z* must be the reciprocal of a velocity. 211 APPENDIX. First Report of the Committee for the Selection and Nomenclature of Dynamical and Electrical Units, the Committee consisting of SIR W. THOMSON, F.K.S., PROFESSOR G. C. FOSTER, F.R.S. PROFESSOR J. C. MAXWELL, F.E.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). [1873.] 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 recommendation. 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 called "absolute" measure, to mention the three fundamental 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 funda- mental 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. * Mr. Stoney objected to the selection of the centimetre as the unit of length. 212 C.G.S. UNITS AND CONSTANTS. 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 electromag- netic elements must be fixed at one common level that level, namely, which is determined 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 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 Centi- metre, 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 magnetic 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. APPENDIX. 213 unit of . . . ." Several lists of names have already been suggested ; and attentive consideration will be given to any further suggestions which we may receive from persons interested in electrical nomenclature. The "ohm," as represented by the original standard coil, is approximately 10 9 C.G.S. units of resistance; the "volt" is approximately 10 8 C.G.S. units of electromotive force ; and the " farad " is approximately - 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. Stoney, a system which has already been extensively employed for electrical pur- poses. It consists in denoting the exponent of the power of 10, which serves as a multiplier, by an appended cardinal number, if the exponent be positive, and by a prefixed ordinal number if the exponent be negative. Thus 10 9 grammes constitute a gramme-nine;-^ of a gramme constitutes a ninth-yramme : 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, hecto, deca, deci, 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 5iW/us. The form dynamy appears to be the most satisfactory 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 Before a vowel, either meg or megal, as euphony may suggest, may be employed instead of mega. 214 C.G.S. UNITS AND CONSTANTS. system with a view to its becoming international, we think that the termination of the word should for the 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 deriva- tive of the Greek fyyov. 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 statement will be useful : The weight of a gramme, at any part of the earth's surface, is about 980 dynes, or rather less than a kilodyne. The iveight of a kilogramme is rather less than a megadyne, being about 980,000 dynes. Conversely, the dyne is about 1-02 times the weight of a milli- gramme at any part of the earth's surface ; and the megadyne is about 1'02 times the weight of a kilogramme. 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. APPENDIX. 215 One horse-power is about three-quarters of an erg-ten per second. More nearly, it is 7*46 erg -nines per second, and oneforce-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 Nomenclative of Dynamical and Electrical Units, the Committee consisting of PROFESSOR SIR W. THOMSON, F.E.S., PROFESSOR G. C. FOSTER, F.K.S., PROFESSOR J. CLERK MAXWELL, F.K.S., G. J. STONEY, F.E.S., PROFESSOR FLEEMING JENKIN, F.E.S., DR. C. W. SIEMENS, F.E.S., F. J. BRAMWELL, F.E.S., PROFESSOR W. G. ADAMS, F.E.S., PROFESSOR BALFOUR STEWART, F.B.S., and PROFESSOR EVERETT (Secretary). [1874.] THE Committee on the Nomenclature of Dynamical and Electrical Units have circulated numerous copies of their last year's Eeport among scientific men both at home and abroad. They believe, however, that, in order to render their recom- mendations 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 illustra- tions here referred to.] 216 O.G.S. UNITS AND CONSTANTS. Resolution* adopted by the International Congress of Electrician* at Paris at the Hitting of September 22nd, 1881. 1. For electrical measurements, the fundamental units, the centimetre (for length), the gramme (for mass), and the second (for time), are adopted. 2. The ohm and the volt (for practical measures of resistance and electromotive force or potential) are to keep their existing definitions, 10 9 for the ohm, and 10 8 for the volt. 3. The ohm is to be represented by a column of mercury of a square millimetre section at the temperature of zero centigrade. 4. An International Commission is to be appointed to deter- mine, for practical purposes, by fresh experiments, the length of a column of mercury of a square millimetre section which is to represent the ohm. 5. The current produced by a volt through an ohm is to be called an ampere. 6. The quantity of electricity given by an ampere in a second is to be called a coulomb. 7. The capacity defined by the condition that a coulomb charges it to the potential of a volt is to be called a farad. 217 INDEX. The numbers refer to the pages. Absolute scale of temperature, 103. Acceleration, 4, 21. Adiabatic compression, 138- 140. Air, collected data for, 143. , conductivity of, 123, 131. , density of, 40. , expansion of, 114. , specific heat of, 109-110. thermometer, 102. Alternating currents, 206. Ampere as unit, 171. Ampere's formula, 203. Aqueous vapour, 115-117. Arc, electric, resistance of, 181. Astronomy, 70-75. Atmosphere, standard, 38-39. , its density aloft, 45. Atomic weights, 198. Attraction, constant of, 72. at a point, 13. B. A. unit, 175. Barometer, 38. , capillarity in, 49. , heights by, 45. Batteries, 184-186, 190, 199. Boiling points, 113. of water, 115-116. Boyle's law, departures from, 41. Bullet melted by impact, 27. Calorie, 104. Candle, standard, 94. Capacity, electrical, 162, 164. , spec, inductive, 107-170. , thermal, 98, 105. Capillarity, 47-49. Carcel, 94. Cells, e.m.f. of, 184-186, 190. , heat of combination of, 199. Centimetre, why selected, 19, 212. Centre of attraction, 13. Centrifugal force, 28-30. C.G.S. system, 19, 212. Change-ratio, 5. Clark's standard cell, 186. Cobalt, magnetisation of, 149- 151. Coil, revolving, 203. Common scale needed, 18. Comparison of standards, xiii. Compressibility of liquids, 61- 65. solids, 58-65. , adiabatic, 138-140. Conductivity, thermal, 118-131. of air, 123, 131. of gases, 131. of liquids, 130-131. of solids, 123-130. Congress of electricians, 216. 218 C.G.S. UNITS AND CONSTANTS. Contact electricity, 186-190. Cooling, 131-135. Critical points of vapours, 118. Crystals, refraction in, 89. Current, unit of, 162, 163, 171. Curvature, 13-14. Daniell cell, 184-185, 190, 199. Day, sidereal, 71. Decimal multiples, 20, 213. Declination, magnetic, 158-160. Densities, table of, 36-37. of gases, 41. of water, 34-36. Density as a unit, 167. Derived units, 1-2. Dew point, 117. Diamagnetic substances, 147. Diamond, specific heat of, 106. Diffusion, 120-122. Diffusivity, thermal, 120. Dimensional equations, 5. Dimensions, 3-6. , deductions from, 30. in terms of K and fj., 208- 210. Dip, 158-160. Dispersive power, 92. Diversity of scales, 18. " Division " extended, 6. Double refraction, 89. Dynamical quantities, 11-13. Dyne, 23, 213. Earth as a magnet, 156-158. , size of, etc., 70. " Effective " current, 206. Elasticity, 52-65. , coefficients of, 52-53. , adiabatic, 140. Electric units, practical, 171. Electricity, 161-210. Electrochemistry, 197-198. Electrodynamics, 203-204. Electromagnetics, 163-104. Electromagnets, 153-154. Electromotive force, 184-200. Electromotive force of cells, 184-186. Electrostatics, 161-163. Emission of heat, 131-135. Energy, 12, 25. Equations, dimensional, 5, 30- 33. , physical, 8. Equivalent of heat, 96-102. Equivalents, electrochemic, 197- Erg, 25, 214. Expansion, 141-143. of gases, 114. of mercury, 142. of water, 35. Farad, 171. compared with earth, 201. Field-intensity, 145. Films, tension of, 47. , thickness of, 48. Flux of induction, 147. Foot-pound(al), 26. Force, 11, 13, 23. Freezing-point changed, 137. Frequencies, 85. Fundamental units, 2, 15. reduced to two, 73. (7, value of, 21-23. Gases at high pressure, 42. , conductivity of, 131. , densities of, 41. , expansion of, 114. , induct, capacities of, 170. , refraction of, 9 1 . , sp. heats of, 109-111. , two sp. heats of, 136. Gauss' magnetic units, 157. Geometrical quantities, 11-14. Glaisher's factors, 117. Gramme-degree, 104. Gravitation, 21, 72. measures, 24, 26. Grove's cell, 185, 190, 200. Heat, 96-143. INDEX. 219 Heat, mech. equiv. of, 96-102. of combination, 135, 199. , units of, 96, 104, 105. Heating by compression, 138. by current, 164, 184. and cooling by current, 194. Heights by barometer, 45. Homogeneous atmosphere, 43. Horse-power, 26. Hydrostatics, 34-49. Ice, conductivity of, 129-130. -, density of, 36, 112. , resistance of, 181. , specific heat of, 107. Illuminating power, 94. Impedance, 206. Indices of refraction, 85-92. Inductive capacity, 167-170. Induction, magnetic, 146. Insulators, resistance of, 182, 183. Iron, critical temperature of, 150. , magnetic properties of, 148-153. J, dimensions of, 97. Joule as unit, 27. Joule's determinations, 101-102. equivalent, 96-102. Kilogramme and pound, xiii. Kinetic energy, 25. Large numbers, 20, 213. Latent heat, 111, 113, 114. Latimer Clark's cell, 186. Lengthening by magnetisation, 152. Lifting-power of magnet, 154. Light, 82-95. , velocity of, 82. , wave-lengths of, 83-85. Liquids, conductivity of, 130. , magnetisation of, 151. , refraction of, 90. , resistance of, 179-180. , viscosity of, 67. Magnetic elements, 158-160. force, 146. induction, 146, 147. permeability, 147, 148-152. susceptibility, 147, 148-152. Magnetisation, int. of, 145. Magnetism, 144-160. , terrestrial, 156-160. Magneto-optic rotation, 160. Mass, standards of, 16. Mechanical quantities, 11. equivalent of heat, 96-102. Mega, as prefix, 38, 213. Melting, expansion in, 112. Melting points, 111. Mercury, density of, 37. , expansion of, 142. , stan. resistance, 175-176. , temp, coeff. of, 179. Metre and yard, xiii. Micro as prefix, 213. Microfarad, 171. Modern views on dimens., 208. Moist air, density of, 143. Moment of magnet, 145. Moon, 71-72. " Multiplication" extended, 6. Mutual induction, 204-206. Neutral point, 193. Nickel, magnetis. of, 149-151. Ohm, 171, 175, 176. Paramagnetic, 147. Peltier effect, 194-196. Pendulum, seconds', 21-22. Pen tan e standard, 94. " Per," meaning of, 6. Platinoid, 178. 220 C.G.S. UNITS AND CONSTANTS. Platinum, resistance of, 177. , specific heat of, 106. Pojsson's ratio, 59-60. Potential, electric, 161, 163. , magnetic, 144. Poundal, 24. Powers of ten, 20, 213. Practical units, 171. Pressure, 13. of vapours, 115-118. Pressure-height, 44. Problems on units, 8-11, 74,206. Quantity of electricity, 161, 163. Radian, 12. Radiation, 133. Ratios of electric units, 165. Refraction indices, 85-92. Report of Units Committee, 211. Resilience, 52. Resistance, 174-184. , ternp. coeffs. of, 177-179. Rigidity, simple, 58. Rotating. coil, 203. plane of polarisation, 93, 160. Saturated air, 143. Saturation, magnetic, 149. Self-induction, 204-206. Shear, 53. Shearing stress, 56. Siemens' unit, 175. Soap films, 48. Solenoid, force in, 153. Sound, faintest, 80. , velocity of, 76-79. Spark, 172-174,201. Specific gravities, 36-37. heat, 98, 106-111. v, determinations of, 166. Vapours, critical points, 118. , pressures of, 117. , spec, heats of, 110. , volumes of, 113. Velocity of light, 82-83. of sound, 76-79. Vibrations per sec. of light, 85. Villari critical point, 149. Volt, 171. Volume, change in melting, 112-113. -, elasticity of, 53, 61-65. 41. Water, compression of, 61-65. , density of, 34-36. , expansion of, 35-36. , spec, heat of, 98, 100. , weighing in, 37. Watt as unit, 26. Wave-lengths, 83-85. Weight, 24. Wire gauge, 183. , resistance of, 183-184. Work, 25-27. Working, rate of, 26. Year, 71. Young's modulus, 53. PRINTED BY ROBERT MACLEHOSB, UNIVERSITY PRESS, GLASGOW. 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