&-i LIBRARY 1VERSITY OF CALIFORNIA. Received *-/1/-44 Yo, I fwrflW ^ ELECTROSTATICS AT THE CLARENDON PRESS M DCCC LXXXV [ All rights reserved ] N> I. PREFACE. THE exhaustive character of the late Professor Maxwell's work on Electricity and Magnetism has necessarily reduced all subsequent treatises on these subjects to the rank of commentaries. Hardly any advances have been made in the theory of these branches of physics during the last thirteen years of which the first suggestions may not be found in Maxwell's book. But the very excellence of the work, regarded from the highest physical point of view, is in some respects a hindrance to its efficiency as a student's text-book. Written as it is under the conviction of the para- mount importance of the physical as contrasted with the purely mathematical aspects of the subject, and therefore with the determination not to be diverted from the immediate contemplation of experimental facts to the development of any theory however fas- cinating, the style is suggestive rather than didactic, and the mathematical treatment is occasionally some- what unfinished and obscure. It is possible, therefore, that the present work, of which the first volume is now offered to students of the mathematical theory of electricity, may be of service as an introduction to, or commentary upon, Maxwell's book. Its aim is to state the provisionally accepted two-fluid theory, and to develop it into its mathematical consequences, VI PREFACE. regarding that theory simply as an hypothesis, valuable so far as it gives formal expression and unity to experimental facts, but not as embodying an accepted physical truth. The greater part of this volume is accordingly occupied with the treatment of this two-fluid theory as developed by Poisson, Green, and others, and as Maxwell himself has dealt with it. The success of this theory in formally explaining and co-ordinating experimental results is only equalled by the artificial and unreal character of the postulates upon which it is based. The electrical fluids are physical impossi- bilities, tolerable only as the basis of mathematical calculations, and as supplying a language in which the facts of experience have been expressed and results calculated and anticipated. These results being afterwards stated in more general terms may serve to suggest a sounder hypothesis, such for instance as we have offered to us in the displace- ment theory of Maxwell. In the arrangement of the treatise the first three chapters are devoted to propositions of a purely mathematical character, but of special and constantly recurring application to electrical theory. By such an arrangement it is hoped that the reader may be able to proceed with the development of the theory in due course with as little interruption as possible from the intervention of purely mathematical processes. Few, if any, of the results arrived at in these three chapters contain anything new or original in them, and the methods of proof have been selected with a PEEFACE. Vll view to brevity and clearness, and with no attempt at any unnecessary modifications of demonstrations already generally accepted. All investigations appear to point irresistibly to a state of polarisation of some kind or other, as the accompaniment of electrical action, and accordingly the physical properties of a field of polarised molecules have been considered at considerable length, especially in Chapter XI, in connection with the subject of specific induction and Faraday's hypothesis of a com- posite dielectric, and in Chapter XIV, with reference to Maxwell's displacement theory. The value of the last-mentioned hypothesis is now universally recog- nised, and it is generally regarded as of more promise than any other which has hitherto been suggested in the way of placing electrical theory upon a sound physical basis. CONTENTS. CHAPTER I. GREEN S THEOREM. ABT. PAGE 1-2. Green's Theorem 1-2 3. Generalisation of Green's Theorem ... . . . 3-5 5. Correction for Cyclosis 5-6 6-17. Deductions from Green's Theorem . . . 6-19 CHAPTER II. SPHEEICAL HAEMONICS. 18-19. Definition of Spherical Harmonics . . . . . . 20-21 20-24. General Propositions relating to Spherical Harmonics . . 21-26 25-34. Zonal Spherical Harmonics 27<-38 35-37. Expansion of Zonal Spherical Harmonics .... 39-40 38-39. Expansion of Spherical Harmonics in general .... 41-44 CHAPTER III. POTENTIAL. 40-46. Definition of Potential 45-51 47-50. Equations of Poisson and Laplace 51-54 51-63. Theorems concerning Potential 54-67 64-68. Application of Spherical Harmonics to the Potential . . 67-73 CHAPTER IV. DESCRIPTION OF PHENOMENA. 69. Electrification by Friction 74 70. Electrification by Induction 75 71. Electrification by Conduction 76 72-79. Further Experiments with Conductors and Insulators . . 77-83 80-81. Electrical Theory 83-86 CONTENTS. CHAPTER Y. ELECTRICAL THEORY. ART. PAGE 82-87. Properties of Conductors and Dielectric Media according to the Two-fluid Theory 87-90 88. Principle of Superposition 90 89. Case of Single Conductor and Electrified Point . . 90 90-91. Electrified System inside of Conducting Shell . . . 91-92 92-94. Explanation of Experiments II, V, VI, and VII . . 93-95 95. Case of given Potentials 96 96. General Problem of Electric Equilibrium .... 96-97 97-98. Experimental Proof of the Law of Inverse Square . . 97-100 99-103. Lines, Tubes, and Fluxes of Force 100-107 CHAPTER VI. APPLICATION TO PARTICULAE CASES. 105. Case of Infinite Conducting Plane 108 106. Two Infinite Parallel Planes; two Infinite Coaxal Cylin- ders; two Concentric Spheres 109-110 107-117. Sphere in Uniform Field; Conducting Sphere and Uni- formly Charged Sphere; Infinite Conducting Cylinder 110-123 CHAPTER VII. THE THEORY OF INVERSION AS APPLIED TO ELECTRICAL PROBLEMS. 118-122. Theory of Inversion; Geometrical Consequences; Trans- formation of an Electric Field 124-129 123-124. Problem of Conducting Sphere and Electrified Point solved by Inversion 129-130 125. The converse Problem 130 126. Case of two Infinite Intersecting Planes .... 131-132 127. Case of n Intersecting Planes 132-133 128-129. Case of Hemisphere and Diametral Plane . . . 133-134 130. Two Spheres external to each other 134 181. Distribution on Circular Disc at Unit Potential . . 135-136 132-133. Deductions therefrom 136-138 134-135. Case of Infinite Conducting Plane and Circular Aperture 138-140 136-140. Case of Spherical Bowl 140-143 141-142. Effect of small hole in Spherical or Plane Conductor . 143-146 CONTENTS. XI CHAPTER VIII. ELECTBICAL SYSTEMS IN TWO DIMENSIONS. ART. 143-144. 145-146. 147-150. 151-152. 153-154. Definition of Field Definition and Properties of Conjugate Functions Transformation of Electric Field Example of Infinite Cylinder .... Inversion in two Dimensions PAGE 147-148 148-152 152-154 154-156 156-158 CHAPTER IX. SYSTEMS OF CONDUCTOES. 155-156. Co-efficients of Potential and Capacity 157-158. Properties of Co-efficients of Potential 159-164. Properties of Co-efficients of Capacity and Induction 165. Comparison of similar Electrified Systems 159-160 160-162 162-164 165 CHAPTER X. ELECTEICAL ENEKGY. 166-168. The Intrinsic Energy of any Electrical System . . 166-170 169-170. The Mechanical Action between Electrified Bodies . . 170-171 171-174. The change in Energy consequent on the connexion of Conductors, or the variation in size of Conductors . 171-173 175. Earnshaw's Theorem 174-176 176-181. A System of Insulated Conductors without Charge fixed in a field of uniform force 176-181 182. Electrical Polarisation 181 CHAPTER XI. SPECIFIC INDUCTIVE CAPACITY. 183-185. Specific Inductive Capacity 186-192. Mathematical investigation of Faraday's theory of Specific Induction 193. Lines, Tubes, and Fluxes of Force on this theory . 194-195. Application of the theory to special cases 195a-198. Extension of the theory to Anisotropic Dielectric Media 199-200. Determination of the Co-efficient of Induction in special 183-184 184-194 195 196-200 200-205 205-207 Xll CONTENTS. CHAPTER XII. THE ELECTRIC CURRENT. ART. 201-203. General description of the Electric Current . . . 208-210 204-206. Laws of the Steady Current in a single metal at Uniform Temperature ........ 210-213 207-211. Determination of the Resistance in special cases . . 214-218 212-215. Systems of Linear Conductors ...... 218-221 216-218. Generation of Heat in Electric Currents .... 221-224 219-221. Electromotive Force of Contact ...... 224-226 222-225. Currents through Heterogeneous Conductors . . . 226-228 CHAPTER XIII. VOLTAIC AND THERMOELECTRIC CURRENTS. 226-229. The Voltaic Current. 229-232 230-238. The Electromotive Force of any given Voltaic Circuit . 232-236 239-240. Clausius's theory of Electrolysis 236-238 241. Electrolytic Polarisation 238 242-243. Thermoelectric Currents 239-240 244-248. Laws of a Thermoelectric Circuit . . . ... 240-244 248-250. The Energy of a Thermoelectric Circuit .... 244-246 251. Systems of Linear Conductors with Wires of different Metals and Temperatures 246-247 CHAPTER XIV. POLARISATION OP THE DIELECTRIC. 252-253. Polarisation of the Dielectric 248-254 254. Stresses in a Polarised Dielectric 254-257 255. Explanation of Superficial Electrification of Conductors . 257-258 256-257. The Relation between Force and Polarisation at each point of. a Dielectric 258-259 258-262. Theory of Electrical Displacement 259-264 263-264. Displacement Currents 264-266 265. All Electric Currents flow in Closed Circuits . , 266-268 CHAPTER I. GREEN'S THEOREM. ARTICLE 1.] LET 8 be any closed surface, u and w' any two functions of a?, y, and #, which are continuous and single-valued everywhere within 8. Then shall du du' du du' du du' = /YV |^- ds- /7YV = ff w ?p4#_ f/T w in which the triple integrals are taken throughout tfye space enclosed by S, and the double integrals over the surface, dv is an element of the normal to the surface inside of S 9 but measured outwards in direction, and V 2 stands for (~dtf + "dy* + &?)' For let a line parallel to x cut the surface in the points v lt y, z and a? 2 , y, 2;. Then integrating by parts between x = #! and a? = ir 2 , we have /**a . -=- > -7- > or one of them, are infinite, the theorem da? dy d0 requires modification as follows : It still remains true, u being always finite, that d lv du\, . T .du\ . T ,du\ C x * dudu' u- r (K-=-)dne = (uK-=-) (uK ) / Kdx\ dx^ dx' dx' Xli dx' Xl J Xl dx dx and from this we may deduce Green's theorem in the form du du = ffuK ^-dS- fflu^u'dxdydz. But we cannot assert the truth of the theorem in the al- ternative form . du du K(- - |- &c. ) dxdydz ^dx dx ' y \udxdydz. If u become infinite at any point within $, we cannot include in the integration the point at which the infinite value occurs. But we may describe a surface S' completely enclosing, and very near to, that point, and apply the theorem to the space between S and /S", regarding u' and its differential coefficients as constant throughout $'. For instance, let u become infinite at a point P within S. Let S' be a small sphere described about P, and let ft' -=-2 and K p be the values of ft', -=- , and K in or on the dv av surface of S'. Then we obtain (fudS f - In this form the theorem can be made use of whenever the two surface integrals relating to ', namely //^ J/S'and // -g-dff, 5.] THE CORRECTION FOR CYCLOSIS. 5 are finite or zero. For instance, if u = -, where r is the distance of any point from P, sr=0 and ^ = - and the equation becomes Correction for Cyclosis. 5.] We have assumed also that u and u' are single-valued functions of #, y, z ; that is, that for any such function the line integral / -~- dl) taken round any closed curve that can be drawn within the space S to which Green's theorem is applied, is zero. The functions with which we shall have to deal in this treatise will generally satisfy this condition. If however for any function u the condition dl = be not satisfied for certain closed curves drawn within $, the state- ment of Green's theorem requires modification in the manner pointed out by Helmholz and Sir W. Thomson. The reader will find the subject fully treated in Maxwell's Electricity and Magnetism, Second Edition, Arts. 96 b 96 d. It will be sufficient here to shew the modification required in a simple case. Suppose, for instance, S con- sist of an anchor-ring, Fig. I, and that for any closed curve drawn within it, so as to embrace the axis, as OPQO, I -jj dl= H, but for closed curves not embracing the axis / dl = 0. Let us suppose u to be mea- do sured from a section $ of the ring. Let be a point in the 6 THE CORRECTION FOR CYCLOSIS. [5. section . Then, if we start from 0, with u for the value of u, and proceed round the curve OPQO, u will, on arriving again at 0, have assumed by continuous variation the value U Q + H. Let S l be any other section of the ring. Then $ and 8 divide the space within the ring into two parts, /S P/S f i and S 1 QS . No curve embracing the axis can be drawn wholly within either S PS l or S l QS . Therefore Green's theorem may be applied to either space. Applying it to S Q PS lt we have (1) in which the first double integral relates to the surface of the ring, and the other two to the sections S Q and S 1 respectively. Again, applying the theorem to and regarding the normals to S and S 1 as measured in the same direction as in the former case, that is inwards as regards the space now in question, we have du du ( -T- + &c. ) dxdydz duf_ ~d>> (2) If we now add the two equations (l) and (2) together, we obtain for the whole space within the ring .du du fffz(- J J J ~ r r fj f Htnce HI I K^dS Q is the correction for cyclosis in this case. Its value depends on the section of the ring arbitrarily chosen as the starting-point from which u is measured. 7-] DEDUCTIONS FKOM GREEN'S THEOREM. Deductions from Green's Theorem. mi du' e a constant. Then, since -= > - 0$ severally zero, we obtain the result that for any function u, A -i T j > i mi du' du' , du' o.J Let u be a constant. Then, since -= > -y- > and -7- are 0$ ^ ^ the integrals being taken over any closed surface 8 and the enclosed space. 7.] (a) There exists one function u of x, y, and z which has arbi- trarily assigned values at each point on a closed surface 8, and satisfies the condition V z K u = at each point within S, K being everywhere positive. For evidently an infinite number of forms of the function u exist satisfying the condition that u has the assigned value at each point of S, irrespective of the value of V* K u within 8. For any function u let the integral throughout the space enclosed by 8 be denoted by Q^ This integral is necessarily positive, and cannot be zero for any of the functions in question, unless the assigned values are the same at every point of S 9 in which case a function having that same constant value within S satisfies all the conditions of the problem. If the assigned values of u be not the same at each point of S, then of all the functions which satisfy the surface conditions, there must be some one, or more, for which Q U) being necessarily positive, is not greater than for any of the others. Let u be such function. Let u + u' be any other function which satisfies the surface con- ditions, so that u f at each point of S. Then also u + On' satisfies the surface conditions, if be any numerical quantity whatever, positive or negative. 8 DEDUCTIONS FROM GREEN' S THEOREM. [7. Then du du du du' du by Green's theorem, = Qu+0 2 Q U '-2 because u' = on 8. Now Qu+eu r is by hypothesis not less than Q U} and therefore 0* Quf-20 fffu' V^ u dxdydz cannot be negative for any value of 0, or any value of u'. But unless V 2 ^ be zero at each point within S, it is possible to assign such values to #', consistently with its being zero on S, as to make / uV 2 K u dxdydz differ from zero. Therefore, it is possible to assign such a value to as to make 0* Qu>- 2 \\\u' V\u dxdydz negative. It follows that V^ = at each point within 8, when u is a function satisfying the surface conditions for which Q u is not greater than for any other function satisfying these conditions. COROLLARY. If u + u' be any other function satisfying the surface condition, but such that V Z K u' is not zero at all points within S, evidently (b) The theorem can also be applied to the infinite space out- side of S with a certain modification, namely, There exists a function u of #, y, and z which has arbitrarily assigned values at each point on S, and satisfies the condition V z K u = at each point outside of S, K being positive, and such that Ku 2 is of lower degree than 1. 9.] DEDUCTIONS FEOM GREEKS THEOREM. 9 For of all the functions which satisfy the surface conditions on S and the condition as to degree, there must be some one or more for which the integral Q u extended through the infinite ex- ternal space is not greater than for any of the others. Let 11 be another function which is zero on S, and satisfies the condition as to degree. Then Green's theorem may be applied to the infinite external space with u arid n for functions. And it can be proved by the same process as used above that, unless V 2 K u = at every point in the external space, some value may be given to u' which will make Q u+u ' less than Q u . Therefore when Q u has its least possible value for all functions satisfying the conditions, V Z K u must be zero at all points outside 8.] The theorems can be extended to the case where V*KU, instead of being zero, has any given value p, a function of or, y> z, at each point within the limits of the triple integral, i.e. within or outside of S as the case may be. For let V be a function of the required degree which satisfies VK V = p at all points within the limits of the triple integral. Such a function always exists independently of the surface con- ditions *. Then if + 2 + &c. dxdydz uV 2 u dxdydz > since u' is constant on each of the surfaces S 1 . . . S n , and is zero on each of the surfaces S^ ... S m '. 12 DEDUCTIONS FROM GREEN' S THEOREM. [ll. Now Q U+0U > cannot be less than Q u , whatever u' may be, and whatever may be. But unless the factor of 20 in the last expression be zero, there must be some value of 6 which makes Q U + 0U ' less than Q u . The quantity multiplied by 20 must therefore be zero for all values of u f consistently with its conditions. V 2 & must therefore be zero at all points within the triple integral, and for all values of %', # 2 ', &c. consistent with u l 'e 1 + u 2 'e 2 + &c. = 0. Therefore we must have where p is some constant, the same for all the surfaces S-^ ... S n . If the function u be found for any value of E, then jx is known from (a), and is proportional to E. There must, therefore, be some value of E for which p is unity, and the function u determined for that value of E satisfies (1), (2), (3), and (4). 11.] The theorem can be extended to the case where V%, instead of being zero at every point within the limits of the triple integral, has any assigned value p, a function of x, y, z. For let V be a function of the required degree which has constant but arbitrary values on each of the surfaces S 1 . . . S n , has the given constant values on S^ ... m ', and satisfies V 2 F= p at all points external to both series of surfaces. The existence of such a function is proved in Art. 8. 7 being so determined, let Then, as we have proved, there exists a function W of the required degree having, some constant values on S 1 . . . S m , the value zero on 8^ ... S m ', and satisfying S 2 = e 2 -e.;, &c., and V 2 /F= at all points external to all the surfaces. 12.] DEDUCTIONS FORM GREEN*S THEOREM. 13 Let u = W+ V. Then u has some constant values on each of the surfaces S l ...S ni and the given constant values on each of the surfaces K...8.'. Also Similarly, &c. = &c., and V 2 w= V 2 JF+V 2 F=p at all external points. 12.] If u be a function which satisfies the conditions (l), (2), and (3) of Art. 10, and for which V 2 u has any assigned value, zero or otherwise, at every point not within any of the surfaces, then u has single and determinate value at each point in external space. For let u and u' be two functions, each of degree less than J, satisfying the surface conditions, so that u and u' are both constant on each surface, and rr JJ J( di* 3 " or and so on for each of the surfaces. Also V 2 ^ and VV both have the same given value at each point in the external space, and therefore V 2 (^ #') = at every point in that space. Then - fff(u- u') V 2 (u-u) dxdydz , rrd(u-u) . + ^-u, )JJ -^-^dS. 2 u-V*u'} dxdydz jjj = 0. 14: DEDUCTIONS FROM GREEN'S THEOREM. [13. du du' du du , du du Therefore j- = T~ > T- = ;r~' and :r = T~' efo? cfo or V* K u dx^ dx' dy^ dy' dz^ dz' instead of V 2 u are given within the limits of the triple integrals, and 1 4.] DEDUCTIONS FROM GREENES THEOREM. 15 where K is positive and constant for each surface, and Ku* of lower degree than 1 . For, we have only to replace V 2 u by the more general ex- pression d . Tr du . d . du. d . Tr . du^ (K ) + (K ) + (K ), or V*KU, dx^ dx' dy^ dy' dz^ dz' and Q u by and every step in the process applies as before. 14.] Again, if S be a closed surface, or series of closed surfaces external to each other, and if o- be a function having arbitrarily assigned values at each point on S, there always exists a function u satisfying the condition (1) = a- at each point on S t (2) V 2 u = at each point in external space, (3) u is of lower degree than J. For there must be an infinite variety of functions U which satisfy the conditions (4) / / UcrdS = JE, where E is any arbi- trary quantity differing from zero, and (5) U is of lower degree than J. For any such function the integral Qu must be greater than zero. There must therefore be some one or more of such functions for which this integral is not greater than for any other. Let u be any such function. Let u + u' be any other function satisfying (4) and (5), and for which therefore Tlu'trdS = 0. Then it can be shewn by the same process as in Art. 10 that -,- oc o-, and V 2 u = at all points external to S, and that by properly choosing E we may make -= = cr and V 2 ^ = as CuV before, and that Q U + U ' = Qu + Q u f > This theorem also, as in the preceding, may be extended to the case in which y 2 u, instead of being zero, has assigned values at all points in the external space. 16 DEDUCTIONS FROM GREEN'S THEOREM. [15. Again, as there always exists a function u satisfying the conditions, so it can be shewn that it has single and determinate value at all external points. For, if possible, let there be two functions u and u' of degree 7 sJ f less than i, both satisfying the conditions, so that -7- = -7 dv dv at each point on 8, and V 2 u = V 2 */, or V 2 (u u) = at all external points. Then = 0, and therefore -=- = -=, &c., and u = u', since both vanish at da dx an infinite distance. 15.] We can shew also by the same process that there exists a function u satisfying the condition that y 2 ^ = at all points in the internal space, and = o- at all points on 8, provided (JvV adS = 0. ff- For if that condition were not satisfied, the condition ua-dS=E might be satisfied by making u a constant, in which case Q u would not have a minimum value greater than zero, and the proof would fail. In fact, if V 2 u everywhere within 8, / / -r- dS ; and therefore -=- cannot be equal to will also satisfy the condition y.. ^ + ^ -Q dx*dydz v ~ For since the order of partial differentiation is indifferent, it follows that *** dM^ = 0. 19.] Let any point be taken as origin of rectangular co- ordinates, and let the coordinates of H P be x, y, z. Let $ (#, y, z) be any function of #, y, z. Let OH be any axis drawn from and designated by Ji, and let Q be any point in this axis, and let OQ = p. I je ^ f> *7j C b e the coordinates of P referred to Q as origin, with axes parallel to the axes through 0. Then the limiting value of the ratio 1 9.] DEFINITION. 21 as p is indefinitely diminished, is denoted by IJ#fe*;4 It is clear that d d(j> ^ U For let 1 19 m lt % be the direction cosines of the axis h v Then by definition du , du du du i ~ n \ ~J~ dx l dy 1 dz But by hypothesis V 2 u = 0. Therefore vi*!, v 2 -, v 2 - dx dy dz are severally equal to zero. Therefore and therefore by successive steps v *^_* J* o. 22 SPHERICAL HAEMONICS. [20. 20.] If - is a spherical harmonic function of degree 1 . d ,l For ( *1 (I) -_.! + !* 1 5 Similarly (-) = - -^ + -^- d* 1 1 ^ 3z 2 and *-"* whence * ^ d! 1_ 3 3(s + J + 2 ' " 3 "* _ __ __ " "*" " __ __ W ^ J cte 2 ' r " r 3 " r 5 " r 3 "" r 3 21.] Whatever be the directions of the i-axes k lt ^ 2 , ... /^, the function JL d \ M ~dh^""dh^ \ r where M is any constant, is a spherical harmonic function of degree (i+ 1). For it is evidently a homogeneous function of that degree, and since it follows that If we write this function in the form 1 1 -^ > Y i is a function of M, the direction cosines of the axes fi l} & 2 , t .*h it and those of r. To fix the ideas we may conceive a sphere from the centre of which are drawn in arbitrarily given directions the ^-axes OHi, Off 2 , ...OHi cutting the sphere mff l9 H 2 , ... H^ Then if OQ be any radius, at every point P on OQ or OQ produced 1^. has a definite numerical value, being a function of the di- rection cosines of Off ly ... OH^ and of OQ, and independent of r or OP. If h lt / 2 , ... hi be the fixed axes of any harmonic, P any 22.] SPHERICAL HARMONICS. 23 variable point, Y i at P is spoken of as the harmonic at P with axes h-fr ^ 2 5 %i- Since each axis requires for the determination of its direction two independent quantities, Y i will be a function of the two variable magnitudes determining the direction of r and the 2i arbitrary constant magnitudes determining the directions of the 2- axes. Y t may also be expressed in terms of the ^-cosines fjiiy H 2 , ... fa of the angles made by r with the 2-axes and the - cosines of the angles made by the axes with each other, 2 and an expression for Y i in this form may be found without much difficulty. 22.] If T t be a spherical harmonic function of degree (i+ I), and if r = \/ 2 +^ 2 + ^ 2 , then r 2i+l V i will be a spherical har- monic function of degree i. For by differentiation = (2 * + 1) r 2 '- 1 x V i + r" +1 Similar expressions hold for Adding these expressions, and remembering that we obtain V 2 (r** 1 V t ) = (2i+ 1) (2 1 + 2) r 2 *'- 1 V i j 4 -t- \ 24 SPHERICAL HARMONICS. [23. 5+^-5 =-+>". and V 2 V i = 0. Therefore = 0, and r* i+ ^ Fi is a homogeneous function of as, y, z of degree i: and is therefore a spherical harmonic function of degree i. Y. We have seen that -~j , as above defined, is a spherical har- monic function of degree (i + 1). It follows then that or r i Y. is a spherical harmonic function of degree i. 23.] Every possible spherical harmonic function of integral positive degree, i, can be expressed in the form r i Y i if suitable directions be given to the axes fi l9 ^ 2 , ... & t determining Y t . For if Hi be a homogeneous function of the i ih degree it contains - - arbitrary constants. Therefore V 2 /^ being 2 of the degree i 2 contains - * arbitrary constants. In order that V 2 ^ may be zero for all values of #, y, and z, the coefficient of each term in V 2 ^ must be separately zero. This involves - - relations between the constants in H^ leaving ^ - ^ - '- -- L- *- or 2i+l of them independent. 2 2 Therefore every possible harmonic function of degree i is to be found by attributing proper values to these 2i-f 1 constants. But the directions of the ^-axes ^ , ^ 2 , ... ^ involve 2 i arbitrary constants, making with the constant M, 2 i -f 1 in all. It is therefore always possible to choose the e-axes ^ , h 2 , . . . li i and the constant M, so as to make .... d d d M . r TnT'Twr -:/* ' or r Y dh^ dfi 2 dhi r 24-] SPHERICAL HARMONICS. 25 equal to any given spherical harmonic function of degree i. Therefore r i Y i is a perfectly general form of the spherical har- monic function of positive integral degree i. Again, every possible spherical harmonic function of negative Y integral degree (i+l) can be expressed in the form -j~ For if V i be any spherical harmonic function of degree (i+l), it follows from Art. 22 that r 2i+l 7i is a spherical harmonic function of degree i. Hence, i being integral, it follows by the former part of this proposition that r 2i+l 7i can always be expressed in the form r i Y i by suitably choosing the axes of Y it and therefore that V i may be expressed in the form , Therefore r i Y i and -~^ are the most general forms of the spherical harmonic functions of the integral degrees i and (i + 1 ) respectively. Y i is defined as the surface spherical harmonic of the order i, 'Y- where i is always positive and integral; r i Y i and -^ are called the solid harmonics of the order i. 24.] If J^ and Yj be any two surface spherical harmonics with the same origin 0, and referred to the same or different axes, and of orders i and j respectively, and if / / Y i Y j dS be found over the surface of any sphere with centre 0, then YjdS= 0, unless i = j. Let H. and Hj be the solid spherical harmonics of degrees i and j respectively corresponding to the surface harmonics Y i and J,, so that H t = r*Y it ffj = rJYj. Make U and U' equal to ff t and Hj respectively in the equation of Green's theorem taken for the space bounded by the aforesaid spherical surface, then 26 ZONAL SPHEKICAL HARMONICS. [25. rrH ^' d ^ d _Ei dffjd JJJ ( dx dx dy dy dz d because V 2 ^ and V 2 ffj are each zero at every point within the sphere ; and similarly, r being the radius of the sphere ; that is or (i j ) therefore either i=j, or 25.] Definition. The points in which the axes h^ h^...h i drawn from any origin meet the spherical surface of radius unity round as centre are called the poles of the axes h^ ^ 2 , ... h it When all these poles coincide, the corresponding spherical har- monics are called zonal spherical harmonics solid and superficial respectively, referred to the common axis, and the surface sphe- rical harmonic of order i is in this case written Q t > If ju be the cosine of the angle between r and the common axis in the case of the surface zonal harmonic Q t of order i, then Qi is the coefficient of e i in the expansion of in ascending powers of e. 25.] ZONAL SPHERICAL HARMONICS. 27 Let OA be the common axis, and let OP be r and the angle POA be 0. In OA take a point M at the distance p from 0. Then if V i be the solid zonal harmonic of degree (i + 1) corresponding to the sur- face zonal harmonic Q i} it follows from definition that o F-/*vJL Fi e-3. i ~dp PM when p is made equal to zero after differentiation. Let p = er and let cos = ju. Then F< = ()* =L== with e = 0. < 1 But = and is constant ; therefore ap r ? whene = 0. But if - be expanded in ascending powers of e. the coefficient of e 1 in the expansion is, by Maclaurin's theorem, 1 .d* 1 T (-J-) . - > when e=0. X * Let it be denoted by A t . Therefore V i = -^A 4 . But T'-pMii Therefore Q i = A t . Hence Q = ^ = 1 and 1 = ^1 = ju. Also when ju = 1 1 and therefore each coefficient Q is unity. 28 ZONAL SPHERICAL HARMONICS. [26, It is evident from definition that the zonal surface harmonic at P referred to OQ as axis is equal to the zonal surface harmonic at Q referred to OP as axis. 26.] Let a be the radius of a spherical surface 8 described round as centre. Let P be any point within or without 8. Let OP =/. And, E being any point on the surface, let PE = D, Fig- 4- /.EOP = 6. Then 1 7 i a Va? 1+ 7^- or - 1~ * or a a 1 -;COS0 according as/ > or < a. according as / > or < a. Therefore, if/ > a, But 2f * _ " 27.] ZONAL SPHERICAL HARMONICS. 29 Therefore and similarly iff < a, 27.] With the same notation as before we can prove that / / ~ = j when P is without , f % 3 and / / = when P is within S, \j J J the integrations being taken over the surface S. Let EOP = 0, and let $ be the angle between the plane of EOP and a fixed plane through OP ; then d(r = a 2 Also D 2 = 2 -2a/cos^+/ 2 ; /* r^o- _ 27ra CdD ' JJ &~ : ~TJ ~D*' the limits on the right-hand side being f a and f+a when P is external, and af and +/ when P is internal ; TAZo- 2wa f 1 1 ) , ' 77 s 5 =: T" i/r -7?^ I when p 1S externa1 ' 1 when P is internal ; or in the respective cases. Hence and f[ a -^[-i~- if- 30 ZONAL SPHEKICAL HARMONICS. [28. for external and internal positions of P respectively, and for both cases ltf=aj I ^ da- = 4^a. 28.] In the last case let F(E) be any function of the position of E on the surface which does not vanish at the point in which OP cuts the surface, nor become infinite at any other point on the surface, let Q be the surface zonal harmonic at E of order i, the common axis being OP, then, if P be made to approach the surface, ultimately shall For with the notation of the last Article let =// then when P approaches the surface and / is indefinitely nearly equal to a, every element of the integral vanishes except when D is indefinitely small. In this case P is ultimately on the surface, and the integral has the same value as if F(E) were equal to F(P), its value at the point of S with which P ulti- mately coincides, or u = (P)- da- = F(P) -dff when/= a ultimately. Therefore = >naF(P) by the last Article. Suppose that/ is originally greater than #, then da; & i i and 2Q.] ZONAL SPHERICAL HARMONICS. 31 And, by Art. 25, i=f{&+ (at P) = And, by Art. 24, each double integral vanishes except or if (7^) denote the value of Y i at P. By putting Y t = Qi we obtain since, by Art. 25, 4 = 1 at the pole, 32 ZONAL SPHERICAL HARMONICS. [30. 30.] If F(E) be a spherical surface harmonic, i.e. F(N) = Y^ then, whether P be on the surface or outside of it, F.or by Arts. 24 and 26, where (1^) denotes the value of Y i at the common axis of the zonal harmonics, that is, along OP. Therefore rr-pinZ a f+i ;?> m- 31.] Considered as a function of //, derived by the expansion of - . the zonal harmonic Q* is called the Legendre's 2 coefficient of order i, and is frequently written P { . We can prove the following properties of the coefficients P. (a) As proved above, if p = 1, l-e Hence, if p = 1, P t = 1 for all values of i ; if IJL = 1, 1 1 Hence, if ^ = 1, P i = + or 1 according as i is even or odd. If u < 1 == is always finite, and is finite if e = 1 . ' Hence the series P 1 + P 2 +... is a convergent series. (V) It is evident from the formation of P^ as the coefficient of e i in the expansion of that P t must contain /u% ju*~ 2 , jut*" 4 , &c., but can contain no higher powers of jut than //, and no powers of which the index differs from i by an odd number. Hence if i be even, P i has the 31.] . ZONAL SPHERICAL HARMONICS. 33 same value for -f /x as for p, and if i be odd the same value with opposite sign. Hence also // can be expressed in terms of P it P f _ 2 , &c (c) f 1 p i P j d fJ . = ifi=j, J i = : - if i = ?. 2t+l For since ^ = cos 0, dfji = sm0d0. Also P i and P^ are both functions of p, and therefore of 0. Hence PP< Pj dp = Fpi Pj sm0d0 J-i Jo over the surface of a sphere of radius a ; = by Art. 24, unless |a/ And if i = /, ri (d) / PipSdp = tf i >j, or if J i is odd. J-i For expanding ^ in terms of the P's, the integral is resolved into a number of integrals of the form / P { Pj dp, in each of ^-i which i =/, and is therefore zero. (e) To find the value of / /x'P^, where K is any positive ^o number integral or fractional *. Let P i = a Then K ., ., , if i be even. ' r- , if i be odd. ' * See Todhunter's Functions of Laplace, Lam^, and Bessel, Art. 34, 35. VOL. I. D 34 ZONAL SPHERICAL HARMONICS. [31. Let i be even. Then if K has any of the values i 2, J 4, &c., or zero, the left-hand member ri I ri / pftp f *fft = i/ ^ Jo *J-i = 0, and therefore K = 0. It follows that ^T = A.K.K 2K 4...K i + 2. Also A is the coefficient of the highest power of K ; therefore P i (fji) when fj. = 1 = 1. Hence, if i be even, K.K 2 ... K i Similarly, if i be odd, / ^ K P i du,= Jo If K be either an integer or a fraction whose denominator when reduced to its lowest terms is odd, then -l JO if ju* P i does not change sign with /u, = 0, if /x"P < does change sign with /x. Hence any function, f(^)^ which can be expanded in a series of positive powers of ju,, whether integral or fractional, can be expanded in a series of the form For we have 1 2 2i+T- * or 32.] ZONAL SPHERICAL HARMONICS. 35 which determines A { , if /(/n) is known in terms of positive powers of JJL. It is perhaps necessary to show that the series converges, if /(ju) can be expanded in a converging series of ascending powers of ju. For let c K fji K be any term in the expansion of f(^). Then the term in A i derived from this term in ^ is and the corresponding term in A i+2 is 2 from which it is easily seen from the expressions for / . ^-i above obtained that, if i be large enough, A i+2 < A { . Now the series P l + P 2 ~f P 3 . . . converges. Hence A Q + A 1 P 1 -{- A 2 P 2 -f &c. converges. 32.] We have hitherto regarded the coefficients Q or P as functions of /x derived from the expansion of Vl We may however take for initial radius any line OC not coinciding with the common axis, and the direction of the common axis OH of the zonal harmonics may be defined with reference to this line by the usual angular coordinates, namely, 6' = Z.HOC, and $' the angle between the plane HOC and a fixed plane through OC. In this case the angular coordinates defining the direction of OP or r will be and 0, and the cosine of the angle HOP will be cos 6 cos tf + sin sin 0' cos (0 - '). Now Q t is, as we have seen, a function of cos HOP, and is therefore a function of cos cos 6' + sin sin 0' cos (< , where x = r sin 6 cos (f> t y = r sin 6 sin , z r cos Q, the equation becomes d z u 2 du 1 d z u I d z u cot# du _ . ~d^ + rd^ + ^d + r 2 sin 2 6 dfi 4 ^~ 5^ ~ 7 Let u = -r^j Then u is a spherical harmonic function of degree ~(i-t- 1), and satisfies the above differential equation. Now Y i = r i+l u, where T i is independent of r, therefore r i+l u is independent of r, whence and t (+ l)^'" 1 ^ + 2 (i+ 1) r + r i+1 = 0, dr dr* d*u 2 du ii and - + = Hence the difierential equation becomes 1 d 2 u Let us now change the variable from to cos#, and let cos 9 = y. Then d . . du Substituting in the differential equation, we obtain 34-] ZONAL SPHERICAL HARMONICS. 37 or restoring -^ for &, This is true for any spherical surface harmonic Y^ and therefore for the zonal harmonic Q t as a particular case. In the case of the zonal harmonic, if the common axis be taken for the initial line from which is measured, Qi * S J as above mentioned, written P if and P i is independent of . Hence P t satisfies the equation 34.] If we differentiate equation (4) of last Article k times, we obtain the equation From (4) and (5) above it appears that P i and -~* respectively satisfy the differential equations We may also prove that P t and ^ are the only solutions of (6) both finite and integral in p. For if in the former of equations (6) we write P^ for y, we obtain a differential equation in u which gives on integration where A and A are arbitrary constants and the integral com- mences from some fixed limit ; If A' = 0, y = AP t , an integral finite solution in 38 ZONAL SPHERICAL HARMONICS. [35. If ^'=^0, the expression fory contains the term ., T d^ A P- I 3 and therefore can be neither finite nor integral. Hence P t or A P. is the only finite integral solution in /u of the former of equations (6). And in the same way it may be d k P- . proved that * is the only finite integral solution in p. of the (I JU. second of equations (6). 35.] By means of equation (5) of the last Article we may generalise the proposition of Art. 24 by proving that T-l) ... (i-k+l) when i =j. For if we multiply the left-hand side of (5) of Art. 34 by (1 /m 2 )*, it may be written and changing Jc into k 1 this becomes But integrating by parts, we get since the integrated terms vanish ; and therefore / + ! ^ 1 < 1 - and therefore by successive reductions, -i 36.] EXPANSION OF ZONAL SPHERICAL HARMONICS. 39 and therefore by Art. 31, /+! /-7& ~p Jlc ~p ( 1 M ) r^ ' d ju ~~ if i ~T^* j 36.] To expand Q* in a series of cosines of multiples of (*-*') Since Q f is the coefficient of e i in the expansion of { 1 2e (cos cos 0' + sin sin 0' cos (0 <')) + e 2 }"*, it follows that the term in Q f which involves cos ((/> <') must contain (sin 0) fc as a factor, or in other words, that the required expansion of Q t must be of the form g + q l cos (<#> ') + &c. + q k cos k ($ ') + &c., where q k = (sin 0) k f (cos 0), and the function denoted by f is rational and integral. If we perform the requisite differentiations on Q iy substitute in (6) of last Article and equate to zero the coefficients of cos ( $'), we obtain the equation where y = cos 0. And since / is a finite integral function of y, it follows from Art. 34 that where A k is independent of y or of 0. Now Qf is a symmetrical function of and r , if therefore we denote cos 6' by y' it will follow that A k must be of the form where P i is the same function of y that P i is of y, and therefore that ft = a P, P / + a x sin sin 0' ^ . ^- cos (0 - ') + &c. ^/tp d k P / -h a^ sin 0^" sin / * -j-^ --- -=-7^- cos k (< ^>') + &c. 40 EXPANSION OF ZONAL SPHERICAL HARMONICS. [37. For most of the applications of Q the actual values of the numerical coefficients a , a l5 a 2 are not required, they may however be determined without much difficulty as follows. 37.] To prove that 2 For it k p' - sin 0* sin 0'* cos Square both sides and integrate with respect to $ from to 27T, remembering that the integrals of all terms containing products of cosines of unequal multiples of <$>' are zero, and that the integrals of all quantities of the form (cosm($ are equal to TT and the integral ofafPfPj* is 27ra 2 P i 2 P/ 2 ; Again, integrate both sides with regard to y from 1 to + 1, remembering that and we get { 2 = <|>', Q ( = 1 ; For in this case ^ becomes The two expressions on the right-hand sides of (a) and (/3) cannot be equal for all values of 0' unless the corresponding terms are separately equal ; 39-] EXPANSION OF ANY SPHEEICAL HAEMONIC. 41 '* - /; rfkp 38:] If YI be any surface spherical harmonic of the i tb order, then Y { is a rational and integral function of cos 0, sin d, cos (b, and sin , i/ = r sin sin , = r cos ; A sin A cos 0^ cos d) v sin rf) 71 ^ therefore J^ is of the form stated above. 39.1 Y, is of the form d k P 2* a cos ^() + 3 sin A;() sin 7i: - where a, and ^,. are numerical constants. K ' tC It is clear that we may assume the coefficients of cosk

) sin 0* If now we multiply Y i by J} and integrate with regard to < from to 2 TT, all the terms will vanish except those in which the multiples of < are the same, and the result therefore will be of the form ^XLIFOR^ 39.] EXPANSION OF ANY SPHERICAL HARMONIC. 43 If we again integrate with regard to y from 1 to +1, the result will be of the form and by Art. 24 each of these terms is zero unless i =/.* It does not follow that /i rz* / Y.Yj i Jo is always finite, inasmuch as the values of the ^'s may be such that although each term in the integral is finite, their sum may be equal to zero. The values of the A 9 a depend upon the inclinations of the two sets of 2-axes of the Y i and J/, and when these axes are so related that /i PS* \ Y.Yf -i Jo is zero, the two spherical harmonics are said to be conjugate. For example, take two spherical harmonics of the first order Y l and F/. If & and $' be the polar coordinates determining the axis of Y lt and 0" and <" those for the axis of 7/, then Y l may be easily seen to be cos 6 cos 'tf + sin sin 0' cos ($ $'), and similarly 7/ is cos cos 0" -f sin sin 0" cos (<$><$>"}) and [ 2 "Y 1 TfdydQ = ^ (cos 6' cos 0" + sin tf sin 6" cos ' cos " -l JO + sin 0' sin 0" sin <' sin c/>") if ^, ?#, # be direction cosines of the axis of Y, l' t m' y n' those of Y'. * In a similar manner the proposition of Art. 26 II-" may be deduced from the form of Yi proved in this article. 44 EXPANSION OF ANY SPHEEICAL HARMONIC. [39. If therefore these axes are at right angles to one another, / - -i or two spherical harmonics of the first order are conjugate when their axes are perpendicular to each other. For the second and higher orders there is no such simple geometrical relation. CHAPTEE III. POTENTIAL. ARTICLE 40.] IF the forces acting on a material system he such that the work done by them upon the system in its motion from an initial to a final position is, whatever those positions may be, a function of the coordinates defining those positions only, and independent of the course taken between them, the system is said to be Conservative. The w6rk done by the forces on the system ill its motion from any position S'to any given j? position which may be chosen as a position of reference, is defined to be the potential energy r , or shortly the potential, of the system in the position S in relation to the forces in question. If we denote by U the potential, and by T the kinetic, energy of the system, then, as shown in treatises on dynamics, T+ U is constant throughout any motion of the system under the influence of the forces in question. If q be any one of the generalised coordinates defining the position of the system, it follows from definition that -7- <>q is the work done by the dq forces on the system as q becomes q + <), and therefore the force tending to increase the coordinate q is j- If the system be a material particle of unit mass, situated at the point P, we may without inaccuracy speak of the potential as the potential of the forces at P. 41.] We are in this chapter concerned only with forces of attraction and repulsion to or from fixed centres, the force varying inversely as the square of the distance from the centre. Now if the central force be any continuous function of the distance, whether varying according to the law of the inverse square or any other law, a potential exists. 46 THE POTENTIAL. [4'. For let there be at a particle of matter of mass m which repels any other particle of mass m' with the force mm'f(r\ where f (r) represents any continuous function of r, the distance between' m and m' ; then it can be shown that if be fixed, the work done by the force upon m' as m moves from a point at the distance r from 0, to another at the distance r 2 from 0, is a function of r^ and r 2 , the initial and final values of r, and of these quantities only, and is independent of the form of the curve de- scribed by m' between these initial and final positions, and of the directions from in which the distances ^ and r 2 are measured. For at any instant during the motion let m be at P, and let Q be a point in the course indefinitely near to P. Let PQ = ds, the angle OPQ = 4>, OP = r, OQ = r + dr. In the limit, if Q be taken near enough to P, the force of repulsion may be considered constant, as m' moves from P to Q, and equal to mm'f(r)._ Therefore the work done by the force in moving the repelled particle from P to Q is mm'f(r) eoaQds, or mm'f(r)dr, and is independent of c/> if dr be given. Therefore the whole work done by the force in the motion from distance r to distance r from is Fig. 5- mm' f(r\dr, and depends upon r x and r 2 , and these quantities only. We have for simplicity considered m fixed at 0, but the proof evidently holds if both m and m' be moveable, and move from a distance r^ to a distance r 2 apart under the influence of the mutual repulsive force mm'f(r). If the mutual force had been attractive instead of repulsive, in other respects following the same law, the expression for the work done would be the same as that for the repulsive force, but with reversed sign. If in any case on effecting the integrations the expression for the work done prove to be negative, this result must be interpreted as expressing the fact that positive work is done against, and not by, the force in the motion considered. 42.] THE POTENTIAL. 47 In either case, whether the force be repulsive or attractive, the work done is proved to be a function of r- and r 2 only, and independent of the course taken between the initial and final positions of m. We have thus shown that if f(r) be any continuous function of the distance between the two particles m and m', a potential exists. At present, as above stated, we are concerned only with the case in which f(r) =-g In that case the work done by the mutual force between m and m', as their distance varies from r (*'2 1 ^dr, that is '-# mm and if the force be attractive ,(1 1 } mm < J-- 42.] We shall now consider two kinds of matter, such that two particles, both of the same kind, repel one another with a mutual force varying directly as the masses of the particles, and inversely as the square of the distance between them, and two particles of different kinds attract one another according to the same law. Then the work done by the mutual force between two par- ticles m and m\ as they move from a distance r to a distance r 2 apart, is, if the masses be of the same kind, and therefore the force repulsive, . ^ ^ . mm' < ; and if they be of different kinds, and the force attractive, /u n mm < > t*i r *> If now we agree to regard all particles of one kind of matter as positive, and all particles of the other kind as negative, we can combine both results under one formula mm i"^r in which m or m' may have either sign, expressing the work 48 THE POTENTIAL. [43. done by the mutual force between m and m' in the motion from distance r to r 2 apart. Finally, we will take for the position of reference to which potential is measured, the position in which the two particles are at an infinite distance apart, that is, in which r 2 is infinite- Then we shall arrive at the following definition. The potential of two material particles m and m\ distant r from each other, is the work done by the force of mutual repulsion as / r zl ~2 dr, r when r 2 is infinite, that is - j and is positive or negative according as m and m' are of the same or different kinds of matter. In physics a body which is within the range of the action of another body is said to be in the field of that other body, and when it is so distant from that other body as to be sensibly out of the range of its action it is said to be out of the field. The following definition is therefore equivalent to the one above adopted. The potential of two material particles distant r apart is the work done by their mutual repulsion as they move from the distance r apart to such a distance as to be out of the field of one another's action^ attraction being included as negative repulsion. 4MU Taking m 1, we define -- to be the potential of m at a point distant r from m. 43.] The potential at any point of any mass occupying a finite portion of space is evidently the sum of the potentials at that point of all the particles of which the mass is composed. If m be any particle of this mass, and r the distance of m from P, the potential of the mass at P is 2 , where the summation T extends throughout the mass, or if p be the density of the mass at #, y, z, the potential is rrr Let this potential be denoted by F. 44.] The repulsion at P of a mass at resolved in any direction is the rate of diminution of the potential of the mass 45-] THE POTENTIAL. 49 per unit of length in that direction. This is a particular case of the general theorem proved above, that the force tending to clV increase any coordinate a is dq If V be the potential of the particle m, and ds the given direction, _dV_ __dV dr ds ~ dr ds m dr m = the repulsion resolved in ds. And this proposition being true of every particle of which the mass is composed is evidently true of the whole mass. Hence, if V be the potential at P of any mass M, the re- pulsion of the mass in the direction indicated by ds is =- ds 45.] If 8 be any closed surface, dS an element of its area, N the repulsive force at dS resolved along the normal to dS measured outwards arising from a particle of matter of mass m placed at the point 0, then if the integration extend over the whole surface ff NdS = 47rm, if m be within S; and ffNdS = 0, if m be without S. Let a line drawn from in any direction cut the surface S at the point P distant r from 0, and let this line make the angle $ with the surface 8 at P. Let a small cone with solid angle da> be described about OP as axis, cutting off from 8 in the neighbourhood of P the ele- mentary surface dS. The area of dS is equal to , also the repulsion at P m Bm * from is , and the resolved part N of this repulsion in the direction of the normal to S at P drawn outwards from S is in . in + sm (f> or sm c/>, according as OP is passing out of S from within, or into 8 from without ; NdS = +mco>, or mdco in the two cases respectively. VOL. I. E 50 THE POTENTIAL. [46. But if be within S, the line drawn from it in any direction as above must emerge from S one time more than it enters it, and therefore the sum of all the values of NdS for this line Taking the corresponding sum for all lines drawn from we get the integral yy^V^-tf, and therefore since the sum of the solid angles about is 4 TT. If be without S the line drawn from it in any direction must meet S in an even number of points, and therefore the sum of all the values of NdS for every such line must be zero ; therefore in this case This proposition is true for any particle within or without S respectively. Therefore it follows that if any quantity of matter of mass M be distributed in any manner within a closed surface S, and if N be the repulsive force of that matter at any point on S resolved in the direction of the normal at that point drawn outwards, then And, similarly, that if M be without 8, then and writing -j- for N, by Art. 44 we have in the two cases respectively. 46.] It follows from Art. 45, that if p, the density of matter, be finite in any portion of space, the first differential coefficients of V cannot be discontinuous in that portion of space. For consider a cylinder whose axis is parallel to x and of length I. Let the proposition be applied to this cylinder. If I be very small compared with the dimensions of the base, we may neglect that portion of the surface integral which relates to the curved surface, and the proposition becomes Jj ~fa dydz = - 47-] EQUATIONS OF LAPLACE AND POISSON. 51 in which the surface integral is taken over the ends of the cylinder, and the triple integral throughout the interior space. dV Also in the surface integral -7 is the rate of increase of V with the normal measured outwards from the enclosed space, in the case of both ends of the cylinder. If it be measured in the same direction in space for both ends, the surface integral may be written Now if p be finite, the triple integral ultimately vanishes when I, and therefore the enclosed space, become infinitely small; and therefore the left-hand member also vanishes, and (-j-) cannot 777 JTT V #'l differ by any finite quantity from (-7-) , or -= cannot be dis- ^#'2 dx continuous. Therefore also V cannot be discontinuous. Equations of Poisson and Laplace. 47.] In the equation of Green's theorem let Fbe the potential of any distribution of matter of which the density p is every- where finite, and therefore such that -7 ? -7 j and -7- are con- dx ay dz tinuous, let 8 be any closed surface, and let u'= unity. Since du f du' , du' -7 , -7 > and -= are zero, the equation becomes dx dy dz dV But 7 is the repulsive force of the matter referred to dv resolved in the normal to 8 outwards from the surface element dS. And therefore by Art. 45 -dS~\ **?***,. Therefore also = j E 2 52 EQUATIONS OF LAPLACE AND POISSON. [48. Since this equation holde for every possible closed surface, it follows that v 2 F+47ip = at every point. This is called Poisson's equation. At a point in free space p = 0, and the equation becomes V 2 F=0. This is called Laplace's equation. It follows as a corollary from Poisson's equation that if V be the potential of any material system at #, y, z, where r 2 = ( x -x'y + (y-yj + (z-zj* ; and the integral is throughout all space. 48.] Laplace's equation can be deduced by direct differentiation of - For if the density of matter at #', /, / is p, the potential at #, y, z is = r/r JJJ y(^Z = rrr (z-zj Now if 0, or x, y, z, be any point not within the mass, the limits of the integration are not altered by any infinitely small change of position of 0. Hence we may place the symbol V 2 under the integral sign, and obtain VT = /YT/> V 2 i dx'dy'dz' = 0. But if be within the mass, we cannot, in forming the triple integral for F, include in integration the point at which the element function - becomes infinite. It is necessary in this * It may be proved by Green's theorem to be identically true for all functions ( F) vanishing at infinity that the integration being extended over all space, and r being the distance from the point at which V is estimated to the element dxdydz; and this proposition may, of course, be made the foundation of an independent proof of Poisson's equation 0. 49-] EQUATIONS OF LAPLACE AND POISSON. 53 case to take for the limits of integration some surface inclosing and infinitely near to it, and to form 7^ as the sum of two separate integrals, one on each side of that surface. Hence any infinitely small change of position of involves in this case a change in the limits of integration, and we are not at liberty in forming V^V to insert V 2 under the sign of integration. This is the reason why Laplace's equation fails at a point occupied by matter. 49.] Definition. We have hitherto supposed the matter with which we have been concerned to be distributed in such a manner that the density p is finite, or in other words that the mass vanishes with the volume of the space in which it is contained. According to this conception the mass of a small volume dv of density p, is pdv, i. e. p is the limiting ratio of the mass to the containing volume when that volume is indefinitely diminished. At all parts of space for which this condition is satisfied we have obtained the equation if V be the potential of any distribution at the point at which the density is p. It may, however, happen that p becomes indefinitely great at certain points. The distribution may be such that although the volume becomes infinitely small the mass comprised in it may remain finite. Suppose such a state of things to hold at all the points on a certain surface $, so that the mass of matter comprised between any portion of this surface, an adjacent surface S' infinitely near to it, and a cylindrical surface whose generating lines are the normals to S along its bounding curve, remains finite however close S' is taken to S, then if the mass vanishes with the area of S, inclosed by this bounding curve, we call the distribution superficial in distinction from the volume distribution hitherto considered. In this conception of superficial distribution we disregard the distance between S and S' altogether, and we say that the mass corresponding to an element of surface dS is crdS, where a- is the superficial density, o- being in other words defined as the limiting ratio of the mass corresponding to, or as we say on, the surface dS to the area of dS, when dS is indefinitely small. 54 EQUATIONS OF LAPLACE AND POISSON. [50. Still further there may be points for which not only p, but n- also, is infinite, and such that if a line I be drawn through these points, the mass of the superficially distributed matter comprised between this line /, an adjacent indefinitely near and parallel line I', and perpendiculars to I at its extremities remains finite, however near I' be taken to I. In such cases the distri- bution is said to be linear ', and neglecting as before the distance between I and ^, we say that the quantity of matter corresponding to, or on the element ds of I is \ds, where A. is the linear density at ds. 50.] On the modification of Poisson's equation at points of superficial distribution of matter. Let dS be an element of the surface,, and let us form on dS a cylindrical surface like that mentioned in the definition of the last article. Let p be the uniform density of matter within that cylindrical surface. If dS 1 denote any element of that surface, including its bases, we have by Art. 45 In the limit, when the bases of that cylinder become infinitely near each other, the right-hand member of this equation becomes 47T// vdS. And if dv, dv' be elements of the normal on either side of 8 t measured in each case from $, the left-hand member becomes CCdV dV "//*>//'' dV dV or + - + 4770- = 0*. dv dv * The cases of finite and infinite p have been considered separately, with the view to their physical interpretations. There is no exception in any case to the equation v 2 F + 47rp = 0, because, v 2 F becomes infinite whenever -, &c. are dx discontinuous, i.e. when p is infinite. 51.] THEOREMS CONCERNING THE POTENTIAL. 55 51.] The mean value over the surface of any sphere of the potential due to any matter entirely without the sphere is equal to the potential at the centre. For let a be the radius of the sphere, r the distance of any point in space from the centre, a 2 do) an element of the surface. Then denoting by V the mean value of V over the sphere, we have = /y 47T = - - 5 / / -j- 47rJJ dr ivtpJJ dr n n JTT but -=-a 2 d v dV I dV -, + 4 TT o- = 0, or or = dv 477 dv But whether the potential be constant or not, the algebraic sum of the distribution over S is dS, by Art. 45. dv 57.] It is always possible to form one, and only one, distribution of matter over a closed surface 8, the potential of which shall have any arbitrarily given value at each point of that surface. For, as we have proved in Art. 7, there exists one determinate function u which has the given value at each point of S, and satisfies V 2 &=0 at each point in the infinite external space, and vanishes at an infinite distance. And there exists one determinate function u' which has the 60 THEOKEMS CONCERNING THE POTENTIAL. [57. given value at each point of S t and satisfies V 2 #'= at each point within S. Then a distribution over S, whose density is 1 ( du du' ) 47T ( dv dv' \ the normals being measured from S, dv on the inside, and dv on the outside of the surface, is the required distribution. For let a small sphere S' be described about any external point, Q, as centre. Let V - where r is the distance of any point from Q. Then, applying Green's theorem to the space outside of S and S', we have with the given meaning of dv, ds + V *r Now V% = and V 2 V = at all points within the limits of the triple integral, and if u denote the value of u at O. Also \\V -^-dS vanishes. JJ dv Therefore the equation becomes Again, applying Green's theorem to the space within S, we have with the given meaning of dv ff or since both V 2 F= and V 2 u' ' everywhere within S, Now -- - = -j-y if Q be not actually on S, however near to 58.] THEOREMS CONCEENING THE POTENTIAL. 61 S it may be, and u = u on S. Hence, subtracting A from B t we have But if P be any point on S, V at P = - du Therefore ._ Now the right-hand member is the potential at Q of the supposed distribution whose density is J_5^ ** M/ ] 4:17 \dv dv 3 It follows that this potential is equal to UQ at every point outside of S, however near to S ; and therefore, since the potential is a continuous function, has the value of , or the given value, at each point on S. Similarly, if Q be an internal, instead of an external, point, we can prove that the distribution over /S whose -density is _l_(du_ <&/) ~4*t = e 2 over $, dv and so on, and u vanishes at an infinite distance. 62 THEOEEMS CONCERNING THE POTENTIAL. [59. For we have proved in Art. 10 that there exists one deter- minate function u satisfying the above conditions. It follows that -=- has a single and determinate value at each point of each of the surfaces. Then if we take for density of the distribution at each point - - -j- , we can prove exactly as before that the 4 7T UV potential of the distribution so formed at any point external to the surface is u, and therefore satisfies all the conditions. 59.] The proposition of Art. 57 may be extended to an unclosed surface thus. Let S be an unclosed surface, S' a similar and equal surface so placed as that each point on ' shall be very near to the corresponding point on 8. If we now connect the boundaries of S and S' by a diaphragm we obtain a closed surface. Let a distribution be formed on this closed surface having potential V on S, and at each point on S' the same potential as at the corre- sponding point on S. Let where a is the radius. a We have then = F, or - = a. * V t ^ 63.] THEOREMS CONCERNING THE POTENTIAL. If $ be an equipotential surface to a system of matter wholly within it, and V be the potential of the system on S, the capacity M of 8 is -= j where Jf is the algebraic sum of the matter of the enclosed system. For, by Art. 60, M is also the algebraic sum of a distribution over S which has potential V at every point on it. 62.] If V be the potential of any distribution of matter over a closed surface S, and if tribution of matter over S whose density is , and the poten- tial at any internal point due to the external portion differs VOL. I. F 66 THEOREMS CONCERNING THE POTENTIAL. [63. 7? from that of a distribution over S whose density is --- , by the potential of the surface. For if we take for origin any point outside of 8 9 and if V be the potential of the entire system, we have by applying Green's theorem to the space inside of S, with u = V> and u' = - > where V t is the constant value of V on S, = 0, since T--&8-**9 9 Jby Art. 45, and V 2 - is zero at all points within S. The equation therefore becomes But =-8, and which proves the first part of the proposition. Secondly, if we take for origin a point inside of S, and apply Green's theorem to the external space, with V and - for u and u', we obtain snce 64.] SPHERICAL HARMONICS APPLIED TO POTENTIAL. 67 in this case, and as before V 2 - = 0. Also in this case, the nor- mal is measured inwards from S, and therefore dV -r=R, also V 2 F=-47rp'. dv Hence the potential at any internal point of the distribution T> 4 over ^ differs by a constant quantity from that of the external portion M', and therefore the force due to the distribu- -JP tion -- over 8 is equal to that due to the external portion 4lT Hence it follows that the force at any external point due to the -n internal portion is equal to that due to the distribution - 4?r over 8, and the force at any internal point due to the external 73 portion is equal to that of the distribution -- 64.] To express the potential at any point P of any distribution of matter in a series of spherical solid harmonics. Take as origin any point 0. Let OP = f. Let M be any point in the distribution. Let the coordinates of M referred to OP as axis, be r, 0, <, where is the angle POM. Let \L = cos 0. Then sin Odd = dp, and an element of volume in the neighbourhood of M is r^dyidfydr. If p be the density of the given distribution in this element of volume, its potential at P is pr*dfJLd(bdr _ ,, _ C 1 , .. r r* ) -yrjjp - = pr*dnddr . j- + Q^ + Q z + ... J if r i / / p-j-dudtidr + I I I prdftdtydr iJo jo / J-iJo Jj ri rzir rs ^ ri r^ r< + I I QiP-^du.d2ir /* du } / d / drp + d drp~ +/ 3*/ .1 (Jo Jo / 2 Jo // + &C., in which the quantities within brackets are known if the given distribution is known. If we denote these quantities by A , A^ A 2 , &c., we have 7= rdp{A +Q 1 A 1 + Q,A 9 +...} 9 J-i in which the A's are generally functions of //. 65.] To find the density of a distribution of matter over a spherical surface, whose potential at any point on that surface shall be equal and opposite to that of a mass e, placed at an external point. Let be the centre of the sphere, a its radius, C the point outside of it, OC=f. Let a- be the required density. It is evident that the density of this distribution on the sphere must be symmetrical about OC, and must therefore be expressible in a series of zonal harmonics with OC as axis. Let this be Let E be any point on the surface, W any other point. Let us denote by Qf the zonal harmonic of order i referred to OE as axis. Then - And the potential at E due to the distribution is VE = H A " SS Q<> Qt ' ds + Ai ff Qi Qi ' ds+ &c ' because every term of the form / IQ i Qj'dS, where i =j, is zero; that is, being the value of Q+ at E. 66.] TO THE POTENTIAL. 69 But by hypothesis the potential at E of the distribution is to be the same as that of the mass e at C with reversed sign; that is, We have therefore and equating coefficients of Q { , 2t+l and if P be external. 70 APPLICATION OF SPHERICAL HAEMONICS [67. But Jf where Y { is the value of Y t at the point where OP, produced if necessary, cuts the sphere. _ ~Y And therefore r i Y i , or <+ * t , according as P is internal or external, is the spherical solid harmonic at P corresponding to 7 i . If we denote this by H it we have in the two cases respectively. The following proposition may easily be deduced from this, but we prefer to prove it independ- ently thus. 67.] If the potential of any material system wholly within a spherical surface S be given at each point of that surface in a series of spherical surface harmonics, then the potential of the same system at any point on the outside of the surface is found by substituting for each surface harmonic the corre- sponding solid harmonic. For let the given potential be 2 A i Y^ and let p be the density of the superficial distribution on S whose potential at every point of S is equal to 2 A i Y^ Let P be any point distant / from the centre on the outside of 4. Then the potential at P of the given system is equal to that of the surface distribution. But, as shewn in Art. 62, if p' be the density of a distribution over S whose potential at any point of S is equal to that of unit of matter situated at P, then is the potential at P of the superficial distribution whose potential is 2AiI i9 and therefore of the given system. Now - P - 68.] TO THE POTENTIAL. 71 where a is the radius of the sphere. Therefore where J^ is the value of J^ at P. The potential of the given system is also equal to 2 ( ) A i Y t J for a certain distance within the given spherical surface S. a \ i+1 For V and 2( ) A i Y i both satisfy Laplace's equation throughout all external space, and are identical at all points out- side of S. They must therefore be identical throughout all space which can be reached from S without passing through attracting matter so long as 2 ( ) A i J i is a convergent series. 68.] To express in zonal solid harmonics the potential of any material system symmetrical about an axis. Let us take for origin any point on the axis. Let / be the distance from of any point in space. Then we can first shew that the potential at any point P on the axis, if more distant from the origin than any point in the system, can be expressed in the form 2 B i -r-y > and if less distant from the origin than any part of the system can be expressed in the form C'+S-^r*, where the functions B and A are determinate if the given system is known, and are inde- pendent of r. For let be the origin, P the point on the axis, M any point in space at which there is matter belonging to the system of density p, Then since the system is symmetrical about the axis, we may take for an element of its volume the space between the two cones whose vertices are at and semivertical angles cos" 1 ju and cos" 1 (p + dp) and whose distance from is between a and a + da. 72 APPLICATION OF SPHEKICAL HARMONICS [68. If p be the density of matter within this element its potential at Pis 2ira*pdtJ.da. , that is, 2ira*pdu,da.-\ 1 + &- + } if r > a, r ( r ) or 2 if a?p dfJi da . - < 1 + Q l - + . .. > if r < a. Then if 1} a 2 be the greatest and least distances from of any matter between the two cones, the potential of all the matter between them is I* ftp ./"*>. \j+Q^ in which the first integral will be omitted when r < 2 , and the second will be omitted if r > a . Finally, the potential at P of the whole system is found by integrating the above expression according to ^ from p I to fji = 1, remembering that a^ and 2 and p are generally func- tions of /u. Let a\ and a\ denote the greatest and least values of r for any point in the system. Then the result, if the integrations can be effected, must appear in the form sn-Lif OP><, and C+2A t r* if OP<<; and if r > a\ < a\. We can now find the potential of the system at any point R not in the axis and distant r from 0, by multiplying each term by the corresponding zonal harmonic referred to OP as axis. For instance, suppose r > a\. Let V be the potential and let 68-] TO THE POTENTIAL. 73 Then since on the axis Q t = 1, and V and V are identical throughout a finite length of the axis. Now both V and V satisfy Laplace's equation at all points not occupied by matter belonging to the system. And therefore since they are identical throughout some finite length on the axis, and are symmetrical about the axis, they must by Arts. 53 and 54 be identical at all points in space which can be reached from that part of the axis without passing either through the system, or through any part of space where 2 B ri i _^ l does not converge. Similarly, the potential at any point R' in space distant r from 0, where r < a 2 ', is C+ 2^ f Q^*, provided It' can be reached from the part of the axis whose distance from is less than a with- out passing, either through the system, or through any part of space where "SA t Q^ does not converge. CHAPTEE IV. , i DESCRIPTION OF PHENOMENA. Electrification ly Friction, ARTICLE 69.] EXPERIMENT I*. Let a piece of glass and a piece of resin be rubbed together and then separated ; they will attract each other. If a second piece of glass and a second piece of resin be similarly treated and suspended in the neighbourhood of the former pieces of glass and resin, it may be observed that (1) The two pieces of glass repel each other. (2) Each piece of glass attracts each piece of resin. (3) The two pieces of resin repel each other. These phenomena of attraction and repulsion are called elec- trical phenomena, and the bodies which exhibit them are said to be electrified or to be charged with electricity. The electrical properties of the two pieces of glass are similar to each other but opposite to those of the two pieces of resin, the glass attracts what the resin repels, and repels what the resin attracts. Bodies may be electrified in many other ways as well as by friction. If a body electrified in any manner whatever behaves as the glass does in the experiment above described, that is, if it repels the glass and attracts the resin, it is said to be vitreously elec- trified, and if it attracts the glass and repels the resin, it is said to be resinously electrified. All electrified bodies are found to be either vitreously or resinously electrified. When the electrified state is produced by the friction of dis- similar bodies, as above described, it is found that so long as the * The description of these experiments is taken almost verbatim from Maxwell's Electricity. 70.] DESCRIPTION OF PHENOMENA. 75 rubbed surfaces of the two excited bodies are in contact the combined mass does not exhibit electrical properties, but behaves towards other bodies in its neighbourhood precisely as if no friction had taken place. The exactly opposite properties of bodies vitreously and resin- ously electrified respectively, and the fact that they neutralise each other, has given rise to the terms 'positive' and 'negative* electrification, the term positive being by a perfectly arbitrary, but now universal convention among men of science, applied to the vitreous, and the term negative to the resinous electri- fication. Electric actions similar to those above described may be ob- served between a body electrified in any manner and another body not previously electrified when brought into the neigh- bourhood of the electrified body, but in all such cases it will be found that the body so acted upon itself exhibits evidence of the electrification. This electrification is said to be produced by induction, a process which will be illustrated in the second ex- periment. No force, either of attraction or repulsion, can be observed between an electrified body and a body manifesting no signs of electrification. Electrification by Induction. 70.] EXPEBIMENT II. Let a hollow vessel of metal, furnished with a close-fitting metal lid, be suspended by white silk threads, and let a similar thread be attached to the lid, so that the vessel may be opened or closed without touching it ; suppose also that the vessel and lid are perfectly free from electrification. Let the pieces of glass and resin of Experiment I be suspended in the same manner as the vessel and lid, and be electrified as before. If then the electrified piece of glass be hung up within the sus- pended vessel by its thread, without touching the vessel, and the lid closed, the outside of the vessel will be found to be vitreously electrified, and it may be shown that the electrification outside of the vessel, as indicated by the attractive or repulsive forces on 76 DESCRIPTION OF PHENOMENA. [71. electrified bodies in its neighbourhood, is exactly the same in whatever part of the interior the glass be suspended. If the glass be now taken out of the vessel without touching it, the electrification of the glass will be found to be the same as before it was put in, and that of the vessel will have disappeared. This electrification of the vessel, which depends on the glass being within it, and which vanishes when the glass is removed, is called Electrification by Induction. If the piece of electrified resin of Experiment I were sub- stituted for the glass within the vessel, exactly opposite effects would be produced. If both the pieces of glass and resin, after the friction of Experiment I, were suspended within the vessel, whether in contact with each other or not, no electrical effects whatever would be manifested. Similar effects would be produced if the glass were suspended near the vessel on the outside, but in that case we should find an electrification vitreous in one part of the outside of the vessel and resinous in another part. Whereas, as has been just now mentioned, when the glass is inside the vessel the whole of the outside is vitreously electrified. In this case, as in the case of internal suspension, the electrification disappears on removal of the exciting body. Experiment proves that throughout the inside of the closed vessel there is an electrification of the opposite kind to that of the outside, that is, when the electrified piece of glass is suspended within the vessel, and the latter is therefore vitre- ously electrified on the outside, as just now explained, the in- side will be resinously electrified, and vice versa when the resin is substituted for the glass. Experiment proves also that the electrification on the outside is equal in quantity to that of the glass, and the electrification on the inside equal and opposite to that of the glass. Electrification by Conduction. 71.] EXPERIMENT III. The metal vessel being electrified by induction, as in the last experiment, let a second metallic body be suspended by white silk threads near it, and let a metal wire 72.] DESCRIPTION OF PHENOMENA. 77 similarly suspended be brought so as to touch simultaneously the electrified vessel and the second body. The second body will now be found to be vitreously electrified and the vitreous electrification of the vessel will have diminished. The electrical condition has been transferred from the vessel to the second body by means of the wire. The wire is called a conductor of electricity, and the second body is said to be electrified by conduction. Conductors and Insulators. If a glass rod, a stick of resin or gutta-percha, or a white silk thread had been used instead of the metal wire, no transfer of electricity would have taken place. Hence these latter substances are called non-conductors of electricity. A non-conducting sup- port or handle employed in electrical apparatus is called an Insulator, and the body thus supported is said to be insulated. Thus the lid and vessel of Experiment II are insulated. The metals are good conductors ; air, glass, resins, gutta- percha, vulcanite, paraffin, &c., are good insulators ; but all sub- stances resist the passage of electricity, and all substances allow it to pass although in exceedingly different degrees. For the present we shall, in speaking of conductors or non-conductors, imagine that the bodies spoken of possess these properties in perfection, a conception exactly similar to that of perfectly fluid or perfectly rigid bodies, although such conceptions cannot be realised in nature. In Experiment II an electrified body produced electrifica- tion in the metal vessel while separated from it by air, a non- conducting medium. Such a medium, considered as transmitting these electrical effects without conduction, is called a Dielectric medium, and the action which takes place through it is called, as has been said, Induction. 72.] EXPEEIMENT IV. In Experiment III the electrified vessel produced electrification in the second metallic body through the medium of the wire. Let us suppose the wire removed and the electrified piece of glass taken out of the vessel without touching it and removed to a sufficient distance. The second body will 78 DESCRIPTION OF PHENOMENA. [74. still exhibit vitreous electrification, but the vessel when the glass is removed will have resinous electrification. If we now bring the wire into contact with both bodies, conduction will take place along the wire, and all electrification will disappear from both bodies, from which we infer that the electrification of the two bodies was equal and opposite. 73.] EXPERIMENT V. In Experiment II it was shown that if a piece of glass, electrified by rubbing it with resin, is hung up in an insulated metal vessel, the electrification observed outside does not depend upon the position of the glass. If we now introduce the piece of resin with which the glass was rubbed into the same vessel without touching it or the vessel, it will be found, as stated in Art. 70, that there is no electrification on the outside of the vessel. From this we conclude that the electrification of the resin is exactly equal and opposite to that of the glass. By putting in any number of electrified bodies, some vitreous and others resinous, and taking account of the amount of electrification of each, we shall find that the whole electrification of the outside of the vessel is that due to the algebraic sum of the electrifications of all the inserted bodies, the signs being used in accordance with the convention already described. We have thus a practical method of adding the electrical effects of several bodies without altering the elec- trification of any of them. 74.] EXPERIMENT VI. Let a second insulated metallic vessel B be provided, and let the electrified piece of glass of Experiment I be placed in the first vessel A, and the electrified piece of resin in the second vessel B. Let the two vessels be then put in com- munication by the metal wire, as in Experiment III. All signs of electrification will disappear. Next, let the wire be removed, and let the pieces of glass and resin be taken out of the vessels without touching them. It will be found that A is electrified resinously and B vitreously. If now the glass and the vessel A be introduced together (the glass being no longer within A) into a larger insulated vessel ^ 4. -u (mn)(m'n'} that is, by '-\ But m n is the number of units of positive electrification on A, and m'n' is the same for B, so that with the notation used above the force between the electricities in A and B is re- ee' presented by -o-, and is repulsive when ee is positive. 86 ELECTRICAL THEORY. [8 1. The unit of electricity in this measurement is such that the repulsive force between two units of positive electricity at the distance unity apart is unit force. It appears therefore that the two-fluid theory involves the existence of a force between the electric fluids in two charged bodies in all respects following the law which has been experi- mentally proved to be obeyed by the mechanical forces between the bodies themselves. But the bodies are either non-conductors or else conductors in an insulating medium, and on either hypothesis the fluids cannot move without the containing bodies accompanying them. Whatever force therefore is proved to exist between the fluids becomes phenomenally a corresponding force between the bodies. We thus see that the results of Ex- periments I, III, and IV, and of Experiments VIII and IX are explained qualitatively and quantitatively by the two-fluid hypothesis. The application of the theory to the Induction Experiments II, V, VI, and VII, is not so obvious, and can only be demonstrated after some further development. CHAPTER V. ELECTRICAL THEORY. ARTICLE 82.] WE proceed now to develop the two-fluid theory as before enunciated, regarding for the present all substances as divided into two classes, namely, (i) perfect insulators, called generally dielectrics, throughout which there is an absolute bar to the motion of the fluids from one particle to another, and (2) perfect conductors, throughout which the fluids are free to move with no resistance whatever from one particle to another. And it is assumed for the present that the repulsion between two masses, ee' e and /, of electricity placed at distance r apart is ^ . The phenomena with which we have at present to deal are those of repulsion and attraction between particles at a distance ac- cording to the above law. The investigations of Chap. Ill are therefore applicable. It will be understood that we do not assert the actual exist- ence of the fluids, or that direct action at a distance actually takes place. It is proposed merely to show how the phenomena of Electrostatics may be explained on this hypothesis. In like manner the conception of space as divided into perfect con- ductors and perfect insulators, will have to be materially modified hereafter. 83.] It follows from the above definition of a conductor, that when the electricities are in equilibrium, the resultant force is zero at each point within the conductor. For if there be any force, it must tend to move one kind of electricity at the point in one direction, and the other in the opposite direction, and therefore to separate them. And since the substance of the conductor opposes no resistance to their motion, such separation will in fact take place until equilibrium is attained ; that is, until the 88 ELECTRICAL THEORY. [84. mutual attraction of the separated electricities, tending to re- unite them, becomes equal and opposite to the force which tends to separate them, and so the resultant force becomes zero. Now it follows from the reasoning- of Chap. Ill that in a field of electric fluid distribution a potential function V exists such that dV dV dV , ? are the component iorces parallel to the axes dm dy dz of a?, y, and z at any point. And since the resultant force is zero, each of these components is zero at every point within the conductor, and therefore V has some constant value throughout the substance of the conductor. This is true whatever the law of force, provided there be a potential. 84.] It follows further from the law of the inverse square, that there can be no free electricity within the substance of the con- ductor. For whatever closed surface be described wholly within it, the normal force N at every point of that surface is zero. Therefore / / Nds = over the surface. That is, by Art. 45, the algebraic sum of all the free electricity within the surface is zero, and this being true for every closed surface that can be described within the substance of the conductor, it follows that there can be no free electricity, of either volume or superficial density, within the substance of the conductor. It follows that, in order to insure the constancy of V through- out the conductor, it is sufficient to make it constant at all points on the surface. For we have seen that if V be constant at all points on a closed surface, within which is no attracting matter, it has the same constant value throughout the interior. 85.] Whatever free electricity is formed by the separation of the two kinds of electricity within the conductor, since it cannot exist within the substance of the conductor, and cannot penetrate the surrounding dielectric, must be found upon the surface in the form of a superficial distribution. And such superficial distribution must be in the aggregate zero for the whole surface ; because since the two kinds of electricity are supposed to exist in equal quantities at all points, for every quantity of positive electricity resulting from their 87.] ELECTRICAL THEORY. 89 separation, there must be an equal quantity of negative elec- tricity, and each must be found somewhere on the surface. But it is possible to place upon the conductor from external sources a quantity of electricity of either sign. This also, for the same reason, can only exist in the form of a superficial distribution. It follows then that if a conductor be in equilibrium its electrification is wholly on the surface, and the algebraic sum of all the superficial distribution upon it is equal to that of the electricity placed upon it from external sources. 86.] Definition. The algebraic sum of all the electricity on the surface of a conductor is called the charge on the conductor. If o- be the density of the superficial distribution at any point, dV -j the rate of increase of V per unit of length of the normal measured outwards in direction, immediately outside of the dis- tribution, -^7 the same thing measured inwards in direction, d v immediately inside of the distribution, Poisson's equation gives dV dV - -- h , = 47TO-. dv dv But -- j,} being the force within the substance of the con- dv ductor, is in this case zero. We have therefore at every point of the surface dv and the charge upon the conductor or 87.] We have seen that when an electrical system is in equi- librium, the potential must have a constant value throughout each conductor. Conversely, if the potential have a constant value throughout each conductor, the electricity on fixed conductors is in equilibrium. For the potential being constant throughout the conductor, there can be no tangential or other force to move the superficial distribution along the surface or through the substance 90 ELECTRICAL THEORY. [88. of the conductor. And since by the hypothesis concerning- the nature of the dielectric medium there can be no motion of elec- tricity in the medium, all the electricity in the field must be at rest. The constancy of the potential throughout each conductor is thus the sufficient and necessary condition of equilibrium. Hence can be established the following principle. The Principle of Superposition. 88.] If or be the density of the superficial distribution on a con- ductor when in equilibrium in presence of any electrified system E, which may include a charge on the conductor itself, and if density is ) where R is the normal force on S due to the 4-7T whole electrification ; that is zero potential, because R = on >$'. Hence the enclosed system and the distribution on the inner surface have together zero potential at all points outside of S. Similarly the external system and induced distribution on the outer surface have together constant potential V at all points inside of S ', and since S may be made to coincide with either the inner or the outer surface of the shell, this proves the proposition. It follows that if the enclosed system, together with the distribution on the inner surface, were both removed, or allowed to communicate and neutralise each other, the distribution on the outer surface would remain in equilibrium. Its density is therefore independent of the position of the enclosed distribution within the shell. It follows further that any charge placed on the conductor will assume a position of equilibrium on the outer surface without causing any electrification on the inner surface. Again, if the external electrification and the distribution on the outer surface were removed, that on the inner surface and the enclosed system would remain in equilibrium. The agreement with experiment of the above proposition, that a charge of electricity upon a hollow conducting shell causes no electrification on its inner surface or on a conductor placed within it, has been employed, as we shall hereafter see, to establish the most conclusive proof of the law of the inverse square in electric action. 92.] In Chap. IV it was shown that the qualitative results of Experiment I, and the qualitative and quantitative results of Experiments I, III, IV, VIII and IX, were completely explained by the two-fluid theory of electricity. We are now in a position to do the same with reference to the results of Experiments II, V, VI, and VII. For it has been proved (Art. 84), that there can be no free electricity within the substance of conducting bodies, but that 94 ELECTRICAL THEORY. [93. in the case of such bodies the charges, if any, are entirely super- ficial. It has been also proved (in Arts. 90, 91) that in the case of the electrical equilibrium of a hollow conducting shell in the presence of any given electrical distributions, whether internal or external, (1) There is a superficial electrical distribution on the inner surface of the shell equal in amount, but of opposite algebraic sign to, the algebraic sum of the given internal system. (2) That the given internal system, with the last-mentioned superficial electrification of the inner surface, constitute a system producing electrical equilibrium throughout the surface of the shell and the whole of external space ; and that the given external system, with any superficial electrification on the outer surface of the shell, constitute a system producing electrical equilibrium throughout the shell and the whole of the internal space. It follows therefore that in the case of the closed insulated metal vessel of Experiment II, containing an electrified piece of glass as therein described, (1) There will be a superficial electrification on the inner surface, the total amount of which will be resinous, and equal to the vitreous electricity of the glass, but the intensity of which at different points will depend upon the position of the glass. (2) That inasmuch as the vessel is insulated, and the total charge zero, and as all the electrification must be superficial, there will be a superficial distribution on the external surface equal in amount to, and of the same sign as, the vitreous electricity of the glass. Since however the external and internal distributions are in equilibrium separately by Art. 91, it follows that ttte intensity of the external superficial electrification at any point, unlike that of the corresponding internal electrification, will be entirely independent of the position of the glass, and will be determined by the given distributions in the field external to the vessel and the shape of the vessel. 93.] In Experiment VI the external electrifications of the vessels A and B are equal and opposite before the introduction of the wire. When the two vessels are connected by the wire, 95-] ELECTRICAL THEORY. 95 the two equal and opposite distributions coalesce, producing evi- dently by that means external equilibrium. The effect on either vessel is the same as if, there being- no introduction of the wire, it received an independent charge equal in amount to, and of the same sign as, that of the glass or resin in the other vessel, and therefore equal and opposite to that of the resin or glass within itself. ' These charges remain when the wire, and afterwards the glass and resin, are removed, as the experiment shows. 94.] The result of Experiment VII also follows at once from the same reasoning. For the external superficial charge on C is the same in whatever part of its interior IB be situated, and is equal to that of B in magnitude and of the same sign. If therefore B be made to touch C, the external electrification of the latter will not be affected, but inasmuch as C and B after contact may be regarded as constituting one conducting body, the vessel C with B in contact constitutes a metallic shell with a given internal distribution zero. Hence the internal superficial electrification must be zero, and there is no free electricity within the compound conductor C and B, and therefore the whole of B is discharged. 95.] We have hitherto considered cases of equilibrium in which certain conductors have given charges. It is sometimes required to determine the density of the induced distribution on a conductor or system of conductors placed in a known field of force ; as, for instance, when the force before the introduction of the conductors is uniform throughout the field, such as may be conceived to be due to an infinite quantity of electricity placed at an infinite distance from the conductors. Another class of problems is found when the potentials of certain conductors are given. When two conductors of known shapes are joined together by any conducting connection, the conductors with their connection of course form one compound conductor, and must be treated as such. In the particular case however of the connection be- tween them being a very thin wire, the total amount of elec- tricity on the surface of the wire must be very small, and generally is inappreciable in its effect upon the field. As far therefore as the electricity on the connection is con- 96 ELECTRICAL THEORY. [96. cerned, such conductors may be regarded as two separate and independent conductors of known form. ; the existence however of the connection will ensure that they are of the same potential. If in the case last mentioned, of two conducting- bodies joined by a thin wire, one of them be removed to a great distance from the field, the charge upon the one so removed will at length cease to exercise any appreciable effect, and may be neglected. If, at the same time, the potential of this removed conductor be maintained at any given value, we may by this contrivance regard the remaining conductor as an insulated conductor at a given potential. In order to effect this object the charge upon the conductor must be capable of variation. In fact, the distant conductor, or some other body connected with it, must be a reservoir containing infinite quantities of either kind of elec- tricity, and so large that the withdrawal of electricity necessary to maintain the given conductor at the required potential has no appreciable effect upon it. A very common case of such an arrangement occurs when one or more of the conductors of the field are connected by a thin wire with the earth, for this latter is an infinite conductor always at the same potential*, which is taken as zero, the potentials of all bodies being measured by their excess or defect above or below that of the earth. A conductor connected with the earth is said to be uninsulated. 98.] It follows from what has gone before that the most general problem of electrical equilibrium, in such a dielectric medium as we have described, is reduced to that of given electrical distributions in the presence of given insulated con- ductors with given charges, or at given potentials, in a dielectric medium of infinite extent. The solution of any such problem, that is, the determination of the electric density and potential at any point, involves the determination of a function F, the potential of the system, to satisfy the following conditions : * The earth for any distances within the limits of any experiment is at the same potential. But there may be differences in the potential of the earth between distant points, as England and America. 97-] ELECTRICAL THEORY. 97 (1) V has some (not given) constant values over each of the surfaces S ... S n bounding the conductors on which the charges are given. ( 2 ) - J/Jj ds i taken over S t = \ ; &c.; i (3) F" has given constant value over each of the surfaces $/. . . S' m bounding the conductors on which the potentials are given. (4) V 2 F-f 4?rp = at any point where there is fixed elec- tricity of density p. and of course, if such fixed electricity be what is called superficial, this may be put in the form dV dV _ + __ +4 ^=:0. (5) Y vanishes at an infinite distance. It was proved in Art. 1 that one such function always exists, and if it be T t a distribution of electricity over the surfaces of density JL^E 4-7T dv satisfies all the conditions of the problem. Then the equation dV +47TO-= dv determines the density of electricity at any point of the surface of any conductor, and the problem is completely solved. 97.] It was stated in Art. 91 that the fact of a charge of electricity on a hollow conducting shell causing no electrification on a conductor placed within it furnishes the most conclusive proof of the law of the inverse square in electric action. By hypothesis there is internal equilibrium when a distribution itself in equilibrium is placed on the outer surface of the shell. Let the outer surface be a sphere. Then by symmetry this distribution must be uniform. Let us take a- for the superficial VOL. i. H 98 ELECTRICAL THEORY. [98. density at any point, and since there must be a potential function, let it be ^ T ' at the distance r from a particle of unit elec- r tricity. Let P be any point within the shell at the distance p from the centre 0. Let the radius of the shell be a, and let be the angle between OP and the line drawn from to any point Q on the surface of the shell. Let dS be an elementary area of that surface in the neighbourhood of Q, and let V be the potential of the whole charge at P. Then r^f/V/W = 2wf* Jo o Also r 2 = a 2 2 ap cos +> 2 , rdr = apsm6d9; ra+p =2TT .'. 0=f(a+p)-f(a-p)', and the force = -- - = - dr r 2 Hence the inverse square must be the law of force necessary to satisfy the experimental data. 98.] It may be of interest to enquire within what degrees of accuracy the experiments which have been made may be depended upon. Let there be an insulated conducting spherical shell within and concentric with the given spherical shell, and of radius I. If the law of force were that mentioned, the charge on the 98.] ELECTRICAL THEORY. 99 smaller sphere would be accurately zero, even with the two spheres in conducting communication ; and, conversely, if the charge were accurately zero, the law of force must be that of the inverse square. If, however, the law of force differed slightly from that of the inverse square, there might be a small charge on the inner shell, and we propose to investigate the amount of this charge with any assumed small deviation from the above-mentioned law. Let the metallic communication between the surface of the inner sphere and the external surface of the outer sphere be made by a very thin wire, then the electricity on this wire may be neglected, and therefore, by symmetry, the charges on the two spheres must be uniformly distributed. And if the shells be very thin, we may, whatever be the law of force, regard the charges as superficial. Let E be that on the outer sphere, and W that on the inner. Lety(r) = C+m$(r) where m is small; i.e. let the law of force be where m is small compared with C. At any point P the potential from the two charges will be ~ I f(r)dr + - f(r)dr, J and this must be the same at the two ends of the wire. Therefore That is, But 100 LINES OF FORCE. [99. Therefore, substituting 1 and neglecting the products of the small magnitudes E' and m, we get a+b For example, suppose the law of force to be -g+^, where q is small. Then and / = rr7-H^ logr - Therefore C = > m = $ (r) = log r. Substituting in the expression for E', and remembering that / logrdr = rlogr r, we get This is the theoretical basis of the experiment by which Cavendish demonstrated the law of the inverse square. The experiment is given in great detail in the second edition of Maxwell's Electricity and Magnetism, pp. 76-82 ; and it appears, from what is there stated, that we may regard it as absolutely demonstrated that the arithmetical value of q cannot Lines of Force. 99.] The state of the electric field under any given distribution of charges and arrangement of conductors is completely known when the value of the potential at each point of the field has been determined. It is obvious however that the direct subject of experimental investigation in any case must be the magnitude and direction of the force at any point of the field, and hence has * See Senate House Questions, 1877. 99-] LINES OF FORCE. 101 arisen the conception of lines, tubes > and fluxes of force, originally suggested by Faraday and developed by subsequent writers. Line of Force. Suppose a sphere of indefinitely small radius to be charged with unit mass of positive electricity and placed with its centre at any given point P in an electric field, and suppose the electrical distribution of the rest of the field to be unaffected by the presence of this charged sphere, and suppose further the inertia of the sphere to be always neglected, then the centre of the small sphere would move through the field under the action of the electric forces of the field in a definite line, generally curved, this line is defined as the line of force in the field through P. tf */ When the electricity of the field consists of an electrified mass of very small volume, inclosing a point and therefore all sensibly situated at the point 0, the lines of force are clearly straight lines radiating from if the charge at be positive, and terminating in if the charge at be negative. If the point moved off to an infinite distance, and the charge at were infinitely increased, the field would become what is called a uniform field, and the lines of force would be parallel straight lines. So also if the distribution consisted of an infinite plane with a charge of uniform density over its surface, the lines of force would be parallel straight lines normal to the plane and proceeding from or towards that plane, according as the density thereon was positive or negative. If the distribution were that of uniform density on the surface of an infinite circular cylinder, the lines of force would be in parallel planes perpendicular to the axis of the cylinder, radiating from or converging to the point in which that axis met each of these planes according as the electrification of the cylinder was positive or negative. For less simple cases of distribution the lines of force are not capable of any such immediate determination ; they are generally curved lines, their direction at every point coinciding with the normal to the equipotential surface through that point and pro- ceeding towards the region of lower potential. It follows that 102 TUBES OF FORCE. [lOO. no line of force can be drawn between points at the same poten- tial, and that all lines of force in the immediate neighbourhood of an electrical particle, i. e. a very small volume with a charge of infinite density, must radiate from or to the point with which that volume sensibly coincides, according as the density of the charge is positive or negative, because the potential in the immediate neighbourhood of such point is positive or negative infinity in the respective cases. 100.] Tubes of Force. A region of space in the field bounded laterally by lines of force, as above described, is called a tube of force. See Fig. 6. When the transverse section of the region is indefinitely small it is called an elementary tube of force. Flux of Force. Suppose any transverse section dS made through any point P in the surface of an elementary tube of force, as in the figure, the angle between the normal to dS and the bounding lines of force being i. If the intensity of the force at dS be F 9 and the area of the orthogonal section of the tube at the point P be a, the force resolved perpendicular to dS will be .Fcos i, and if this be denoted by F n , the product F n dS will be equal to FdScosi, or Fa, and will be the same Fig. 7. for every transverse section of the tube in the neighbourhood of dS. This product, from its analogy to the flux of a fluid flowing through a small tube with velocity M=F, is called the flux of force across dS; the limiting value of the ratio of the flax offeree across any elementary area to the area is the intensity of the force in the field at that elementary area and perpendicular to it. When the distribution arises from a so-called charged particle, the tubes of force are conical surfaces with their vertex at the particle; when in a uniform field they are surfaces limited laterally by parallel straight lines, and so forth. 101.] Let a charge of electricity of either kind, and with mass 1 01.] FLUX OF FOKCE. 103 numerically equal to m, be situated at a given point 0. Let a sphere of any radius be described about as centre. Then the fluxes of force across all equal elementary areas of the sphere's surface will be equal to one another, and will take place from within outwards, or from without inwards, according as the electricity at is positive or negative, the total flux over the whole sphere being 4 irm. Faraday regarded the charge at as a source from which, or a sink towards which, lines of force proceed symmetrically in all directions, and he further regarded the density of these lines of force, or the number contained in each unit of solid angle at 0, as proportional to m. The number of lines of force therefore, which, in this view, traverse any surface, corresponds to the flux of force across that surface, and the force in any given direction at a point P in the field is the limiting value of the ratio which the number of lines traversing a small plane at P perpendicular to the given direction bears to the area of that plane when the latter is indefinitely diminished. If the point were eccentric, the equality of flux over all equal elementary areas would no longer be maintained, but the flux over the whole surface would, as we know from Art. 45, or as would result at once from the equality of flux over every transverse section at any point of an elementary tube of force, proved in Art. 100, still remain equal to lirm. We know also from Art. 45, or we might prove at once from Art. 100, that the total flux across a closed surface of any form surrounding would be 4 urn. If there were any number of sources or sinks within the closed surface, the traversing flux across the whole surface from each such source or sink would be 4irm, where m is the numerical value of the charge at such source or sink, and the flux is outwards or inwards according to the sign. The total flux in this case across the inclosing surface would be 4-7T (2jt? <**> Zn), where Sjt? and 2n are the sums of the charges of the sources and sinks respectively, and would be outwards or inwards according as 2/? was greater or less than 2#. If there were any number of sources or sinks in the field 104 FLUX OF FORCE. [lO2. external to the aforesaid surface, their existence would not affect the value of the total flux across the whole surface. 102.] Suppose that a tube of force, elementary or otherwise, in any electric field, is limited by transverse surfaces S and S' } and that it contains electrical distributions, such that the difference of the sums of the masses of the positive and negative charges is m, then the flux offeree across the whole surface of the tube thus closed from within outwards will exceed that from without in- wards by the quantity Ivm if the preponderating included electricity be positive, and the former flux will fall short of the latter by 4?m if the preponderating electricity be negative. But the flux of force across that portion of the tube's surface which contains the lines of force is zero. If therefore the direction of the lines of force be from S to f (see Fig. 8), the flux of force across S' will exceed or fall short of that across S by the quantity 4irm, according to the sign of the preponderating included electricity. If F and F' be the forces normal to S and & at any points in them respectively, and if m be now taken to represent the alge- braical sum of the included electricity, these statements are expressed by the equation / / FdS = 4:7rm. The portions of any surfaces in an electric field intercepted by the same tube of force are called corresponding surfaces, and there- fore in proceeding along any tube of force, finite or elementary, the fluxes across corresponding surfaces are continually increased by the quantity 4 irm, where m is the algebraic sum of the elec- tricities included in the tube in its passage from any one surface to any other, such increase being a numerical decrease when m is negative. And if there is no such included electricity, or if its algebraic sum is zero, then the fluxes across the corresponding surfaces are all equal to one another. 103.] Suppose that there is in the field a surface 8 charged with electricity, the density at any point P being a. 1 03-] FLUX OF FORCE. 105 Let dS be an element of 8 about the point P, and conceive a small cylinder to be drawn with its generating 1 lines passing through the contour of dS and perpendicular to that element. The total flux across this cylinder must be equal to the included elec- Fig. 9. tricity, i.e. to adS. Also, if the length of the cylinder's axis be indefinitely dimin- ished, the flux across the curved surface will become infinitely less than either of the fluxes across the bounding planes, and these fluxes therefore must ultimately differ from one another by 4770-^$, so that if N and N' be the forces in the field normal to dS and on opposite sides of it, we have N'dS-NdS=4.Tt OP 1 I d I therefore ,=___._; and if k be the distance of from the plane, r the distance of a point P in the plane from the intersection of 0(7 with the plane, J_ ~ 2irdh 1 h 27T OP 3 ' which determines the density at any point in the plane. 106.] In certain very simple cases the value of Fmay be deter- mined by the integration of Laplace's equation. For instance Two infinite conducting planes at given potentials. Let the planes be parallel to the plane of xy. Then, since the density is uniform throughout each plane, T is in this case a function of d 2 V z only, and Laplace's equation becomes -^ = 0, from which V can be found with two arbitrary constants, and the constants are to be determined by the given conditions on the planes. 110 PARTICULAR CASES. [107. (1) Two infinite coaxal cylinders. In like manner, if we have two infinite coaxal conducting cylinders at given potentials, the density is uniform throughout the surface of each cylinder, and V is a function of r, the distance from the axis. Laplace's equation is in this case IdV _ ~dr*+~r~dr" which admits of integration. (2) Two concentric spheres. Again, if there be two concentric conducting spheres at given potentials, the density is uniform throughout the surface of each sphere, and V is a function of r, the distance from the centre. Laplace's equation becomes in this case dW 2^T_ dr* + r dr = which admits of integration. In this problem, as in the last, the two arbitrary constants which enter into V in solving the differential equation must be determined with reference to the given conditions on the cylinders or spheres. 107.] Case of an insulated Conducting Sphere in a Field of Uniform Force. Let us take the direction of the force for axis of #. Let X be the force, a the radius of the sphere, V the potential. Then V must satisfy the conditions, (1) F is constant and = C on the surface of the sphere ; (2) V 2 F = at all points outside of it j (3) ^ r== Xx + C at a sufficiently great distance from the sphere ; (4) The total electrification on the sphere is zero. The function where / is the distance of any point from the centre, satisfies all these conditions. I08.] PARTICULAR CASES. Ill / t The density on the sphere is - ^-, that is, 4?r dr ' 4-na It is easily seen that 108.] Case of an uninsulated Conducting Sphere and another Sphere outside of it uniformly filled with electricity of density p. This is the same problem as that treated in Chap. Ill, Art. 65. We give another method of solution. Let C be the centre, a the radius, of the conducting sphere ; and let be the centre, b the radius, of the other sphere. Let OC=f. Let V be the potential of Fig. n. the whole system. It is required to find the density of the induced distribution on the conducting sphere which gives zero potential on that sphere, and the general value of V in this case. 7 has to satisfy the conditions, (1) 7 at all points on the conducting sphere ; (2) V 2 7= at all points external to both spheres ; (3) V 2 F+ 4-7T/) = within the non-conducting sphere. a z Take a point E in CO such that EC = -^ t/ Let OP = r, EP = /, where P is any point. Let V Q be the potential of the charged sphere at P. Then if e - 3 p, or the total charge of electricity in the charged 3 sphere, the function satisfies all the conditions, and must therefore be the required potential. 112 PARTICULAR CASES. [108. For outside of the ch above equation becomes M For outside of the charged sphere F = - , and therefore the r*j _-(. Now by a known property of the sphere, if the point P be on its surface, EP _ r' _ a OP~ r ""/" Therefore for a point on the conducting sphere F= 0. - Also for a point outside of both spheres V 2 - = and V 2 4 = ; therefore V 2 F= 0. For a point inside of the charged sphere V 2 F+4irp = V 2 -^ + 47r/3 = 0. The density at any point on the conducting sphere is In dv ( r fr' Also r z =f* + v*-2fvcos0, where the angle PCO = 6, and v denotes the distance of a point from C\ also and in the expression for cr, v is to be made equal to a after differentiation. We have therefore dr afcoa0 ___ == - , dv r a? _ dv a -- cos 6 e (afcosO a f but -^{ f / " a Q -P& j2 Therefore = 0. The density at any point on the conducting, cylinder is found from R c d d I , 27T ( dv dv ) where r 2 =/ 2 + v 2 -2fv cos (9, and i; is to be made equal to a after differentiation. The result is /2_ a 2 "~ 110.] Ow Electric Images. We have seen in Art. 108 that if a sphere be at zero potential under the influence of a charged point outside of it, the induced distribution has at all external points the same potential as that due to a certain charge placed at a point within the sphere, and the last-mentioned charged point is defined to be the image of the influencing point in the sphere. An infinite plane is for this purpose a particular case of the sphere. III.] ELECTRICAL IMAGES. 115 Every electrical system outside of a sphere, inasmuch as it may be regarded as consisting of a number of charged points, is re- presented by a series of images in the sphere, and these together may be said to form the image of the external system. In like manner, if the sphere be at zero potential under the influence of a charged point within it, the induced distribution has the same potential at all internal points as that due to a certain charge at a certain point without the sphere. The external point is called the image of the internal point. Every electrified system within the sphere has its image outside of the sphere. It can easily be shewn that no closed surface except a sphere or infinite plane generally gives rise to an image. For let 8 be any uninsulated closed surface, and let E be an external point at which a charge e is placed. If the induced distribution on S have at all points on S the same potential as that of a charge / at a point .F within S 9 that is, if F be an image of E within S, we must have EP _e FP ~?' P being any point on 8. Thus the locus of P is a sphere, that is, S is a sphere. 111.] By the method of electric images many problems re- lating to the distribution of electricity on spherical or plane surfaces can be solved. The case of two spheres cutting each other orthogonally (Max- well's Electricity and Magnetism, p. 168). Let C lt C 2 be the centres, a l} a 2 the radii of the spheres. Let AB represent the circle of inter- section, E the point in which the line C L C 2 intersects the plane of that circle. Then C^AC^ C^BC^ are right angles, and f = a 1 > + a . Also = ) 116 ELECTRICAL IMAGES. [l I 2. or E is the image of C 2 in the sphere C 1 , and the image of (7j in the sphere C 2 . If therefore we place at C l a quantity of electricity a^ , at (? 2 a quantity a 2 , and at E a quantity the potential at any point on either sphere will be unity, because if the point be, for instance, on the sphere whose distance from A is -7^, and its distance from H is Ji -j^ That produces on the plane a distribution whose density we may denote by ^ ; and the effect of this distri- bution over the plane at all points on the left side of the plane rt is the same as that of its image, namely, a charge r-=; e placed at a point distant k ^ to the right of the plane. Let i " = *~A; From this distribution, or its equivalent image, we derive in the same way a second distribution on the sphere equivalent to a charge c c ~~AE' ' 120 SUCCESSIVE IMAGES. c 2 at a point distant -7 - from A, and from this again a second n-\- X-^ distribution of density p 2 on the plane. We shall then have a series of images to the right of the plane, whose distances from IT are # 15 # 2 , &c. And c 2 and generally n? w+1 = h -t &c. n-rx n It is easily seen that # n+1 > # n , and every a? is less than _ C 2 > ip ne success i ve images continually approach A'. The charges at these images are successively c c c c c c _ _ _ AE ) AE \+*C AE and the ratio between two successive charges continually approaches -T-J> A.A. Again, the charge at E induces on the plane a primary dis- tribution which is equivalent to the image of E in the plane. This original image is at a point distant from H 9 os\ HE, and the distances from H of the derived images are &c., &c. which continually approach HA'. The charges at these images are successively e at the first image, = - e at the second image, h + x^ and so on. Hence the density of the induced distribution at any point M on the plane, where HM = r t is "50 e c f SUCCESSIVE IMAGES. yX QGn 121 \ + &C. Each series converges rapidly, and the terms soon cease to differ sensibly from those of a geometric series whose common /> ratio is -, Hence the actual density at M can be calculated A.A. to any required degree of accuracy. The integral charge on the plane is the sum of both series of images irrespective of their position. That is AE -e\l + 115.] Another very interesting case is that of two concentric spheres and an electrified point placed between them, treated in Maxwell's Electricity. In that case the distances of the images from the common centre are in geometrical progres- sion. Also the charges are in geometrical progression, and their sum can be accu- rately determined. Let be the common centre, a the radius of the inner sphere, b the radius Fig. 15-. of the outer sphere, E the point where the charge e is placed, OE = k\ all the images are in the line OE produced. We have then 6 2 an image at P lt where OP 1 = T an image at Q lt where OQj_ = -^p = -TT ' an image at P z , where OP Z = -^TJ an image at Q t , where 00 2 = -^ = Tj-A. UJTn 122 SUCCESSIVE IMAGES. [1-16. We see then that the distances from of the successive images derived from the primary distribution on the outer sphere are and the charges at those images are o a a* b e ' v e ' &c - Again, if we start with the primary distribution on the inner sphere, represented by -7 e a ^ Qi, we obtain a second series of fa images whose distances from are a in b*h and whose values are a 3 Hence the total charge on the inner sphere, or the sum of the images within it, is f a ab v VT ~ ~" zTTi ~\ ) e > or b a h(b a) h b a h b-a ' and that on the outer sphere is hb a ha e. h ba h ba 116.] Another class of cases is that in which the number of images is finite. For instance, let us consider Y two infinite conducting planes at right angles to each other, in presence of an electrified point. Let the projections of the X' ^ X planes on the plane of the paper ' be XEX\ YET, and let O l be an electrified point. The image of O l in XEX f is Q. Y' The image of (7 X in YEY' is 2 . Fig. 16. The image of 2 in XEX' is C 2 . The image of <7 2 in YEY' is . o, 1 1 7.] SUCCESSIVE IMAGES. 123 The two planes are at zero potential under the influence of + e at O l and 2 , and e at Q and C 2 . If we now substitute for dr be the quantity of electricity in the space-element of the original system, we will place in the corresponding space element of the inverted system the quantity r Since as we have seen the element of volume dv in the original K 6 system becomes dv in the inverted system, it follows that r r 5 K 5 the volume density in the new system is r p = . o. J J K 6 r 5 122.] INVERSION GENERAL THEORY. 127 In like manner if of the original system is represented geo- metrically by a surface S' in the new system, and every quantity of electricity in the original system is represented by a corre- sponding quantity in the new system. 121.] We now proceed to find the relation between the potential at any point Q of the original system and that at the corresponding point Q'j due to the electricity which we have supposed placed on the inverted system. Let s denote an element of volume at P in the original system, ps the quantity of electricity in it. Then the potential at Q of s\ a the element is v = In the inverted system, ps at P becomes -^p P s its potential at Q' is- K v = but 1 _ OP. OQ 1 P'Q' = ~^" PQ 5 , OQ ps OQ whence v = ^- -77= K fty K As this is independent of the position of P and P', it holds true for the whole potential of the original system at Q, and of the inverted system at Q'. That is, if 7 and V denote the potentials at Q and Q', 122.] It follows that if Tbe zero for any conductor whose bound- ing surface is S in the original system, 7' is zero throughout the 128 INVERSION GENERAL THEORY. [l22. corresponding surface S' in the inverted system. Therefore if the space S' be occupied by a conductor, the assumed distribution of electricity throughout the inverted system will, as regards such conductor, be in equilibrium with zero potential. And if any electrical system consists of conductors all at zero poten- tial in presence of fixed charges of electricity, the inverted system will also be in equilibrium with all its conductors at zero po- tential. Again, let the original system be one in which the potential of a distribution over a closed surface S is equal at each point on S to that of any electrification enclosed within S. Then if we invert with respect to an external point, and S becomes S', the potential of the corresponding surface distribution over S' will be equal at each point of S' to that of the corresponding enclosed electrification. If, for instance, the distribution on S have the same potential in all external space as if it were col- lected at a point C within S, that is, if the original system be a centrobaric shell, the surface distribution over S' will have the same potential in all space outside of &, as if it were collected at (f, the point corresponding to C ; that is, the new system will be a centrobaric shell too. If in any system V be not zero for the conductor S, V is not generally constant over ', and the inverted system will not be in equilibrium with & for a conductor. But, as we have seen, F' F = If therefore we place at the centre of inversion a charge * F, the potential of this charge, together with that of the inverted system, will be zero at each point on &. If therefore we have given a conductor S, and know the density at every point on its surface of an equipotential dis- tribution giving potential F, we can, by inverting the conductor so electrified with any point for centre and K for radius of inversion, find the density of the distribution over S' required to give zero potential in presence of a charge K V at ; namely, if a be the density at any point P on the original conductor, the > 124.] INVERSION PARTICULAR CASES. density at the corresponding point J? of the distribution giving zero potential is ^ if C> t> e external, or ^ -^- a if be internal. And f a a / according to the general result above proved, the distribution on the new sphere will be such as together with a charge K V> or liTKaa, at will give zero potential at each point of the inverted sphere. But if dA be an elementary area of the original sphere distant r from 0, ad A is the charge upon it in the original system. The charge upon the corresponding area in the new sphere will be adA 9 and dA becomes -^dA. There- fore the density at the corresponding point of the new sphere is r 3 K 3 adA or -^- o-dA, that is, it varies inversely as the cube of K* 7* the distance from 0. Again, being without the original sphere, let /c 2 =f 2 a 2 , then the sphere does not change its position. Let the charge at 0, or 4 TT K a a = e, or o- = - .- . 47TKa Then the density for zero potential is 7T 6 ^2 __ ft '* or as we have already found by different methods. 124.] Again, if at the centre of the original sphere there be placed a quantity of electricity 477& 2 the four points from the plane XEX', x the distance of any one of them from the plane YEY'. If we invert the system with O x for centre and K for radius, the two infinite planes become two orthogonally intersecting spheres. The common section of the planes becomes the circle of intersection of the spheres and passes through O lt The plane XEX' becomes a sphere whose centre is /, the point K 2 corresponding to C and whose radius is a 2 = - %/ Similarly the plane YEY becomes a sphere whose centre is K 2 C 2 ' and whose radius is a = The portion XEY' of the two infinite planes becomes on inver- sion the figure formed of the two outer segments of the spheres. Similarly X'EY becomes the lens formed of the two inner seg- ments, and XEY, or X'EY', becomes a meniscus formed of the outer segment of one and the inner segment of the other sphere. (See Fig. 19.) K 2 K 2 The charge at <7/ is or a^ , and the charge at C{ is or / a 2 , and the charge at 2 is that is, 132 INVERSION PARTICULAR CASES. [127. .Fig. 20. We obtain therefore the system already treated in Art. 112. Further, if before inversion we substitute x for the charges at C^ and 2 their equiva- lent distributions on the plane XEX', and for C 2 its equivalent distributions on YEY', these densities on XEX' and YEY' will in the inverted system give unit po- tential on XEY' ', and are the same which we found by a different method in Art. 112. 127.] Again, instead of two infinite planes, let there be 2n infinite planes, having a common section E and making with each other the angle - n Let there be n negative points : ... O n each having charge K, and n positive points C ... C n each having charge -+-K, all at the same distance from E and placed alternately, so that each negative point is the image of the next positive point in the plane between them. Then all the planes are at zero potential. Let YEY' and SES' be two adjacent planes. Let the n points on the left of YEY' be replaced by the corresponding distri- butions on YEY, and the n\ points on the right of YEY', that Fig. 21. Fig. 22. is, all the points on that side except 1 , be replaced by their corresponding distribution on SES'. Then the portions STEY are at zero potential. When we invert the system with respect to O lt KEY' becomes I2Q.] INVERSION PARTICULAR CASES. 133 the figure formed by the outer segments of two spheres inter- secting at the angle n The density at P', any point on the outer segment of the OP 3 C P 3 sphere corresponding to YEY', is -^-30-, that is, ^- o-, where > K or is the density at P, the corresponding point to P f , of the distribution on YEY', which can be determined without much difficulty. 128.] Returning to the conductor XET'O of Art. 126, with its surface distribution above determined in Art. 112, let us invert the system, taking for centre of inversion a point on the internal segment of the sphere C r The sphere C 2 becomes then another sphere, and the sphere C-L an infinite plane cutting the inverted sphere C 2 orthogonally, that is, a diametral plane, and the external segment OXE be- comes the portion of that infinite plane which lies within the new sphere C 2 . So that the figure XEY'O becomes on inversion the closed surface formed by a hemisphere and its diametral plane. Let P be a point on the outer segment of the sphere C 19 P' the point on the diametral plane which corresponds to P. And o- being the density above found for P, namely the density at P 7 required to give zero potential under the influence of a charge unity at a point within the hemisphere is 2 (T. OP* 3 129.] The construction for finding o- in terms of known quantities on the hemisphere will be as follows. Let C be the centre of the hemisphere, a its radius, the point where the unit charge is placed. Then by inverting the system with respect to 0, we shall reconstruct the original figure of two orthogonally intersecting spheres, which by its inversion gave rise to the existing system of the hemisphere and plane. Let the diameter, and the potential at M due to the disc under the influence of a charge OA at is QA 2 /Q _r, (d) 138 INVERSION PARTICULAR CASES. [134. Again, the potential of the bowl so influenced at any point P on the remaining segment of the sphere whose colatitude is 9 is OA F, where 7 is the potential of the uninverted disc at the point in its plane which on inversion becomes P. And IT 2 . NA 7= -tan l 2~ tan 2 134.] We have thus dealt with the case of a conducting circular disc, which may be regarded as part of an infinite plane of which the infinite external part is non-conducting. We will now take the converse problem, namely, that of a circular non-conducting disc of radius a, the infinite external portion of the plane of the disc being a conductor. Let it be required to find the density at a point on the conducting plane when that plane is at zero potential under the influence of a charge at a point in the non-conducting disc. In order to solve this problem we will invert the conducting disc when at unit potential as before determined, with respect to a point in itself distant /from Cihe centre, and with Va 2 f 2 for radius of in- version. The disc then becomes the infinite external plane, and the infinite plane becomes a disc, the boundary between the two after inversion being a circle of radius a, and whose centre Cf is distant f from on the opposite side to C. Let P be a point in the plane outside of the new circle, P f the point within the original conducting Fig. 26. disc which on inversion becomes P. Then the density at P, when the infinite plane is at zero potential under the influence of a charge K at 0, is K* ! Let C'P - r, LPG'O = 0. I3S-] INVERSION PARTICULAR CASES. 139 Then a ^ and the density at P is a 2 -/ 2 1 1 ~ r ' The density at P due to a uniform ring of electricity of density 1 in the plane of the disc distant from C', /.../-f df is 2 >/^7* t _f.df (*" d0 V* ' yV-a 2 Vo r 2 +/ 2 -2r/cos0 _ 2 >/a 2 -/ 2 /rff The aggregate of the distribution over the plane due to any electricity m in the disc distant /"from the centre is m . T 00 2-nrdr : - m. (h) 135.] If the whole non-conducting disc be covered with electricity of density 1, the density at P in the surrounding plane when at zero potential under that influence is 2 TT vV-a ; 2 C a or w (./ r 2_ a 2 2 Now let the entire plane, including the non-conducting disc, be covered with a uniform stratum of density + 1 . There will then be zero density on the non-conducting disc, which may therefore be regarded as a circular aperture, and the conducting plane will have constant potential, and the density at any point upon it distant r from the centre of the disc is This differs from the constant density + 1 by 2 f a a j - tan ~^^r it If therefore we have an infinite conducting plane with a circular aperture, the total^charge that must be placed upon the plane in 140 INVERSION PARTICULAK CASES. [ I 3^- order to bring it to the same potential as a complete plane would have when coated with density -f 1, is / ardr C> a 4 /-, ^~ 4 / ^tan- 1 == = ^a 2 , (t) j a yr z a 2 ^/a Vr 2 a 2 or the same quantity which would be removed from the infinite plane so coated in the act of making the aperture. 136.] We will now proceed to Sir W. Thomson's problem, to find the density of electricity on a spherical bowl, or portion of a sphere cut off by a circle, when at unit potential under the influence of its own charge alone. In order most easily to effect this, let us recur to the non-conducting disc and infinite external con- ducting plane, and instead of the density of electricity on the disc being uniform, let the density at P be 3 > where Q is a point in the axis of the disc distant Ji from the centre, and P any point within the disc distant f from the centre. Then 1 The density at any point in the conducting plane, when at zero potential under the influence of this distribution, is 2 W TI Vr 2 , 1/1 Let / = h cot > a = K cot r = Ti cot 2 2 . _-" -7T (3 ( 1 77 _^ "2* 2~2' 2~2 are the angles subtended at Q by/, #, and r respectively. The integral then becomes cot cosec 2 - A/ cot 2 cot 2 9 ~ c t 2 ^) cosec 3 - 1 3 7.] INVERSION PARTICULAR CASES. 141 This may be put in the form 2 T sin a A/COS /3 cos a sm *"" cos cos a 2 Let v cos (3 cos a = x, cos /3 cos a = x z , sin a da = Then when a =?r, a; = \/cos /3 + 1, and when a = (3, x = 0. Then [da. sm A/COS ^ - cos ^ _ g f ^ + 1 ^ J/3 cos 6 cos a Jo cos0 cosft + x* . _ ^ 7S ?' cos ^ cos ^3 J Hence the density is 2 . 3 ( / cos/3+1 / cos +1 ) sm 8 -JA / tan" 1 A / > (j) 137.] Let us now again invert the system, taking Q for centre of inversion, and k for radius. The infinite plane becomes a sphere whose diameter is Q(7, or h. The infinite conducting plane outside of the disc becomes a spherical bowl, cut off by a circle at right angles to QC, and whose colatitude measured from the pole Q is /3. (See Fig. 27.) The density on the remaining segment of the sphere, which before inversion was 7,2 or zero. h, fi Therefore p + - \ sin0d50 h = (/3 + sin/3). 27T The capacity of the bowl formed by the other segment of the sphere is h Hence we see that if a sphere be divided by a plane into any two parts, the sum of the capacities of the two parts exceeds the capacity of the sphere by the capacity of a circular disc coin- ciding with the intercepted plane. If the bowl be hemispherical the capacity is | > a being the radius, or the arithmetic mean between the capacities of the sphere and disc of the same radius. 140.] Recurring to Art. 138, let us next place at the centre of the sphere of which the bowl forms part, a charge . This 2 will reduce the potential of the bowl to zero. 141.] INVERSION PARTICULAE CASES. 143 We may now again invert the system so formed, taking for centre of inversion any point whether in the spherical surface or not, distant / from the centre. In that case, if be not on the surface, the sphere becomes a new sphere, and the bowl becomes a new bowl. In the particular case of being on the original spherical surface the new sphere is an infinite plane, and the new bowl a circular disc upon it. The centre of the original sphere becomes (7, the image of in the new sphere; and the charge at the centre becomes a f J) charge - at the image. K 2i If r be the distance from of a point P on the new bowl, o- the density of the equipotential distribution, as found above, at K 3 the corresponding point of the original bowl, then 5- a- is the density at P of the distribution on the new bowl which gives / 7 zero potential in presence of - at (7. K We can therefore give the following rule for finding the density at any point on a spherical bowl under the influence of an electrified point not on the surface of the sphere. First, find (7, the image of in the sphere. Secondly, suppose the system inverted with respect to (7, and a new bowl so formed, and let /3 be the colatitude of the rim of the supposed new bowl, and let be the colatitude of P', the point on the supposed bowl corresponding to P on the given bowl. Then we know cr, the density at P' of the equipotential distribution on the supposed bowl, as a function of (5 and 6. And if r be the distance of P from (7, the density at P is proportional to 141.] On the effect of malting a small hole in a spherical or infinite plane conductor. The above results enable us to estimate some of the effects of making a small circular aperture in a conductor otherwise spherical. 144 INVERSION PARTICULAR CASES. [141. For instance, let /3 = TT y, where y is a very small angle. The spherical bowl becomes then a spherical conductor of radius > with a small aperture whose radius is - y. Its capacity is h . n . , . . Ti h y sin y -(/3 + sm/3), that is --.-.ZZ; The capacity of the complete sphere is - . We see then that 2t the effect of making an aperture, whose radius subtends at the centre the small angle y, is to diminish the capacity by h y sin y 2 ' ~^~ If A be the area of the aperture, we may write, neglecting higher powers than y 3 , h y sin y A% 2 '- = x -' where A = ' Again, the conductor being charged to unit potential, the density at a point whose colatitude is (less than /3) is COS / 3 _ t COS0-COS/3 Now - =- is the uniform density which would give unit 271% potential on the complete sphere. The term J_ 5 / cos/3+1 ^ / cos/3+1 ) -7r 2 A( / V cos^-cos^ " "V cos ^- cos /3) expresses the density due to the existence of the aperture. The total quantity of the distribution due to the aperture on a ring between the parallels of and -f d& is g sm0d0 2 " ( / cos/3+1 _ a / cos /3+1 J IV cos 6- cos ft V cos0-cos/3f Now unless be very nearly equal to TT, not only does cos/3+1 cos cos /3 141.] INVERSION PARTICULAR CASES. 145 itself become very small, but also it tends to vanish in a ratio of equality with / tan" 1 A / cos ^ + 1 V cos cos/3 Hence if O l be a value of which is, and 2 a value which is not, nearly equal to (3, it is easily seen that the quantity of the distribution due to the aperture on the ring between 1 and 2 is very small compared with that on the ring between /3 and 1 . The distribution due to the aperture has therefore the same effect as if it were all collected on the aperture. For instance C* C / cos/3 + 1 / cos +1 J. W COS0-COB/3 - tan V coBtf-eoB **** and is independent of 0. The system is therefore equivalent to a complete sphere charged to unit potential, that is, having a uniform density on its surface, together with the additional charge 2 7f n h y siny 2 TT on the aperture. This quantity h y siny . A% -2 * ' or ~ x i?' shall be called the abnormal charge, since it constitutes the difference between the capacities of the perfect and the im- perfect sphere. Let P be any external point distant r from the centre of the sphere, and / from the centre of the aperture. Then the po- tential at P of the charged sphere is h h y siny 1 or is the potential of the perfect sphere, together with that of the abnormal charge Ti y siny ~2 7T~~ placed on the aperture. VOL. i. ^ 146 INVERSION PARTICULAK CASES. [142. 142.] Let us now invert the charged conductor, taking for centre of inversion a point 0. On the imperfect sphere when charged to unit potential, such point not being very near the aperture. The sphere becomes then an infinite plane with a circular aperture, at zero potential under the influence of unit charge at 0. And the potential at any point P on the opposite side of the plane to 0, instead of being zero, is that due to a small positive charge upon the aperture. These results, which are accurately true in the limit as the aperture vanishes, are approximately true for a sphere whenever the aperture subtends a very small angle at the influencing point. To find the effect of a large aperture it would be necessary to find the potential at any point due to a spherical bowl charged to unit potential, when /3 is not nearly equal to TT. This might be done approximately by the method of Art. 61, or otherwise. CHAPTEE VIII. CONJUGATE FUNCTIONS AND ELECTRICAL SYSTEMS IN TWO DIMENSIONS. ARTICLE 143.] Let there be an infinite cylinder whose axis is parallel to the axis of 2, and whose section is the element of area das dy, cutting the plane of xy in the point #, y. Let this cylinder be charged with electricity of uniform density p, so that p is independent of z, but is a function of as and y. In like manner we may conceive an infinite cylindrical surface whose axis is parallel to that of z, having a- for surface density of electricity, constant along any infinite line parallel to the axis, so that , the potential at Q is and vanishes if Q be removed to an infinite distance. It will be understood in this chapter that the potential does so vanish, and therefore that the algebraic sum of all the electricity in the system of which we treat is zero. 144.] Let us suppose then that in such a system there are certain conductors whose equations are /i (*, y) = 0, f 2 (x, y) = 0, &c., and given charges are placed upon them ; and also certain fixed charges on given points or lines of the system. Let us further suppose that we have by any method obtained the solution of this electrical problem : that is, we have found the single function, V, of x and y, which is constant within all the con- ductors, and satisfies Poisson's equations at every point in free space, and dV t dV 5 -- H ~T~f = 47TO- dv dv at every curve charged with electricity; and by consequence we have determined the density at any point on any of the con- ductors. The solution so found contains implicitly the solution of a I45-] CONJUGATE FUNCTIONS. 149 class of problems ; all those namely that can be formed from the given one by substituting f (oc,y) and *}(%, y] for x and y, (x,y) and 77 (x,y}> or shortly and 77, being functions of x and y having a certain property. 145.] For let f and rj be so chosen that d drj ~T~ ~ ~T~ ' ax ay __. dy dx' and rj are then defined to be CONJUGATE TO x AND y. It follows immediately that __ (Za? dx dy dy By the ordinary formula for change of independent variables we know that dv d dx __ dy_ _ dx __ dy _ d~ddrj dgdrj' drj ~~ d dr] dr] d dx dy dy dx dy dx dy dx -ir 7 / dfdrj = (-^--r 1 -^ ^ dy dx Also =, and ? = - in this case, dy dx dx dy dx and ddr\ }j?dxdy. Again, if F be any given function of x and y, we have by ordinary differentiations, d*V_d*V dj[~ ^dj[d^ 3W ( *n**V_ ^!f,^^, dtf~~~d' l( dx> ' ddi}dxdx* dif'^dx' ^df dx* dr) dx* d*V d*V.d* d*V ddri o*V fy.dV d^ dVd^n dr l *' { dy ) + df df * d^ df 150 CONJUGATE FUNCTIONS. [146. Therefore, remembering that cZf ofy cZf in the curve F(^ rj) at the point P corresponding to P in the second plane, i.e. it is the tangent of the angle between the curve F(, 77) = at P' and the curve 77 = const. through P 7 , since the curves 77 = const.,, f = const., intersect everywhere at right angles. Also, if dA be any elementary area dxdy in the original plane, we have dA = dxdy = ddrj. But from the last article and therefore if dA' be the elementary area in the second plane corresponding to dA in the first plane, we have dA 152 ELECTRICAL SYSTEMS IN TWO DIMENSIONS. [147. And in like manner the length of any elementary line in the second plane corresponding to the element dv in the first plane may be proved to be 147.] Suppose now that we have any given electrical system of two dimensions in equilibrium in the original plane of oo, y^ with conductors whose bounding equations are given by closed curves of the form / (x, y) 0, the algebraic sum of all the electricity being zero. Construct in the new plane of #', y' a system of corresponding curves /(, 77) = 0, and for every linear or superficial charge in the original plane of #,y, place the same linear or superficial charge upon the corresponding lines and areas in the new plane of of, y'\ then the electrical system so formed in the plane of x' , y' will be a system of two dimensions in equilibrium with conductors bounded by the corresponding closed curves to the original curves in the plane of os,y. And the potential V at any point P in the old plane of as, y will be equal to the potential V at the corresponding point P f in the new plane of #', y'. For since the total charges on corresponding superficial areas are the same, but the areas themselves are in the ratio of \j? to 1 , it follows that if p be the surface density at any point in the old plane, and p' that at the corresponding point in the new plane, then p f = /u 2 /o, and similarly if a- and ' 2 r *' By the principle of superposition, if e l were increased in any ratio, T lt V% . . . V n would be increased in the same ratio. It follows that we may express F lt F 2 . . . V n in terms of e 1 in the form V 1 = A n e 1 , r a = A a e lt &Q.; where A n , A 12 , &c. are coefficients depending only on the forms and positions of the conductors. In like manner if C 2 received a charge e 2 , all the others being uncharged, we should have Y _ A t> V A t> 8ra ' i < "*H e 2' r 2 22 e 2> * XlUt > the coefficients being again dependent on the forms and position of the conductors. By the principle of superposition, if at the same time C l receive a charge e lt and C 2 a charge 2 , the others remaining uncharged, we shall have == - e "*" ^ e &c.= &c., 'n == -"in e i "t" -" 160 SYSTEMS OF CONDUCTOKS. And, generally, if the conductors all receive charges e lt e z ... e the potentials will be expressed by the linear equations &c.= . The coefficients A are called the coefficients of potential. 156.] Evidently there exist algebraic values of V corresponding to any assigned values of e lf e 2 ... e ni though we do not assert that it is practically possible to charge the conductors without limit. By solving the above linear equations we should obtain a new set expressing the charges in terms of the potentials, namely, &C.= &C, " e n =B ln V l + B, n V,+ ..+B nn V n -, > in which the coefficients B are functions depending only on the forms and positions of the conductors. Since the equations (B) must give possible and determinate values of e for any assigned values of V lt F" 2 ... V n , it follows that there must exist a set of charges corresponding algebraically to any assigned set of values of the potentials. The capacity of a conductor in presence of any other conductors is the charge upon it required to raise it to unit potential, when all the other conductors have potential zero. Thus, if F 2 ... V n are all zero, we have from equation (B), 4-4*?,; and if Fj = 1, e l = B u , so that B n is the capacity of F = 4 M i. &c.; in which 7 1 is positive, and 7 2 . . . V n lie between 7 l and zero, and are therefore all positive. Hence also. A u is greater than A 12 ... or A ln , and each of these latter is positive. 159.] Properties of the Coefficients of Capacity and Induction, For let V denote the general value of the potential, when C^ is charged and has potential K, and all the other conductors are uninsulated. And let U denote the general value of the potential when C 2 is charged and has potential K, and all the other 162.] OF CAPACITY AND INDUCTION. 163 conductors are uninsulated. The equation of Art. 157 then becomes or 21 . = 12 , or B n = B lt . In other words, the charge on C if uninsulated when C 2 is raised to unit potential is equal to the charge on C 2 if uninsulated when CL is raised to unit potential, all the other conductors being in either case uninsulated. 160.] Each of the coefficients of capacity is positive. For as we have seen it is possible so to charge the conductors as to make 7 2 ...7 n each zero. Then V l must be either the greatest or least potential in the field, viz. the greatest if e l be positive, the least if e-^ be negative. Therefore, we have in this case % = J& u F,p and since e^ and F 1 are of the same sign, _Z? n is positive. 161.] Each of the coefficients of induction is negative. If F 2 ...7 n are all zero and e l not zero, the density on each of the conductors C 2 . . . C n must be of the same sign throughout its surface, viz. opposite to that of e 1 ; for let e 1 be positive, then if the density at any point on any other conductor were positive, there would be a less potential than zero, that is, less than that of any of the conductors, at some point in free space. But the charge on C 2 is in this case * 2 = A 2 ^> and since V^ is positive and e% negative, I> 12 is negative. 162.] The sum of the coefficient of capacity and all the co- efficients of induction relating to the same conductor is positive. For let the conductors be so charged as to be all at the same potential V. Then Now if V be positive, e 1 must be positive, for if it were negative, there would be a greater potential than V somewhere in free space. Therefore B n + B^ + . . . -f B nl is positive. M 2 164 PROPERTIES OF CAPACITY AND INDUCTION. [163. 163.] If two conductors (7 15 where A rs is a coefficient depending, like the coefficients A already investigated, on the forms and positions of the con- ductors and the position of the element in question, and the suffix s relates to that element, and r to the conductor. Similarly the potential at the element due to a charge e on C r will be A sr e, where A sr depends only on the form and position of the conductors and the position of the element. Also the equality of A rs and A sr follows from that of 2 Ve' and S V'e above proved. Thus the systems of equations (A) and (B) of Arts. 155, 156 can be extended to any electrified system, whether consisting exclusively of conductors of finite size or not. Evidently if in the equation E J 2 Ve we express every V in terms of the charges by means of equations (A), E will be a quadratic function of the charges with coefficients depending on the forms and positions of the conductors. In this form we shall write it E e . Similarly if we express every e in terms of the potentials by means of equations (B), E will be a quadratic function of the potentials. In this form we shall write it Ey. It follows from the equality of A J2 and A zl , &c., that **- r d _l-y &c d^ ' ' K " de, ~ or generally, = = V. (JL& For as from E = \ 2 Ve we have dE dE dE dV dE dV, ~ ) ~ + () ~' & and - = \V -- 170 MECHANICAL ACTION BETWEEN t 171 -I -I -I therefore -^ = -V, + -A^e^ -A 12 e 2 + ...&c. dE Similarly = e^ &c. 169.] 0# /$ where e is any quantity of elec- tricity forming part of the system, V the potential at the point where that quantity is situated. If q be any generalised coordinate defining the position of the system, the force tending to produce in the system the displace- ment dq is dE Id -- 7 or - (2 Ve). dq 2 dq x Now if the charges e are invariable in magnitude, the po- tentials V are functions of the coordinates q, and therefore the force is in which every Fis a function of q in respect of the coefficients A. If, on the other hand, the potentials be maintained constant notwithstanding displacement, by proper variation of the charges. the force is in which every e is a function of q as it is involved in the coefficients B. It remains to find the relation between the forces in these two cases. 170.] If q be any one of the generalised coordinates defining the position of the conductors, and if R denote the force tending to in- crease q when the charges are invariable, and R / be that force when all the potentials are maintained constant, then R + R' = 0. For J let e, V) and q all vary. 171.] ELECTRIFIED BODIES. 171 Then we have dE dE H de dq dV dq But *jl.= 7, de and dV hence 2-r ?8i and 2 =^- 61 therefore ?g g+ 8? = 0. ofy efy 7 ^T 7 Now ~ is the rate of variation of E with q when the charges are constant, and is therefore the force tending to diminish q imder those circumstances; that is, it is R. o- i i dE v , bimilarly =R, dq hence 72 + ^=0. 171.] If any group of conductors previously insulated from one another become connected by very thin wires, so as to form one conductor, the energy of the system is thereby diminished ; and the energy lost by it is equal to that of an electrical system in which the superficial density at any point is the difference of the densities at the same point before and after the con- nection is made : that is, is equal to the energy of the system which must be added to the original in order to produce the new system. For let V denote the potential of the system after the con- nection is made, V+V the original potential. Then V and T' are both constant throughout each conductor of the system. The charge on any conductor which retains its insulation, or / / -=- dS. remains unaltered. 4*w dv 172 ENERGY OF AN ELECTRICAL SYSTEM. [172. Any group of conductors which become connected form one combined conducting- surface on which the aggregate charge, or -- 1 1 -j- dS, is unaltered. 4t-nJJ dv V is then a function which satisfies the conditions (i) V 2 V = at all points outside of all the connected conductors, has given values over each of them, (3) V is (3) -j- constant over each of them, while F+ V satisfies conditions (i) and (2), but is not constant over each of the connected con- ductors. Therefore, by Art. 1 3, Qv+v = Qr + Qv, where throughout all space outside of the surfaces, and Q v+ v > and Q y have corresponding values. Now -- Qy+v' is the original energy, -- Q v is the energy 8 77 O7T after the connection is made, and Qp is the energy of the O7T system in which the potential is the difference of the two potentials, and therefore, by the principle of superposition, in which the density at any point is the difference between the densities before and after the connection is made. 172.] If any portion of space S, previously not a conductor, become a conductor, or which is the same thing, if a conductor be brought from outside of the field and made to occupy the space S within the field, the energy of the system is thereby diminished ; and the energy lost by it is equal to that of the system in which the superficial density at any point on any conductor or on the surface of S is the .difference of the densities at the same point before and after S became a conductor ; that is to the energy of the system of densities which must be combined with the original to produce the new system. For let V be the potential of the system when the space S is a conductor, V-\- V the potential when S is non-conducting space. 1 74.] ENERGY OF AN ELECTRICAL SYSTEM. 173 Then V is constant throughout S and throughout each conductor, V+ V is constant over each other conductor but not over S. Let Qy be the energy of the system in the former case, Sir Q v+ v > the energy in the latter case. 8 7T It can then be shewn, as in the last case, that throughout all space outside of the original conductors whether within or without S. But the integral of the second term is the energy of the system in which the potential is T' t that is of the system which must be combined with the original to produce the new system. It follows that a conductor without charge is always attracted by any electrical system if at a sufficient distance from it. Hence also any number of .uncharged conductors in a field of constant force generally attract each other. 173.] It follows as a corollary to the two last propositions that if a conductor increase in size, the energy of the system is thereby diminished. For if C be a conductor, 8 an adjoining space, if S became a conductor insulated from (7, the energy would be diminished, and if the new conductor were then connected with C so as to form one conductor with it, the energy would be further diminished. Hence the resultant force on an element of surface of a conductor is, if the charges be constant, in the direction of the normal outwards. 174.] If S be a surface completely enclosing a conductor (?, then if 8 were itself a conductor, its capacity would be greater than that of C. For let the conductor C be charged to potential unity, all the other conductors being at zero potential. Let M be its capacity, that is, the charge upon it under these circumstances. Since unity is in this case the highest potential in the field, the potential on S is less than unity at every point. Let it be 174 EARNSHAW'S THEOREM. [175. denoted by V. If x (os, y, z), or shortly (f) x . Then X(f> x is the density of the induced distribution which would be formed if the force were X. And the potential due to the distribution X$ x is therefore CX+xX. Definition. The function \\x $ x dS taken over the surface of every conductor of the system is the electric polarisation of the system in direction x due to a unit force in direction x acting at every point of the system. It is evidently independent of the position of the origin. For dS = " since 1 1 $*$$ is the algebraic sum of the induced distribution and is therefore zero. In like manner we may define to be the electric polarisation of the system in the directions of y and z respectively due to unit force in direction x. In like manner if there be forces T and Z parallel to the other 1 7 7.] IN A FIELD OF UNIFORM FORCE. 177 coordinate axes, they will produce in the system induced dis- tributions whose densities are Yfa and Z

C denote the total polarisation parallel to #, y, z respectively, due to the three forces, we shall have y fa dS+ Yy $ y dS+ zy j. dS, dS+ Y^fz v dS + zffz z dS. 177.] We can now prove that For the potential of the system of densities denoted by x is, as we have seen, C+ a?, C being, as before mentioned, a constant for each conductor, but having generally different values for dif- ferent conductors. Therefore is the work done in constructing the distribution whose density is

x dS is the work done in constructing the system (j> x against the repulsion of the system <$) y previously existing. But by the conservation of energy these two quan- tities of work must be equal. Therefore Similarly, ffx y ds- These equations are of the same form as those employed in Thomson and Tait's Natural Philosophy , 2nd Edition, p. 127, for determining the principal axes of a strain. As there shown, the system leads to a cubic equation in , of which, when the three roots are always real. The equation is the same as that treated of in Todhunter's Theory of Equations, 2nd Edition, p. 108. As shown by Thomson and Tait, each of the three values of e corresponds to a fixed line in the system, and the three lines corresponding to the three roots are mutually at right angles. There exist, therefore, for every system of conductors three directions at right angles to each other, and fixed with reference to the system, such that a uniform force in any one of these directions produces no polarisation in either of the others. We might define these directions as the principal axes of polarisation of the system of conductors in question. If we take these three lines for axes we shall have evidently b y dS = 0, / / y fa dS 0, &c. Let us denote / /. // x (j) x dS by Q x , I I y $ v dS by Q y , and f / *$ ^ by ft ; */ / / */ 180 A SYSTEM OF INSULATED CONDUCTORS [l8o. then the polarisation in any line whose direction cosines referred to these axes are a, p, y, due to unit force in that line is 180.] Of the energy of the polarisation of a system of conductors placed in a field of uniform force. Let the system be referred to its principal axes. Let X, Y, Z be the forces parallel to these axes respectively. Let X$ x be the density at any point on any conductor which would be produced by X alone acting, Y$ y and Z(f> z the same for Y and Z. Then X(j> x + Y(j> y -\-Z(f) z is the density when all three forces act. The potential of the three forces is -(Xx+Yy + Zz). The work done in constructing the system against the external forces is then taken over the surface of every conductor. The work done against the mutual forces of the system itself is Yy + Zz) The whole work is therefore taken over the surface of every conductor, and the work done against the mutual forces of the separated electricities is the same expression with the positive sign. But the axes being principal axes, ffx^y dS = 0, Pi x where r is the radius) will be found to depend on / the nature of the dielectric medium in the space between the spheres. And therefore the charge on the sphere necessary to produce a given potential, or the capacity of the sphere, is greater for some of such substances than for others. The dielectric, in Faraday's language, has inductive capacity. It is less for air and the permanent gases than for any solid dielectrics, and rather less for vacuum than for air. 185.1 In order to explain this phenomenon, Faraday adopts the hypothesis that any dielectric medium consists of a great number of very small conducting bodies interspersed in, and separated by, a completely insulating medium impervious to the passage of electricity. In his own words, ' If the space round a charged ' globe were filled with a mixture of an insulating dielectric, as ' oil of turpentine or air, and small globular conductors, as shot, * the latter being at a little distance from each other, so as to be ' insulated, then these in their condition and action exactly ' resemble what I consider to be the condition and action of the ' particles of the insulating dielectric itself. If the globe were ' charged, these little conductors would all be polar ; if the globe 1 were discharged, they would all return to their normal state, to ' be polarised again upon the recharging of the globe/ The properties of such a medium closely resemble, as far as their mechanical action is concerned, those of a magnetic mass, as conceived by Poisson, each of Faraday's ' shot ' being in fact when polarised equivalent to a little magnet, except that in dealing with magnetic masses the polarisation is usually under- stood to be in parallelepipeds instead of in spherical particles, and Poisson's investigations are therefore applicable. (See Memoir es de PInstUut for 1823 and 1824.) The mathematical theory has also been treated by Mossotti with especial reference to Faraday's theory, but by a different method from that here employed. 186.] In accordance with this hypothesis of Faraday's, we will consider the dielectric as consisting of a great number of very 1 86.] SPECIFIC INDUCTIVE CAPACITY. 185 small conducting- bodies, not necessarily spherical, and separated by a perfectly insulating medium. A first object is to find the form assumed by Poisson's equation when the conductors become infinitely small. In this medium take any parallelepiped whose edges, h, k, I, are parallel to the coordinate axes. Let these edges be very small compared with the general dimensions of the electric field, but yet infinitely great compared with the dimensions of any of the little conductors in question. The face Tel of the parallelo- piped will intersect a great number of these little conductors. If the medium be subjected to any electromotive forces, there will be on each conductor an induced distribution of electricity whose superficial density we shall denote by $. Then for each conductor / / ds = 0. Let X be the average value over the face kl of the force parallel to #, and therefore normal to kl, due to the whole electric field, including the induced distributions on all the conductors whether intersected by kl or not. We will first assume that the corresponding forces Y, Z are zero. In this case the average value per unit area of kl of the algebraic sum of the induced distribution on the intersected conductors which lies to the right, or positive, side of kl is, by the principle of superposition, proportional to X. Let it be QX, Q being the value which it would have if X were the unit of force. If the forces Y and Z are not zero, then the quantity of the induced distribution on any individual conductor lying to the right of kl will generally depend on Y and Z as well as on X. But if the conductors be in all manner of orientation indiffer- ently, the quantity of free electricity to the right of kl, due to Y and Z, will disappear on taking the average ; because for any conductor, if the total density of the induced distribution arising from Y or Z for any position of the conductor be calculated, then on turning the conductor through two right angles about an axis parallel to #, the corresponding density in the new position will be equal to that in the former position, but of opposite sign. 186 SPECIFIC INDUCTIVE CAPACITY, We shall for the present confine ourselves to the case in which the conductors are orientated indifferently in all directions, and we shall define a medium in which this is the case to be an isotropic medium, and any other to be a heterotropic medium. It is evident that in an isotropic medium the quantity of the induced electricity in the conductors intersected by the faces hk or h I of the parallel opiped which lies to the positive side of those faces respectively, due to unit force in direction y or z, is the same as the corresponding quantity for the face kl. That is, it is Q. 187.] The total electricity included within the parallelopiped will consist of (l) p 7ikl, the quantity of the given electrical distri- bution, which is supposed to exist independently of the condition of the medium within h Jc I, whether it be vacuum or dielectric ; (2) the sum of the induced superficial distribution on those parts of the conductors intersected by the faces of the parallelopiped which lie within its volume. The part of (2) arising from the two Jcl faces will be UQX and -klQX - Tiki (QX) dx respectively, and their sum is therefore In like manner the part of (2) arising from the faces hk, Jil of the parallelopiped are and -Kkl- respectively. We obtain therefore for the whole electricity within the parallelopiped, which we will call E, Now let N be the normal force at any point on the surface of the parallelopiped measured outwards. Then on the two kl faces the average value of N is -X and +X + h^ dx 1 88.] SPECIFIC INDUCTIVE CAPACITY. 187 respectively. Therefore for these two faces dX Nds = +~ hkl dx Similarly, for the other two faces we shall have ' rl Y Nds = + ~ hkl, dy /YV I = -r Zi = - > cfo v dx ' dy^ dy ' dz ^ dz ' This is the form assumed by Poisson's equation in such an isotropic medium as now under consideration. K evidently depends on the form, number, and position of the conductors, that is, on the nature of the substance. It is called the dielectric constant. 188.] Again, if p +p' be the whole free electricity in the element 188 SPECIFIC INDUCTIVE CAPACITY. [l8 9 . of volume dxdydz, including both that of the general electric field and that of the induced distributions, evidently Hence we have d , dV * dV d_ "fa dV dz and phkl is the sum of those portions of the induced distri- butions on the conductors intersected by the faces of the parallelepiped which lie within its volume. 189.] If over any surfaces there be superficial electricity of the given electric distribution, the equation dx ^ dx ' dy ^ dy ' dz x dz becomes, as in the cases previously investigated, dv (2) ,dV or ( dv where the suffixes relate to the media on either side of the surface of separation. Let (1) and (2) denote two media bounded by a plane surface AS, such that K, Q, and X have the suffixes 1 and 2 in these media respectively. Let C lt C. 2 be two parallel planes on either side of AB. Then, by the preceding, the superficial induced electrification in the space between C 1 and AB, and C 2 and AB, per unit area of the planes, is + Qi-^i an( ^ ~~ Qz^-z respectively, and the total induced electrification in the space between C^ and C 2 is Q 1 X 1 Q 2 X 2 per unit area oiAB. If the two planes Ci and C 2 be made to approach each other till they become infinitely near, this gives a superficial electrifi- B Fig. 28. SPECIFIC INDUCTIVE CAPACITY. 189 cation ', we have seen that Therefore our equations may be written = 0, // and a' are called by Maxwell the apparent electricities solid and superficial respectively, and Faraday's hypothesis of the dielectric medium supplies us therefore with a physical meaning for these quantities. It follows from the equation (2) that at the surface of separation of any two isotropic media, in which the constant K has the values K- l and K 2 respectively, if there be no real electri- fication on that surface, that is, if o- = 0, the normal forces on either side of the surface are to each other in a constant ratio., namely 190.] Now let be the superficial electrifi^llia at any point of the surface of any one of the small conductors in jbhe neigh- bourhood of any point P, (x, y, z) in the dielectric, and let "sffxQdS, or the sum of the integrals 1 1 to $ US taken over the surfaces of all these conductors within uni 4 o volume, be denoted by o- x , assuming that the distribution of electricity on each 190 SPECIFIC INDUCTIVE CAPACITY. [190. conductor and the distribution of the conductors themselves is constant throughout that volume and the same as it is at P ; thus a- x is a physical property of the dielectric at P analogous to the pressure p referred to unit of surface at any points in a fluid mass, and other similar quantities. Let a be the average distance between two planes parallel to y, z, touching any small conductor, i. e. the average breadth of a conductor parallel to #, and let n be the average number of such small conductors in unit of volume. It follows that the number of conductors in an elementary parallelepiped dxdydz is n dxdydz^ and the number intersecting the dydz face must be dx or n a dydz. a Now if x l be the x coordinate of the left-hand plane parallel to yz touching any conductor, the average amount of the elec- tricity lying to the right of any other plane parallel to yz inter- secting the same conductor must be // - tydS, the integration . * rr being over the surface of the conductor, i.e. it is - / / x (/> dS, a 1 1 rr since N)dS = 0. Therefore the amount of electricity on the small conductors intersected by the left-hand dydz face of the parallelepiped dxdydz and lying to the right of that face must be nadydz I - j or n / I axfrdS. dydz. But nilxfydS is the quantity above designated by a x . Therefore jr m is the amount of electricity per unit surface on the rin| hand of a plane at P parallel to yz situated on small conductors intersected by that plane ; it is the same quantity as is denoted by QX in Art. 186; similarly for a- y and 0-3. The quantities IT^, cr y3 " J J dv > JJJ rrr rr ,dv dv. the last term / / / xv*V dxdydz including / / # (;r- + T~)d>3 for surfaces of JJJ ^ JJ <". superficial density or discontinuous , &c. Since the distribution within the planes is algebraically zero, / / ^~dyidz l and / I fl~ dytdz z are separately zero. Also ^- and are I and + i respectively, / / -= dt/i d + i resp and / / / x v 2 V dxdydz - 4tr I I I xtydxdydz if be the density at any point anes. I ir t dy 9 .ds g - / /Fi^efoj = 4w / / / xdxdydz. the mean force in direction of x, and C the area of either plane, - ll' where H is the volume between the planes. Therefore r-- a These conclusions are equally true when the planes are of any magnitude, provided the distance between them is infinitely small compared with their linear dimensions, in which case / J J y becomes the same quantity as that n above denoted by a x and the proposition is proved. 1 9 I.] SPECIFIC INDUCTIVE CAPACITY. 193 small conductors situated between these planes and not inter- sected by them will be in the direction of the electrical density at any point in any molecule, and by o- x) v y , and o- z the sums for unit of volume in the neighbourhood of any point P, (#, ^, z) in the medium, the triple integrals scfydv &c. being replaced by iJx m, n, we get = S / / / ( (lx + my + nz) (f)dv VOL. I. 194 SPECIFIC INDUCTIVE CAPACITY. [ 1 9 2 - and the results arrived at in the preceding articles as to the density, solid p or superficial or briefly the inductive capacity of the medium. The phenomenon observed by Faraday, using two concentric conducting spheres separated by a homogeneous isotropic medium, is a particular case of the general result of this article. 193.] The conception and treatment of lines, tubes, and fluxes of force developed in Arts. 96-103 of Chap. V are equally applicable to an isotropic dielectric with any value of K, either uniform or variable, and might, indeed, have been applied to establish the equations above obtained in such a medium. If we integrate each term of the equation above proved, viz. dz I93-] SPECIFIC INDUCTIVE CAPACITY. 195 over a space inclosed within any closed surface S, we get and if I, m, n be the direction cosines of the normal to any element dS of the surface, this becomes or / 1 KF n dS '= 4 Tim; where F n is the normal force measured outwards at each point of S, and m is the algebraic sum of the included electricities. If, therefore, as in Art. 102, any tube of force be limited by the transverse surfaces S and $', and if F and F' be the normal forces at points on S and S' respectively, and if K and K' be the in- ductive capacities at those points, then the equation becomes ^K'F'dS'- or as it may be written /7Vd,S"- ff'f& = 4w jm- an equation expressing the same physical property as that of Art. 102, inasmuch as is the addition to the electricity included in the limited tube of force arising from the polarisation of the small conductors in the dielectric medium. In a medium with a continuously varying specific inductive capacity and finite volume densities, the force F obviously varies continuously both in direction and magnitude. If however there be an abrupt transition from one dielectric medium to another at any surface S, then, whether there be an actual charge on S or not, there is, as we have seen, a charge over S arising from the polarisation of the small conductors, called O 3 196 SPECIFIC INDUCTIVE CAPACITY. [194. generally the apparent electrification, although both from theory and experiment it is proved to have an existence as real as what is called by distinction the actual charge ; if a' be the density of this charge we have seen that where F l and F 2 are the forces normal to 8 at the point in the two media respectively, each supposed to act from medium K L towards medium K 2 . If therefore there be no actual electri- fication on S 9 we have or - = 0, as above shewn. And if i and i' be the inclination of the lines of force to the normal to S before and after the transit over $, tan i = ( 1 + ~f, - .) tan i' ' as above proved. All the properties of tubes of force, elementary or otherwise, proved in Chap. V, for a dielectric of uniform K hold good in the case of a medium with varying K, if we substitute for F (the force at any point) the quantity KF. This quantity KF is some- times called the induction. Should there be an actual charge a over 8, then the ordinary equations wonld give (/ being determined as before. 194.] As an illustration of the application of the preceding results, let us consider the state of the electric field when two media of different but uniform inductive capacities are separated by an infinite plane surface, and an electric charge is situated at a given point in one of them. Suppose the plane of the paper to be perpendicular to the plane of separation; let YEY' (see Fig. 29) be the line of inter- 1 94.] SPECIFIC INDUCTIVE CAPACITY. 197 section of these planes, and let the charge m be situated at the point m in the medium whose inductive capacity is k l3 that of the other medium being 2 . Let mEm' be drawn perpendicular to the bounding plane, and let m' be so taken that m'E = mE = , suppose. Let the distances of any point P from , m and m' be called r and / respectively. If V and V be the potentials on the left and right of the plane YEY' respectively, then V and V must satisfy the following conditions, ^ and / 2 being uniform in each Fi medium : (1) V = V on the plane and each vanishes at infinity, (3) VF+ = Now r = / over the plane and the differential coefficients of the functions - and along the normal to the plane are pro- portional to -3 and -^ respectively. It follows therefore that the conditions (l) and (4) may be satisfied by assuming A B _, C D V ~ + and V'= + r r r r and properly determining A, B, C, and D. Since p is zero to the left of YEY' except in the neighbourhood of m, the condition (2) requires that A should be equal to -j- > and since p is zero everywhere to the right of YEY', the 198 SPECIFIC INDUCTIVE CAPACITY. [195. condition (3) requires that D should be zero, and therefore ) r m (1) gives T- (4) gives whence we get 2m _ -^ / 2m and the problem is completely determined. If o-' be the superficial electrification of polarisation, or so called apparent electrification, over the plane, then m f^i- 1 /. ^i-^xo *-! 7 CT - ^) 27T7- 3 If k^ = 1 and >^ 2 = oo , we have agreeing, as they should do, with the results obtained for an infinite conducting plane in presence of a charged point, in Art. 105. 195.] As an instance of a similar treatment let us consider the case of a sphere composed of a dielectric medium of specific inductive capacity k y brought into a field of uniform force F in air. Before the introduction of the sphere the force throughout the field was F in a given fixed direction, which we will take for the direction of the axis of x. I95-] SPECIFIC INDUCTIVE CAPACITY. 199 If the origin be measured from any point, as for instance the point with which the centre of the dielectric sphere is made to coincide, then, before the introduction of the latter, the potential of the field was Fx+C. Let a be the radius of the sphere, and let V and V be the potentials outside and inside of the sphere, then the conditions to be satisfied are : (1) V = V at the sphere's surface, and V becomes Fx + C at infinity ; (2) V 2 V = V 2 V = everywhere ; (3) k-j + =0 at the surface. dv l dv Now we know from Art. 107 that a potential To? either of the forms Aas t or -y gives a normal force at any point on the sphere's surface proportional to a?, and satisfies the condition Bx V 2 V 0, provided that in the case of the form being chosen, the point is not infinitely near to the centre. If therefore we make V of the form and make V of the form Bx+C, we shall have condition (2) satisfied identically, and shall be able to satisfy conditions (1) and (3) by properly determining A and B. (1) gives 7' = Bx+C = 7 = Ax+C at the surface, or B = A. .Bx ZFx 2 Ax w 3^ T (3)gives _*_ = _ + _ -, or = ,!=__ And therefore the potential V throughout the region external to the sphere is given by the equation and the potential V within the sphere is given by the equation 3Fx 200 SPECIFIC INDUCTIVE CAPACITY. [195 a. The density as already 4TT a determined for a conducting sphere in a uniform field. The potentials Fand V obtained above are obviously those out- side and inside of the surface due to the superficial electrification BF k-l x ITT " k+2 ' ~a together with the potential of the field Fx+C. If we subtract this latter from V and V respectively, it appears that the potentials of as well as X. By the principle of superposition it must consist of three portions proportional to X, Y y and Z respectively. Let it be denoted by In like manner the quantity of electricity of the induced distributions on the conductors intersected by the hi and hk Of THf 196.] SPECIFIC INDUCTIVE CAPACITY. faces of the parallelepiped, which lies on the positive side of these faces respectively, may be denoted by = 2 j j where x dS+ ry ^ dS + zy . dS, 202 SPECIFIC INDUCTIVE CAPACITY. [197. and therefore ff &c. = &c. The properties of the coefficients Q xx , Q xv , &c. or x 9 d8, &c. were investigated in Chap. X; and as there proved, Q xv = Q yx , &c. If I, m, n be the direction-cosines to the normal to any plane drawn in the medium, the quantity of electricity on the little conductors intersected by unit area of that plane which lies on the positive side of it is l y dS = 0, Uy z dS = 0, and ifz x dS = 0. And x ifc dS =y <\> v dS = z $ z dS. In that case and JT = 1 + 4-7T 197.] If the medium be not isotropic, the integrals / / x $ y dS, &c. are not generally zero for all directions of the coordinate axes. But it was shewn in Art. 179 that for any system of conductors in a field of constant force there exist three directions at right angles to each other, such that if these be taken for axes, each of the integrals \\x (j> y dS, &c. is zero. 197.] SPECIFIC INDUCTIVE CAPACITY. 203 We may therefore describe a small sphere about any point in our medium, and find three perpendicular directions such that, if these be taken for axes, each of these integrals vanishes, if taken throughout the sphere. And we may call these three directions the principal axes of electric polarisation at the point in question. With these directions for axes we have (T = <* = But it will not be generally true that and K & be constant, and the directions of the principal axes also constant/ the medium, whether isotropic or not, is said to be homogeneous. If otherwise, heterogeneous. An isotropic medium is therefore homogeneous if K be constant. 204 SPECIFIC INDUCTIVE CAPACITY. [198. If K x) K y) K z vary continuously, the same reasoning by which in an isotropic medium we obtained the equation d , T ,dV. d , T7 dV. d /r ^dV. (K--) + --(K- r ) + (K) + 4irp = dx v dx ' dy v dy ' dz v dz ' leads, in case of a heterotropic medium if the coordinate axes be the principal axes, to d . dV. d , _ = 0. cfcc v (fcc ' or K x is the ratio between the average force which would exist on the plane Icl if all the intersected conductors were removed and the average force which does exist over that plane in the medium. 198.] As shown in Art. 180, the energy of the medium per unit of volume is -\ and the little conductors, if each free to rotate on any axis, will so use their freedom as to make the expression within brackets the 1 99.] SPECIFIC INDUCTIVE CAPACITY. 205 greatest possible, by placing themselves in suitable positions ; that is, they will endeavour so to place themselves as that the axis of greatest polarisation shall coincide with the resultant force ; so that, for instance, if the conductors will so place themselves as that the principal axis x shall coincide with the direction of the force. If they be perfectly free to move, this object will be effected for any direction of the resultant force ; and as in that case there will be no polarisation in any direction at right angles to the force, the expressions are zero. Such a medium will then have all the properties of an isotropic medium. But unless the conductors be perfectly free to move, or are spheres, the medium will in general be heterotropic. 199.] It appears from the preceding that the numerical value of the dielectric constant K in any isotropic medium must depend upon the form and density of distribution of the small conductors within the medium. Suppose now that these are spherical, and that A is the fraction of any volume within the medium which is occupied by the whole of the small conductors within that volume. Suppose also that the average force within the medium in the neighbourhood of any point is X, parallel to the axis of x. Since the force within each conductor is zero, it follows that the average force in non-conducting space in the neighbourhood X of the point in question must be - . 1 A Now the electrical distribution on the surface of each sphere must be such that the force arising from it within the sphere, together with that from all the other electricity in the field, shall be zero throughout that sphere. If the spherical distribution were very rare it is clear that the force arising from all the electrical distributions except that in 206 SPECIFIC INDUCTIVE CAPACITY. [2OO. any one sphere must be sensibly constant throughout that sphere. In other words, the sphere is situated in a field of jr constant force - - parallel to the axis of x. J. ' A. Therefore the polarisation of any particular sphere must be -- > where v is the volume of the sphere; see Art. 182. 4-77 1 A Hence the polarisation per unit of volume, which we denoted 3A X above by QX, is - - 1 and therefore q \ 1 A or, as A. is supposed very small, K 1 + 3A. This amounts in fact to regarding each sphere as polarised independently of the rest. If A be not very small, so that we have to consider the mutual influences of the spheres, the reasoning is precarious and cannot be insisted upon. 200.] We may also construct a composite medium, portions of which shall consist of a dielectric whose constant is K l9 and other portions consist of a dielectric whose constant is K 2 . If such a medium be uniform throughout any space, that is, if the distribution inter se of the two dielectrics be the same for unit of volume in any part of the space in question, the problem presents itself for consideration, what is the average force in such a composite medium due to the induced distributions within it ; or, as we may otherwise express it, what must be the value of K, in order that a uniform medium with K for dielectric constant may have the same effect as the composite medium. The solution of such a problem depends on the manner in which the two dielectrics are distributed inter se. If, for instance, the medium K 2 is in separate masses bounded by closed surfaces dispersed through the medium K ly the solution of the problem will depend on the shape as well as the number and magnitude of the separate masses. "We might endeavour to determine the density of an induced distribution on those surfaces which, if they were filled with the dielectric K lt would cause the normal forces on opposite sides of 2 CO.] SPECIFIC INDUCTIVE CAPACITY. 207 rr the surfaces to bear to each other the ratio -=? If by that, or by any other method, we could find the value of / / x ds for the composite medium, the value of K is at once known to be rr JJ The problem is substantially the same as that of the deter- mination of the dielectric constant in a single medium. If, for instance, the dielectric K 2 be contained in spheres, and they be so distant from each other that their mutual influence may be neglected, and the whole system be regarded as placed in a field of uniform force X, it will be found that the density of the induced distribution upon them which causes the normal forces within and without the spheres to have the required ratio is 3X proportional to cos 0, X being the external force, and 0, as before, the angle between the radius of any point on a sphere and the direction of #. Let the density be 3X a = n x cos 0, 47T where n is a ratio to be determined. This distribution gives a force n X within any sphere. Consequently the normal force within any sphere is (l-n)XcoaO. The normal force outside of a sphere is We have then by the condition respecting the forces KI (1 + 2 n) = A* 2 (1 n), and from which K or, X being very small, CHAPTER XII. THE ELECTRIC CURRENT. ARTICLE 201.] HITHERTO we have been engaged in the de- velopment of the so-called two-fluid theory of electricity in its application to Electrostatics, or the conditions, on that theory, of the permanence of any electric distribution, one essential condition being that the potential shall have the same value at every point in a conductor. The results arrived at are so far in agreement with experiment as to justify the acceptance of this theory as a formal explanation of electrical phenomena. We now proceed to consider how far the theory can be adapted to the explanation of observed phenomena in another class of cases, those namely in which different regions of the same con- ducting substance are maintained by any means at unequal potentials. Suppose, for example, that two balls of any given metal and at the same temperature, originally at different potentials, are held in insulating supports, and connected together by a wire of the same metal ; then it is found that after an interval of time inappreciably small, the potentials are reduced to an equality at all points of the conductor thus formed of the balls and wire, and that the total charge on the ball of higher potential has been diminished, and that on the ball of lower potential has been in- creased. With the conception and language of the two-fluid theory there has been in this short interval a flow of positive electricity in the one direction along the wire, or of negative electricity in the opposite direction, or both such flows have taken place simultaneously. If a magnetic needle be suspended near to the wire, a slight transitory deflection of this needle may be observed during the process of equalisation of potentials, and it might be possible 203-] THE ELECTRIC CURRENT. 209 with a sufficient length of wire and apparatus of sufficient delicacy to detect a slight rise of temperature in the wire. 202.] Methods exist whereby the inequality of potentials in different parts of a conductor may be restored as fast as it is destroyed, and in such cases certain properties are manifested in the conductor and its neighbourhood so long as this inequality is maintained. For instance, if the conductor be very small in two of its dimensions in comparison with the third, in ordinary language a wire, the deflection of the needle is no longer transitory but persistent, so long as the inequality of potentials is maintained, the amount of such deflection depending upon the amount of the inequality, and the dimensions and constitution ef the wire ; heat also continues to be generated in the wire at a rate depend- ing upon the same circumstances. Also if the wire be severed at any point, and the severed ends connected with a composite conducting liquid, thus forming a heterogeneous conductor of wire and liquid, chemical decomposi- tion of the liquid will ensue at a rate dependent on the difference of potentials, and the nature of the wire and liquid. According to the two-fluid theory, there must be under the given circumstances a permanent flow of one or both electricities between the unequal potential regions, of a like nature to the transitory flow spoken of above, and the wire is, in the language of that theory, spoken of as the seat of an electric current. Of course the existence of such a current is as purely hypothetical as that of the electric fluids themselves. A transference of some kind there must be, for it is indicated by the respective gain and loss of electrification in the two connected conductors, but whether that transference be a material transfer as implied by the two-fluid theory, or a formal transfer like a wave, or the transmission of force as in the case of a tension or thrust, we are not in a position to determine. The current, as it is designated, must be regarded as a phenomenon by itself, called into existence under certain conditions, and subject to laws to be investigated by independent observation. 203.] The question indeed might arise, how far are we VOL. i. P 210 THE ELECTRIC CURRENT. [204. warranted in regarding current phenomena as indicating the absence of electrical equilibrium ? When parts of a conductor are maintained at permanently unequal but constant potentials, a certain state of the field ensues, which is also permanent, and it might be said that we have in such a case a system in equilibrium although not satisfying the conditions required by the two-fluid theory. To this it can only be replied, that in the case of electrostatical equilibrium we have a system permanent of itself ; whereas in a constant current the permanence always necessitates an expendi- ture of energy from some external source. The former case resembles the mechanical equilibrium of a heavy body on a hori- zontal plane. The permanence of the latter case resembles that of a heavy body dragged uniformly up an inclined plane, and requiring at each point of its course the expenditure of external work. Laws of the Steady Current in a Single Metal at Uniform Temperature. 204.] (l) The intensity of the current is the same at every point. We have mentioned certain physical manifestations accom- panying the current, viz. thermal, chemical, and magnetic. These are capable of measurement ; and it is reasonable to re- gard these measurable effects as exhibited in the neighbourhood of different portions of the current as giving a measure of the intensity of the current in those portions. It is found experi- mentally that in the case of a steady current these effects are the same throughout. Wherever the magnetic needle is sus- pended assuming its distance from the wire and other circum- stances to be the same the same deflection results. If the wire be of equal section in every part, then equal portions are heated at the same rate, and in whatever portion of the wire the liquid conductor above described is introduced, chemical action also takes place at the same rate. This law is evidently consistent with the two-fluid theory, according to which we regard the current as a flow of either fluid across any transverse section of the conducting wire. 205-] THE ELECTRIC CURRENT. 211 (2) Ohm's Law. This law, which is universally accepted, asserts that If a uniform current be maintained in a homogeneous wire whose surface is completely enveloped by insulating matter, the intensity of the current in the wire is directly proportional to the electromotive force (i. e. the difference of potentials at its extremities), and inversely proportional to the resistance of the wire; the T? mathematical expression of the law being 7 = - s where I is the JK current intensity ', E is the electromotive force, and R the resistance. The quantity here called the resistance depends upon the length and transverse section of the wire and upon the material of which it is composed. For wires of the same substance it is proportional to the length directly and the transverse section inversely, and Ohm's law asserts that if through a wire W the electromotive force E produces a current I, and through another wire W the electromotive force E' produces a current /', then E E' the fractions -=- and -~r will always bear the same ratio to one another so long as the same wires ^Tand W are employed. If R be the resistance, -^- is called the conductivity, and if this J\i be denoted by K, then Ohm's law may be expressed in the form 205.] If the insulation of the wire is perfect, so that no trans- ference of electricity can take place across its surface, the direction of transference at each point must, in the permanent state, be parallel to the axis of the wire at that point, but this direction must be also perpendicular to the equipotential surface through that point 1 . The wire is in fact a tube of force, and if it be of uniform section the resistance through each element of length ds is proportional to ds, and therefore we have i and V be the potential at any point, the conditions that there shall be no free electricity within the plate, and that the equipotential surfaces are all normal to the plate, lead to the equations V 2 F=0, ^=0. dz Hence the problem may be treated as one in two dimensions only, and the electrodes may be regarded as circles with radii equal to those of the cylinders ; let these radii be a and l > and let the constant potentials be V 9 and V q respectively. The equation in F, or - dx* r df ~ may be satisfied by assuming F=C r +4 1 logr, + 4 2 logr 2 + &c., or C+2A logr, where the quantities r l9 r 2 , &c. are the distances of the point x, y from any assumed fixed points, and these points must be so taken that V is equal to V p and V q respectively at the circum- ferences of the circular electrodes. 211.] THE ELECTRIC CURRENT. 217 Let O p and O q be two points within the circles P and Q such that each is the image of the other in its own circle, and let the potential V at any point be taken equal to C A log - > where r 2 r^ and r 2 are the distances of the point from O p and O q re- spectively, then all the required conditions will be fulfilled, provided C and A be taken to satisfy the conditions f V p = CA log at the circumference of P, r z V q = C A log at the circumference of Q, r 2 inasmuch as is constant over each of these circumferences. **2 Since V is constant whenever is so, it follows that the r 2 equipotential curves are circles each one of which is conjugate to the centres of P and Q. The orthogonal trajectories of such circles, or the lines of current flow, are circular arcs each passing through these centres, and, if and be found at the da do circumferences of these circles respectively, we can find the whole V V current in unit time in terms of V p V q in the form of ^= - ; the quantity R is then called the resistance of the system, its reciprocal being the conductivity. In the particular case of the radii a and 6 being equal, and each very small compared with ./, the distance between the centres, we find from the above equations V V A = \ ~ " Let i be the current in unit time over the arc ds of the circumference of the P electrode, then, since r 2 is sensibly con- stant and the direction v of the line of flow is along the radius ofP, T ,dV . T ^dV , AK i -K^ i -ds=K- T ~ds - --ds, dv dr a 218 SYSTEMS OF LTNEAE CONDUCTORS. [212. where K is the conductivity of a unit length of a prism of the conductor of unit breadth. Therefore the total current in unit v time over the P circumference is 2-nai, or 2irKA. It is clear that K is proportional to the thickness 8 of the plate, and if for it we write Kb, the current per unit time will be where K is now the conductivity through a cube of the substance whose edge is the unit of length. V V Writing for A the value already found - ~ , we get the 2 log- i- current per unit time equal to 3 a and the resistance of the system is log On Systems of Linear Conductors. 212.] A conductor, two of whose dimensions are very small compared with the third, as for instance a wire, is called a linear conductor. We have had occasion to consider certain properties of linear conductors. Firstly, we have seen that if such a conductor be divided into several parts through which a current flows con- secutively, as A, C, &c., the resistance of the whole is the sum of the separate resistances of the several parts. Hence, in case of a homogeneous conductor at uniform temperature, if the potentials at the ends are known we can determine the potential at any intermediate point when a current is flowing. For instance, let APB be a wire the potentials of whose extremities are V a and 7 b . Let P be an intermediate point, and let the resistance of the portion AP be r ap , and that of PB be r pb . Then if i be the current, 2 1 3.] SYSTEMS OF LINEAR CONDUCTORS. 219 Hence which determines F p . Similarly, if i be given, but the potentials are not given, we can determine the differences of potential F a F p and V a V b . Again, in case of two or more wires connected in multiple arc, we have shown that if V m V b be the potentials of the extremities the currents in the several wires are respectively K l (V a V b ] J K z (V a F&), &c., where K v K^ &c. are the conductivities of the wires. And we can therefore determine all the currents if V a and T b are given, or the difference of potentials F a F b , if the sum of the currents is given. It is assumed that the wires are all of the same metal, and at uniform temperature. 213.] The points of junction of the wires are called the electrodes. In the above simple case we have only two electrodes. But we may conceive a system of wires meeting in more than one point. For instance, to take a case a little more complicated, let there be two wires APB, AQB, and the two intermediate points P and Q ^ B connected by a third wire. If the potentials at A and B are given, we may determine those at P and Q, as follows. Let K ap , K pb , K pq be the conductivities of the three wires AP, PB, PQ. Then, since the sum of the currents flowing from P must be zero, we have (r (I -7,) = o. Similarly, *.. (F.- F.J + Z,, (F- F.) + * M (F,- F.) = 0, from which, the conductivities K being known, the two unknown potentials, V p and F q , can be determined; and thence the cur- rents are known. If instead of the potentials, the current C, entering the system at A and leaving it at B, be given, we have three linear equations 220 SYSTEMS OF LINEAR CONDUCTORS. [214. to determine the differences of potential V a f^i, ^o~ ^p V a -V q , namely, C = K ap (V a - V p ) + K aq (V a - V q \ The points P and Q will generally be at different potentials, and a current will pass along PQ or QP. 214.] The case in which P and Q happen to be at the same potential is of special importance. In that case no current passes in PQ, and the potentials at every point in either wire are the same as if there were no metallic connexion between P and Q. Tkali " (V VA ( ' K ^ A current will pass in one or other direction along PQ, unless P and Q are at the same potential, that is, unless This principle is made use of in instruments for measuring resistance. Suppose, for instance, AX is a wire whose resistance r ax is required. Let BX be a con- ductor whose resistance r xb is known. Place AX and XB so as to form one conductor AXB. Let AEB be a uni- form wire, E a point in it. If E and X be joined by a wire, a JB current will pass along it in one or other direction, unless the potential at X is the same as at E. We increase or diminish the distance of E from A until a needle suspended near EX shows no deflection when an electric current is made to pass from A to B. Then we know that the potential at X is the same as that at E; and therefore AE 2 1 5.] SYSTEMS OF LINEAR CONDUCTORS. 221 which determines r ax . This is the principle of the instrument known as Wheatstone's Bridge. 215.] In a more general case, there may be n points, or elec- trodes, connected each to each by wires of known conductivities. Let V l . . . V n be the potentials at the several electrodes, (? p c 2 , ... c n the currents which enter the system from without at these electrodes respectively, taken as negative when a positive current leaves the system. Then the current in AB is and we have for the electrodes P and Q o p = K p (V,- V a ) + K, f (V p - Fi) + &c.) , = *(?,- r.)+r. (v,- r)+&4 ' and so on for each electrode. Now since no electricity can be generated or destroyed within the system, the sum of the currents entering the system at all the electrodes must be zero. That is, e 1 + c 2 +...+o n = 0. Therefore only ft 1 of the c's are independent. Also, since we are only concerned with the differences of the potentials, there are n i independent quantities of the form _ _ In all we have n 1 independent linear equations of the form A subsisting between the 2n2 independent quantities. If therefore any n 1 of the quantities be given, the equations suffice to determine the others. For example, if the entering currents c be given at any n 1 of the electrodes, we can deter- mine all the differences of potential. And if all the differences of potential are given we can determine the currents. If we differentiate the equation A for any electrode, as P, we obtain Similarly, differentiating the equation for A we obtain Since K ap = K pa , it follows that the potential at P due to the introduction of unit current at A is equal to the potential at A due to the introduction of unit current at P, and so on. 222 GENEKATION OF HEAT BY ELECTRIC CURRENTS. \2 1 6. On the Generation of Heat ly Electric Currents. 216.] Suppose a uniform current of intensity / to be existing in a linear conductor AB of resistance R, with terminal potentials V A and V B . There is a transference, per unit time, of electricity / from the extremity A to the extremity of B. Now if e lt 2 , &c. be the charges upon a system of conductors AH A 2) &c., and if T l} V z , &c. be the corresponding potentials and W the electric energy of the system, we have proved that de ' In the case now under consideration, the charge at the extremity A of the conductor, where the potential is T A) is diminished by Idt in the time dt, and that at the extremity J9, where the potential is F B) is correspondingly increased by the same quantity. Hence, since V A is greater than V B , there is by the process a diminution of the electric energy of the system in time dt equal to (V A -V B )Idt. But by Ohm's law, we have R Therefore the diminution of electric energy, owing to the existence of the current, in the time dt is = RPdt. K This is the work done ly the electrical forces in the field in time dt in the passage of the current 7 through the conductor, and this work done, or electric energy lost,, must reappear in heat evolved in the conductor AB in the same time. If therefore / represent the Joule heat- equivalent, the heat evolved per unit time will be RI* ~J" Joule was the first to prove by direct experiment that the rate of evolution of heat in any wire through which a current passes 2l8.] GENEKATION OF HEAT BY ELECTRIC CURRENTS. 223 is proportional to the square of the intensity of the current, and we now see that this result follows directly from Ohm's law and the principle of the conservation of energy. 217.] If the current, C, having been generated in a system, be allowed to decay by the resistance R, the value of the current at time t after the commencement is C*~ Rt . Hence the total quantity of heat generated when the current has ceased is &C* *+**& = 1C*. */o For this reason \C Z is sometimes called the energy of the current. It is supposed here that the current during this process is uninfluenced by any other current, or by any magnetic field, as we shall see later that electric currents in the same field exert mutual action on each other. On the Generation of Heat in a System of Linear Conductors. 218.] In the simple case of a number of wires in multiple arc, we have seen that R^C^ = -S 2 JB, &c. the sum of the entering currents is unaltered. Hence, for any electrode as A, or 2X a = 0. Hence 2^RCX=Q. Therefore the heat generated per unit of time in the con- strained system is 2KC 2 +2RX 2 , and exceeds that generated in the original system by the essentially positive quantity 2 EX 2 . Electromotive Force of Contact. 219.] Up to this point we have introduced the restriction that the conductors with which we are concerned shall be of the same substanee throughout. The reason of this restriction, which in strictness is equally required in electrostatic investigations, will now be considered. Volta believed that when two different metals were placed in contact, the potential of one of them was always higher than that of the other, and this without any disturbance of electric equilibrium. In fact, that instead of the condition of electric equilibrium being V = Constant throughout all continuous con- ducting space, the condition should really be, when such conduct- ing space is composed of substances of different materials, V C 19 V =. C 2 , V =. (? 3 , &c., in the regions occupied by these substances respectively; the values of the constants C lt C 2 , (? 3 , &c. being dependent upon the nature of the substances, and the electric distribution in the field ; subject only to this restriction, that in every case of electrostatic equilibrium of a compound 221.] ELECTROMOTIVE FORCE OF CONTACT. 225 conductor the difference C r C g of the constant potentials of any two given substances should always be the same at the same temperature. 220.] For instance, if a zinc wire and a copper wire were held by insulating- supports, and brought into contact at one end of each, the potential of each wire would be the same throughout, but that of the zinc would exceed that of the copper by a quantity always the same for the same temperature. If platinum were substituted for copper a similar result would be observed, but the difference of potentials (the temperature being the same as before) would be less. If platinum, and copper were similarly connected, the platinum would stand at the higher potential, and the con- stancy of temperature being still maintained, it would be found that the excess of potential of zinc over copper in the first case, supposed above, was equal to the sum of the excesses of the potentials of zinc over platinum and platinum over copper in the two last cases. This difference of potentials is generally called the electromotive contact forces of the two metals, and is for metals A and B denoted by A/B. It is considered as positive if the metal of higher contact potential is placed before the line and negative if the reverse, so that A/B + B/A = 0, and if there were three metals J, B> and C whose electromotive contact forces at any temperature were A/B for A and B and B/C for B and R r and therefore So that in this case also the same expression results as in the homogeneous multiple arc already investigated, provided the external electromotive force be increased by B/A. 224.] The expression for the energy dissipated in the case of wires in series also requires to be modified when the wires are not of the same metal throughout. If as in the last article there be two wires of metals A and B, and the notation of that article be retained, we have The total loss of electric energy per unit time as the current passes from the free extremity of A to that of B = (J^ V^)i. Q2 228 PELTIER EFFECT. [225. Therefore the whole heat generated must be J But by the equation above obtained Therefore the heat generated is that is to say, if B/A be positive the heat generated in the com- pound wire of resistance R by the passage of the current of Hi 2 intensity i is less than =- , or what it would have been had the J wire been homogeneous, by the quantity ' ' , and is greater than j- by T '- if A/E be positive : that is to say, when / / a current in passing through a circuit of heterogeneous metal wires traverses a junction from an electronegative to an electro- positive metal there is absorption of heat at the junction, and on the contrary, there is evolution of heat in the passage from an electropositive to an electronegative metal. 225.] This absorption and evolution of heat at metal junctions was first observed by Peltier, and the phenomenon is called after his name ; it is physically analogous to the absorption and evolution of heat accompanying chemical dissociation and combination respectively, the electricity at the junction being raised to a higher, or sinking to a lower potential in the respective cases, just as the chemical potential of the dissociated or combined elements is raised or depressed. The actual amount of heating or cooling as experimentally observed is always less than the theory requires, and in some cases is of the opposite sign ; indicating, apparently, that the whole electromotive contact force of Volta is not to be sought in the mere metallic contact, but in the action of the surrounding medium. CHAPTER XIII. OF VOLTAIC AND THERMOELECTRIC CURRENTS. ARTICLE 226.] IF any number of wires of different metals 3f l9 M 2 , 1/3, M l are joined together in series, and are kept at the M! M z M 3 M l Fig. 35- same temperature throughout, the wire of metal M l beginning and ending the series, it follows from the laws of contact action above stated that each wire is at the same potential throughout its length, and that the beginning and ending M l wires are also at the same potentials, inasmuch as the sum of the electromotive contact forces M^M^ -f M 2 /M 3 + M 3 /M l is zero ; hence if a circuit be formed by joining the free ends of the M l wires no current will ensue. If however we substitute for the M^ wire a composite liquid conductor L, and thus complete the circuit, the electro- motive contact forces L/M Z and L/M% are modified, the liquid L being at the same time decomposed. According to the extreme views of the Volta contact theory, the last-mentioned electromotive forces disappear with the de- composition, the liquid L and the metals M z and M 2 at their points of immersion in that liquid are reduced to the same potential, the electromotive contact forces Jf 2 /M" 3 , &c. of the metallic junctions are no longer compensated by the forces L/M 2 and L/M Zt and a current ensues through the wires and liquid. Suppose, for instance, the liquid be dilute sulphuric acid and the metals be plates of zinc and copper partially immersed and having their unimmersed ends attached to platinum wires, so long as these platinum wires are not united to each other, the zinc, the copper, and the liquid stand, according to this theory, at the same potential (V suppose), but the platinum wires attached to the zinc and copper plates are at the potentials VZ/P and 230 VOLTAIC CURRENTS. [227. V+P/C respectively. If now the platinum wires be united, electric equilibrium can no longer be maintained, inasmuch as the two portions of the same platinum wire are now at poten- tials differing from each other by Z/P + P/C or Z/C. Hence a flow of electricity must take place through the platinum wire from the copper to the zinc plate, raising the potential of the zinc and depressing that of the copper. The inequality of the potentials thus produced in these im- mersed plates is again destroyed by the action of the liquid, which is at the same time decomposed, oxide of zinc being formed at the zinc plate, which is dissolved as soon as formed, and hydrogen being given off at the copper plate, and thus a permanent current ensues in the closed circuit of copper, plati- num, zinc, liquid, copper, and in the direction indicated by the order of these words. Such an arrangement is called a Voltaic current, the vessel containing the liquid and plates is called a Voltaic cell, the decomposable liquid is called an electrolyte^ and its decomposition on the passage of the current is called electro- lysis. The intermediate platinum wire is in no respect essential to the process, which would have equally taken place if the copper and zinc plates had been in immediate external contact with each other. 227.] According to the theory of the Voltaic circuit, above explained in outline, the potential rises discontinuously at the metallic junction or junctions outside the cell, and falls continu- ously throughout the rest of the circuit ; the whole electromotive force of the current is sought for in the contact force at the junctions, the function of the chemical action in the cell being limited to the continued equalisation of potentials within the cell as fast as the equality is destroyed by the electric flow. According to the chemical theory of the circuit, which is now more generally accepted, a discontinuous change of potential takes place at the junctions between the metals and the liquid, those being the points at which, as we shall see presently, energy for the maintenance of the current is in fact evolved or absorbed. The true theory of the cell is not finally settled, only it is known that the chemical decomposition is an essential 229.] VOLTAIC CURRENTS. 231 part of the phenomenon. It is possible however to develop certain fundamental laws of the action, which are essentially the same whatever be the metals constituting the plates, and whatever be the liquid in the cell, provided it be capable of electrolysis. 228.] The plates by which the current enters and leaves the cell are called electrodes, that by which it enters is called the anode, and that by which it leaves is called the cathode, the two elements into which the electrolyte is decomposed are called the ions, the element appearing at the anode is called the anion, and that appearing at the cathode is called the cation. Let the metal electrodes be called P and N respectively, and the two constituents, or ions, into which the liquid is resolved be called TT and v respectively. On the passage of the current the ion TT will appear at the electrode N t and the ion v at the electrode P. Then it is found that (1) The ratio of the masses in which the two constituents TT and v appear at the electrodes is that of their combining weights. (2) The absolute mass of each ion so deposited per unit of time is proportional to the strength of the current, or in other words, for each unit of positive electricity transmitted a certain mass of each ion is deposited at the corresponding electrode. This is called the electrochemical equivalent of that ion. (3) So long as the electrolyte is the same, the ions into which it is decomposed are the same, whatever the metals constituting the electrodes ; and the same ions appear at the anode and cathode respectively. One or both of the ions may be com- pounds, and the same constituent which in one electrolyte becomes an anion, may in another electrolyte become a cation. (4) The source whence the energy, requisite for the maintenance of the current, is derived, is the arrangement of the elements of the electrolyte and the immersed plates in a combination of lower chemical potential energy than that which existed anterior to the current. 229.] The action of the typical cell described above of zinc and copper plates in diluted sulphuric acid may be supposed to be as follows. The chemical arrangement before the circuit was completed was Zn H^SOl . . . H 2 SO^ . Cu ; 232 VOLTAIC CTJKKENTS. [230. and during the existence of the current it is ZnSOv ... ff^Cu. The zinc first combines with the oxygen of the water (ff 2 0), and the zinc oxide is then replaced by the zinc sulphate Zn S0, which being soluble leaves the zinc plate free for further action. The potential chemical energy of Zn S0 4 is less than that of 7/ 2 0, or, as more practically expressed, the heat evolved by the combinations Zn and Zn 0, S0 3 is greater than that required for the decomposition of H 2 0, the difference furnishing the current energy. 230.] A feeble current might have been obtained with water only in the cell, the chemical arrangements before and after the completion of the circuit being and ZnO.^H^Cu respectively. But in this case, since the oxide Zn is insoluble in water, the zinc plate would soon, by its oxidation, become unfit for action, and the current would cease. We may however use this ideal case as an example. In this case the ions TI and v are and H 2 respectively. Taking unity as the combining number for hydrogen, that of oxygen is 8, and that of zinc is 32-53. Therefore one gramme of zinc takes in combination with oxygen the place of - - grammes 32-53 g of hydrogen, each combining with - , or -246 gramme of o &'5 o oxygen. The heat evolved by the combination of one gramme of zinc with the oxygen is 1310 units *. The heat which would be evolved on the combination of -7^-^ gramme of hydrogen O '53 with the oxygen, and which is therefore absorbed on their dis- sociation, is 1060 units. Therefore for every gramme of zinc * The object in this and the two following articles being illustration only, the absolute numerical values are of less importance. The system of units and the numerical values are those employed in Hospitaller's Formulaire pratique de VElectricien, English Edition, p. 214. 232.] VOLTAIC CURRENTS. 233 oxidised the excess of heat evolved over that absorbed is (1310 1060) units; that is, 250 units. Again, for every unit of current 00034 gramme of zinc is oxidised. In other words, -00034 is the electrochemical equiva- lent of zinc. Therefore for every unit of current the excess of heat evolved over that absorbed is -00034x250. And this is equivalent to an amount of mechanical work 34 x 250 X 100000 ' where / is Joule's factor. Now if F be the electromotive force of the cell, i the current, the amount of heat evolved is Fi. And therefore the amount of heat evolved by unit current is F. That is, 34x250 ' 100000 " 231.] It is usual, as above said, to employ instead of water dilute sulphuric acid, the formula for which is H 2 OS0 3 . In this case we may suppose that the H 2 is decomposed, and in the first place oxide of zinc, ZnO, is formed, and the hydrogen H 2 is set free. Then the Zn combines with the fiO Si forming Zn. 4 . The heat evolved by this last-mentioned combination must be added to the 250 units above mentioned. One gramme of zinc combines with -246 of a gramme of oxygen H 2 being decomposed with the evolution of 250 units of heat. And 1-246 grammes of oxide of zinc combine with S0 3 with the evolution of 360 units. Adding together 360 and 250, we obtain 610 units as the total heat evolved. 232.] In the cell known as Darnell's cell the electrodes are zinc and copper, but there are two liquid electrolytes, one of them saturated solution of sulphate of copper in contact with the copper, and the other dilute sulphuric acid in contact with the zinc, the mixing of the liquids being prevented by a porous diaphragm which does not interfere with the electrolytic con- duction, that is to say, the liberated ions pass through the diaphragm but the liquids do not. The following may be sup- posed to be the action of such a cell. 234 VOLTAIC CURRENTS. [233. The electrolysis of the II 2 S0 in contact with the zinc gives rise to a chemical action identical with that of the last case, but the hydrogen H 2 does not as in that case remain free. It passes through the diaphragm and displaces an equivalent of copper in the sulphate of copper Cu B, and with junction temperatures q' and q be A q '/ q B, then the electromotive force of the couple when the junction temperatures are p and q will be that is, A*/ t B = This also is due to Becquerel. 245-] THERMOELECTRIC FORCES. 241 III. The direction of the current, that is whether it be from A to J3, or from B to A, at the hot junction depends on the mean temperature of the junctions. When the mean temperature of the junctions for a given pair of metals is below a certain temperature T, dependent upon these metals, the current sets in one direction through the hot junc- tion, and when the mean temperature is above T the current sets in the opposite direction, or the electromotive force is reversed. This was discovered by Seebeck. The temperature T is called the neutral temperature for the pair of metals employed. In an iron and copper couple this neutral temperature is, according to Sir W. Thomson, about 280 C. When the mean temperature of the junctions is below this, the current sets from copper to iron through the hot junction, and when it is above this the current sets from iron to copper through that junction. IV. For any constant temperature of the cold junction, the electromotive force is the same when that of the hot junction is T+x, as when it is T x, and is a maximum when it is T. This was established by Gaugain, and results from Tait's ex- periments. It may be expressed thus : The electromotive force of the couple between temperatures t and t is proportional to 245.] The following is a mathematical explanation of these phenomena : If the difference of temperature between the two junctions be very small, as dt, the electromotive force of the couple must be proportional to it, and for the metals A and B may be denoted by (t> ab dt, where ab is for the given metals a function of the mean temperature of the junctions, and is taken as positive when the current sets from A to B at the hot junction. It is called the thermoelectric power of the two metals at temperature t. It follows from II. that if the temperatures of the junctions be t Q and t lt where t^ 1 is finite, the electromotive force is $ ab dt } which for the given metals is a function of t Q and rf x . Again, if we take any particular metal for a standard, and VOL. I. R 242 THERMOELECTRIC FORCES. [246. denote it by the suffix o, it follows from I. that the electromotive force for the couple in which the metals are A and Z?, and the temperatures of the junctions t and 1} is r - Jt 't If the reference to the standard be understood, we may call = 0, and is proportional to as stated in IV. 246.] Adopting a method originally suggested by Thomson, we may represent the thermoelectric powers of different metals at different temperatures by a diagram. Let the abscissa 246.] THERMOELECTRIC FORCES. 243 represent absolute temperature, and for any given metal let the ordinate represent its thermoelectric power, that is, the thermo- electric power of a couple composed of that metal and lead, with the temperatures of the junctions infinitely near that denoted by the abscissa. It follows then from the constancy of , - that the locus of d> at is a right line inclined to the axis of x at the angle tan" 1 -^- 5 and that for any given abscissa, as that corresponding to 50 C., the difference between the ordinates of any two metals represents the thermoelectric power of a circuit of the two metals at that temperature. In the annexed diagram we see that for temper- atures below 50 C. lead is , , jj , ZERO CENT 50C positive to iron and negative to copper ; from 50 to 284 C. copper is positive to iron and negative to lead ; from 284 to 330 iron is positive to copper and negative to lead; above 330 lead is positive to copper and negative to iron. Gene- rally, if for any two metals NM be the difference of the ordinates at temperature t, and M'N' at temperature t', and if E be the neutral point, the ther- moelectric power of the couples with the junctions at t and t' is graphically represented by the area MEN-M'EN', whether M'N' be at temperature below or above E. So long as the lower temperature represented by MN is un- altered, the difference between MEN and M'EN' has its greatest value when the higher temperature is at E, the neutral point. It becomes zero when the mean temperature of the junctions is the neutral temperature. Further, if M'N' and M"N" be taken at equal distances from E 2 Fig. 36. Fig. 37- 244 DISSIPATION OF ENERGY [247. E on either side of it, MEN-M'EN'= MEN -M" EN". These results agree with IV. 247.] Next, let us consider a circuit of three metals AB, BC, and CA, the junction A being at temperature t ly B at temperature t 2 , and C at temperature 3 . We may imagine three lead wires AD^B, BD 2 C, and CD Z A connecting the junctions, and forming three distinct circuits. The electromotive force of the circuit ABC is the sum of the electromotive forces of the three circuits AB^A, BCD 2 B, CAD 3 C, together with that Fi 8 of the circuit composed of the three lead wires AD l BJ) 2 CD^ A. But, by the law of Magnus, the electromotive force of the latter circuit is zero. Hence the electromotive force of the circuit ABC is r^ rt 3 /*! a dt+ b dt + c dt. h Jtz Jt s In like manner we can express the electromotive force due to any circuit of different metals with unequally heated junctions. 248.] We may suppose further a circuit composed of alternate wires of two metals only, A and B, and each alternate junction at the lower temperature t lt and every other junction at the higher temperature t 2 . If there be n pairs, the total electromotive force of such a circuit is, by the last article, The pairs are said to be joined in series. By this means the electromotive force of a thermoelectric couple can be multiplied at pleasure. Such an arrangement is called a thermoelectric pile. Of the energy of the current in a Thermoelectric Circuit. 249.] Energy, as alove shown, is necessary to maintain the current. In the case of thermoelectric circuits, now considered, 250.] IN THERMOELECTRIC CURRENTS. 245 no energy is supplied from without, nor are there, so far as we know, any chemical actions between the metals, or between them and the surrounding medium, from which the requisite energy can be obtained. We infer that the energy required for maintenance of the current is supplied by the conversion of part of the heat of the metals into another form of energy, namely, that of the electric current. This might conceivably be employed to do external work. But if not, it will be reconverted into heat by the resistance of the circuit. As in the working of a heat engine, the entropy of the system must be diminished by the process, that is, there must be equalisation of temperature. It is found that at the neutral temperature for any two metals a current passing the junction has no heating or cooling effect. The Peltier effect changes sign at that point. But' if a couple be formed with the hot junction at the neutral temperature, the cold junction is nevertheless heated, although the heat cannot be derived from the cooling of the hot junction. It is evident, therefore, that the current itself must have a heating or cooling effect. For instance, in an iron and copper circuit, with the hot junctions at the neutral temperature, either a current in iron from hot to cold must cool the iron, or a current in copper from cold to hot must cool the copper, or both these effects take place. And it may be inferred that the heat so gained or lost is compensated by a change in the potential of the current. It was this consideration that led Sir W. Thomson to the discovery of the electromotive force in unequally heated portions of the same metal. 250.] The method adopted by some writers (Mascart and Joubert, Legons sur VElectricite et le Magnetisme ; Briot, Theorie Mecanique de la Ckaleur] is as follows. It is assumed that the heat generated, as unit current passes from potential V a to potential 7 b , is always 7 a F b , whether the fall of potential be gradual as in a single metal, or abrupt as at the junction of two metals. That being the case, the electromotive force of a couple formed 246 GENERAL SYSTEM OF LINEAR CONDUCTORS. [251. of metals A and B whose hot and cold junctions are at ^ and t Q respectively, must be H,-H Q + f\cr a - where x( a ^p) represents the contact electromotive force from the metal (a) to the standard metal at temperature t p -, similarly if F q were the potential of the connecting metal at the electrode Q, where the temperature is t q , the potential at the extremity Q of the wire PQ would be Pg + x(^)' ^ n estimating the currents therefore in terms of the potentials we may regard the potentials of the common extremities of all wires at any electrode as equal provided we increase the electromotive force in any wire as PA by the quantity xK)-xK)- The Thomson effect treated of in Art. 243 will produce a ftp similar increase of electromotive force of the form / vdt, Jtq which may be expressed in the form If therefore E pq be the electromotive force arising from a battery, if any, in the course of the wire PQ, the expression for the current in that wire will be and similarly for each of the remaining wires. Of course the wire PQ may itself be composed of dissimilar metals, or may consist of two wires communicating with the liquids of an interposed battery, in which cases the requisite corrections are obvious. CHAPTEE XIV. POLARISATION OF THE DIELECTRIC. ARTICLE 252]. IN the preceding chapters we have endeavoured to explain elect rostatical phenomena by the method of Poisson and Green as the result of direct attraction and repulsion at a distance, according to the law of the inverse square between the positive and negative electricities, or electric fluids. As explained at the outset, in Chaps. IV and V, we do not assert the actual existence of these fluids. We assert merely that the electro- statical relations between conductors are as they would be if the two fluids existed, and conductors and dielectrics had the properties attributed to them in those Chapters. Faraday and Maxwell made an important step in advance. They assume all non-conducting space to be pervaded by a medium, and refer the force observed to exist at any point in the electric field, not to the direct action of distant bodies, but to the state of the medium itself at the point considered. Faraday was led by his experimental researches to believe in the existence of certain stresses in the dielectric medium in presence of electrified bodies. Maxwell shews that if the dielectric medium consist of molecules with equal and opposite charges of electricity on their opposite sides, or, as we expressed it in Chapters X and XI, polarised, these stresses would in fact exist. See Maxwell's Electricity, Second Edition, Chap. V. There would be at every point in the medium a tension along the lines of force, combined with a pressure at right-angles to them, and by such tensions and pressures all the observed phenomena may be accounted for without assuming the direct action of distant bodies on one another. It is true, as Maxwell says, that some action must be supposed between neighbouring molecules, and that we are no more able to account for that than for action between distant bodies. And if only electro- H XJNTVr X^C4LIF03N^^ 253.] POLARISATION OF THE DIELECTRIC. 249 statical phenomena were concerned, it would be perhaps of little importance whether we attributed them to direct action of distant bodies or to a medium, so long at least as the electric fluids and the medium were equally hypothetical, and had no other duties to perform than to account for the phenomena in question. The advantage of Maxwell's hypothesis is that it connects the phenomena of electricity and magnetism with those of light and radiant heat, both being referred to the vibrations of the same medium. There is, in fact, in the phenomena of light, inde- pendent evidence of the existence of Maxwell's medium, whereas there is no independent evidence of the existence of the two fluids. The medium therefore has better title to be regarded as a vera causa than the two fluids have. No treatment of the subject can, in the present state of know- ledge, be more complete than Maxwell's own in Chapters II and V of his work, and it is necessary to study those chapters in order properly to understand his views. The whole subject of statical electricity has also been treated very fully from Maxwell's point of view in the article ' Electricity ' in the Encyclopaedia Britan- nica, Ninth Edition, by Professor Chrystal. It may, however, be of some advantage to obtain the same results from a slightly different starting-point. 253.] In Chap. XI we had occasion to treat of a particular case of a polarised medium, a medium, namely, in which are interspersed little conductors polarised under the influence of given forces. If the induced distribution on the surface of any conductor be denoted by $, the quantity xtydS, taken over all the con- ductors in unit of volume, was defined to be the polarisation in direction x per unit of volume. We will now adopt a rather more general definition of polari- sation. Let us conceive a region containing an infinite number of molecules, conductors or not, each containing within it, or on its surface, a quantity of positive, and an equal quantity of nega- tive, electricity. Let P be a point in that region, and about P let there be taken a unit of volume, containing a very great number of the molecules in question. Let us further suppose 250 POLARISATION OF THE DIELECTRIC. [253. that throughout this unit of volume the distribution of the molecules in space, as well as the distribution of electricity in individual molecules, may be regarded as constant, and the same as in the immediate neighbourhood of P. Let $ dx dy dz be the quantity of electricity of the molecular distributions within the element of volume dxdydz. Then \\\$dxdydz throughout the unit of volume is zero; and we will define / / / x$ dxdydz taken throughout the unit of volume to be the polarisation in direction x at P. Let I] Ixfydxdydz = a x \ and let v y , a s have corresponding meanings for the axis of y and z. If a plane of unit area be drawn through P parallel to the plane of yz> it will intersect certain of the molecules. And the reasoning of Chap. XI (Art. 190) shews, that the quantity of electricity belonging to these intersected molecules which lies on the positive side of that unit of area is a- x . Similarly if the plane were parallel to az, or xy, the quantity of electricity of the intersected molecules on the positive side of the unit of area would be (T y or tr a in the respective cases. If the direction-cosines of the normal to the plane were , m, n, the quantity of electricity of the intersected molecules lying on the positive side of the unit of area would be lo- x + mo- y + n(r ls . For, by definition, the polarisation in the direction denoted by I, m, n is or = / / / (lx + my + nz) dx dy dz, that is, = I x dxdydz + m III y dxdydz + n z$ dxdydz ', that is, a- = Hence a- xt a- y} and o- z are components of a vector. If the distribution be continuous, so that o- x , a- vy and v z do not change abruptly at the point considered, the same reasoning as employed in Chap. XI shews that the amount of the distri- 2 5 3-] POLARISATION OF THE DIELECTRIC. 251 bution within the elementary parallelepiped dxdydz is where ^ da x 9 ~~ { dx Should the values of o- x , o- y , and v z change abruptly at the point in question, there will be over the unit of area a super- ficial or quasi-superficial distribution a- x a' x , where -=- > -7- vary continuously. d# ^ ds And also ZK-o-'sHw^-o-'^ + ^s-cr's) 1 .dV dV .dV dV dV dV' =r (T over surfaces of discontinuous values of these coefficients. It appears then that such a system of polarised molecules not only produces at all points in space the same potential as the system of volume and superficial distributions for which it was substituted, but also causes the distributions them- selves to reappear. It can be shewn also that the energy is the same in the two cases. For the polarised medium is in a state of constraint, because the separated electricities are not allowed to coalesce and neutralise each other. Work has been done upon it in producing- this state. In ordinary experiments the constrained state of the dielectric is produced by the intro- duction of charged bodies, and the work is the work done in * With the distribution of polarisation assumed in the text, if a small cylindrical region be described in the medium whose generating lines are parallel to the force at the point and infinitely smaller than the linear dimensions of the bounding planes, the force at any point within the cylinder is that arising entirely from the polarisation of the molecules completely included within the cylinder, and the total force from all the rest of the molecules is zero. If the polarisation were magnetic, this result would be expressed by saying that the law of magnetisation is such that the magnetic induction at every point is zero. 253-] POLARISATION OF THE DIELECTRIC. 253 charging' them, but according 1 to this theory the energy resides, not in the charged bodies, but in the dielectric. The energy in unit volume of the polarised system is R 2 , see Chap. V, that is, + h W h The energy of the entire system estimated in the same way is throughout the whole of dielectric space. But this is also the expression for the energy of the originally given system according to the ordinary theory, as shewn in Chap. X. The two systems are therefore for all purposes equivalent. We may conceive that the molecules of all dielectrics are capable of assuming such polarisation as required for this hypo- thesis. If, as we have hitherto supposed, vacuum be a perfect dielectric, it becomes necessary for the hypothesis to conceive it as permeated by a non-material ether, the molecules of which are capable of such electric polarisation. And if the existence of such an ether be assumed, it may be that in case of other dielectrics, the electric polarisation resides in the ether rather than in the molecules of the substance. We may further suppose that the essential property of con- ductors, as distinguished from dielectric media or insulators, is that their molecules are incapable of sustaining electric polarisa- tions, or that the substances of conductors are impermeable by the supposed ether, and therefore that no electric force and no free electricity can exist within them. We might thus construct a theory of electrostatics founded on the polarisation of the dielectric, just as the ordinary theory is founded on the property of conductors. In the ordinary theory the electromotive force at any point is the space differential of a function 7, which is constant through- out any conductor, and satisfies the condition V 2 F+4irp= at all points where there is free electricity of density p. Assuming that no case of electrostatic equilibrium has yet been discovered, which can be proved to be inconsistent with the ordinary theory, 254 STRESSES IN POLARISED DIELECTRIC. [254. it follows that the supposed dielectric polarisation must, when there is equilibrium, be the space differential of a function F, which is constant over and within every closed surface bounding the dielectric, and satisfies V 2 T+ 4 itp = at all points in the dielectric. The Stresses in the Dielectric. 254.] If any closed surface 8 separate one portion ^ of an electri- fied system from the other portion U 2 , as, for instance, if the whole of j^i be inside, and the whole of E 2 outside of S, then this hypo- thesis suggests an explanation of the phenomena without assuming any direct action between E 2 and E. For if the polarisation be given in magnitude and direction at each point in S, -=- is dv given at each point on S. ' Then we know that if the form and charge of every conductor within S be given, and if all fixed electrification within S be given, V has single and determinate value at all points within S. It follows that all electrical phenomena within S, which in the ordinary theory are due to the action of H 2 , are on the polarisa- tion hypothesis deducible from the given polarisation, that is the dV given value of -=- > at each point on 8. We might then always substitute for the external system E 2 a certain polarisation on 8, without affecting the equilibrium of U lm An example of this substitution has already been given (Art. 58) for the case where S is an equipotential surface. For T> then a distribution over S whose density is -- exerts the same 47T force as the external system at any point within S. If S be not equipotential we obtain a corresponding result as follows * : Let Y l be the potential of the external, T 2 that of the internal system, and V V^ + T 2 the whole potential. The whole force in direction x exerted by the external on the internal system is throughout the space within 8. * This investigation is taken from Maxwell's Treatise, Second Edition, Chap. V. 254-] STRESSES IN POLARISED DIELECTRIC. 255 But and within S V 2 F 2 = V 2 F. Hence tlie whole force is The object is to express this in the form of a surface integral over S. If we can find three functions X, J", Z, such that dx dx dy da then evidently, by Green's theorem, - V over the surface S. This is the required surface integral. Let us assume dVdV dVdV dx dy Then ** satisfy the condition. The quantities p xx , p yy , &c. are the six components of the stress on the surface S due to the polarisation of the dielectric. If S be an equipotential surface, we have dV p . dV dV Rl = _ R m , - = Rn, dx dy dz 256 STRESSES IN POLARISED DIELECTRIC. [ 2 54- where E is the normal force, and therefore a - The ^-component of stress is then that is R 2 1. Similarly the y- and ^-components are O7T That is, the stress is normal to 8 t and is equal to that of the -n force R acting on the surface electrified to a density J If S be at right angles to an equipotential surface, we find the stress in any element of it thus, in this case, ,dV dV dV / +m +n = ............... (1) dx dy dz Now 8 IT {lp xx H- mp xy + np xa } dVdV dVdV 3- +2w T--J- .. (2) o?c dy dx dz dV Multiplying (l) by 2 and subtracting from (2), we obtain Sir {ty xx + mp. xy + np x!} } =-lR*. Hence the components of tension perunit area are If therefore these stresses exist at every point of the surface /S, no matter how they arise, they produce on the interior system EI exactly the same effect as, according to the theory of action at 255-] SUPERFICIAL CHARGE ON A CONDUCTOR. 257 a distance, would be produced by the attraction and repulsions due to the external system E z . 255.] According to the theory of dielectric polarisation as explained in Art. 253, the so-called charge on a conductor is to be regarded as the terminal polarisation of the dielectric ; as belonging in fact not to the molecules of the conductor, but to the adjacent molecules of the dielectric. (Maxwell's Electricity^ Art. 111.) So long as we are dealing with a system at rest and in statical equilibrium, it is indifferent for all purposes of calculation whether we regard the charge as belonging to the dielectric or to the conductor. It is however possible to induce in any conductor or other solid body the state which in the ordinary theory is called a charge of electricity; and it is possible to move the body in this state from place to place through air without destroying its charge. It should seem therefore that although the electric force at any point in air may be due to the polarised state of the medium at the point, and not to direct action of the charged body, and although the polarised particles be always those of the dielectric, yet the ultimate cause of the phenomena may be in the body and not in the dielectric. And this appears to be Faraday's view, where he says (1298), 'Induction appears to consist in a certain polarised state of the particles into which they are thrown by the ' electrified body sustaining the action? Certain experiments have been appealed to as shewing that the electrification, whatever it be, is in the dielectric and not in the conductor. If, for instance, a plate of glass be placed between and touching two oppositely charged metallic plates, and these be then removed, it will be found that they exhibit scarcely any trace of electrification. If they be replaced and connected by a wire, a current passes of the same or nearly the same strength as if no removal had taken place. See Jamin, Cours de Physique, Leon 36. A similar result was obtained by Franklin with a Leyden jar, the metallic coatings of which were moveable. VOL. i. s 258 MAXWELL'S DISPLACEMENT THEOEY. [256. 256.] Up to this point we have not dispensed with the two- fluid theory and the law of the inverse square in electric action, because it is only by the use of that theory that we have proved the properties of our medium. All that we have done is to introduce a somewhat different conception of an electric field, and the distributions of which it is composed. If any advance is to be made, it must be in the steps of Faraday and Maxwell as follows : We observe that in the polarised medium the relation between the force at any point and the polarisation at the point is given by the equations JT = 4770-,,., &c. These equations are of the same form as those which express the relation between the force existing at any point in an elastic body in equilibrium and the molecular displacement at the point. In treating of elastic bodies we regard these relations as ulti- mate facts based on experiment. We might then regard the corresponding equations for the dielectric as ultimate facts, without resorting to the two fluid theory for their explanation. We might regard the dielectric as an elastic medium capable of being thrown into a state of strain, and presenting when in that state the phenomena which we call electric force and electric distribution. 257.] According to the theory in this form, no action is exerted by the electricity in any part of the dielectric on that in any other part, unless the two are contiguous. We might thus dispense with the notion of action at a distance, on which the ordinary theory is founded. Another characteristic of the ordinary theory is the instantaneous nature of the actions with which it deals. For, according to that theory, if any change take place in electrical distributions in any one part of space, the corresponding change takes place at the same instant in every other part however distant. The sub- stitution of the medium for the direct action between distant bodies, suggests that these corresponding electrical changes may not take place at the same instant, but that electrical influence may be propagated from molecule to molecule through the 258.] ELECTRIC DISPLACEMENT. 259 medium with a certain velocity. And herein lies the strength of the theory. For, as Maxwell discovered, if electrical effects are propagated with finite velocity through an insulating medium, such velocity is the same as that of light, or so nearly the same as to leave no room for doubt that the two classes of phenomena are physically connected. Again, an elastic medium, if thrown by any forces into a state of strain, does not on removal of those forces immediately recover its original condition. There is a time of relaxation. Certain phenomena, such as the residual charge of a Leyden jar (see Maxwell's Electricity, Chap. X), lend countenance to the supposition that a dielectric medium influenced by electric forces does not immediately, on the removal of those forces, recover its original condition. 258.] We proceed to consider the meaning of the term electric displacement as used by Maxwell, for which purpose we must revert to the conception of the two-fluid theory. If through any point in the medium of polarised particles a plane be drawn perpendicular to the v direction of the re- sultant force R at that point, the density or, per unit area of that plane, of the electricity on the particles intersected by that plane and on the positive side of it according to It's direction is, as we have seen, determined by the equation In Maxwell's view, any field of electromotive force in the dielec- tric is accompanied by a strained state of the particles of the dielectric or of the pervading ether, a displacement or transfer T) of positive electricity equal to - - per unit area of surface of the particles taking place from each particle to the adjacent particle on the positive side, along with an equal displacement of negative electricity to the adjacent particle on the negative side. The + or - electricities do not coalesce or neutralise each other within each particle, but a polarised state is set up throughout the field, each particle being in a strained state owing to the 260 ELECTRIC DISPLACEMENT. [259. separations of the electricities within it, the result being graphic- ally represented thus, Fig- 39- the shaded sides of the particles A, J5, C, &c. indicating positive, and the unshaded sides negative, electrification. According to this view the displacement is the process hy which the polarised state of the particles has been brought about. We shall gene- rally denote the polarisation by cr, and the displacement by/. It is easily seen that in a dielectric medium/ = cr. 259.] If the field were one of uniform force parallel to a line from left to right across the plane of the paper, the total displacement or transfer of electricity across all planes perpendicular to that line would be the same and equal to G> - per unit of area. Fig. 40. If the field were such as corre- sponds to what is called an elec- trified point 0, i.e. a charge within a very small volume about 0, the particles would be polarised as in the figure, the displacement (sup- posing no other charge in the field) taking place concentrically from within outwards, and the quantities o- x , a y , and a- z being so determined that d (r) - , or y = $ (r) ^ , (r) - . d(T x d(T y do-,, Therefore since -^- + -=-^- + -=*- = 0, ax ay az or of actual charge in the medium at P is determined by the equation d(T x d(T y d(r z _ dx dy dz But if K be the specific inductive capacity, we know from Chap. XI that whence we may, as in the preceding case, choose as our solution for o- x , a y) and a-,,, and their resultant o-, the equations K dV K dV K dV K v x = -y- j R. T 4-7T Therefore = /. and X = X e Kr , f and X being the initial values of /and X. These equations express the law of decay of the efficiency of condensers. 265.] According to Maxwell's doctrine, as we have already said, all electric currents flow in closed circuits. 265.] ELECTRIC DISPLACEMENT. 267 Let us recur to the case of the charged particle of Art. 259, which we have hitherto regarded as at rest within the medium, and suppose the charge to be unity. If it move from one position to another we have in effect a current of electricity from to (7. But from another point of view the effect is the same as if the particle in the first position were annihilated, and another similar particle placed in the second position; that is, as if a particle with unit positive charge were placed in the second position, and a particle with unit negative charge superadded to the positively charged particle in the first position. If denote the first, 0' the second position, P any point in space, the displacement at P due to the placing of a negative particle or annihilation of a positive particle at is a displacement - in direction PO. The displacement at P due to the positive particle at 0' is a displacement 47T If O f be infinitely near to 0, and 0(7 = a, we can find the equation to the resultant as follows : Let ZPOO' = 0. Let OQ = 3 a cos 0. Then if PO = r, P0'= r-a cos 0, and P(> = r-3a cos 0, PQ r 3acos0 r 2acos0 and ^ e = r-acosd r r 2 PO 2 Therefore the resultant is parallel to O'Q. Its equation is therefore dy SacosOsinO y a -~ = .> and - = tan 0. ax a 3 a cos 2 x Therefore ^ = dx 268 ELECTRIC DISPLACEMENT. The solution of which is where c is a variable parameter. This is the equation of a system of closed curves having 00" for a common tangent. It thus appears that if any quantity of positive electricity flows from to (/ (for our moving particle is equivalent for the purpose to a flow of positive electricity), we have a flow or current of electricity at every point in space, in direction form- ing closed curves with the line 0(7. From which it would seem, as we have already said, that there is no real change of position of all the positive electricity in space. Or, in other words, either kind of electricity behaves like an incompressible fluid, and the quantity of it within any finite space cannot be increased or diminished. If, for instance, the charge on the moving particle be unity, and it move from to 0', that is a distance a in unit of time, the current in 00' is a. 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