57o 



NATURE 



Oct. ii. 1888 



for a moment these potentials as rectangular components of 

 velocity in a case of liquid motion, taking the spin in this motion 

 as the velocity in the required motion. Applying this solution 

 to find the velocity in our Secondary from the velocity in our Ter- 

 tiary, we see that the three velocity components in our Primary 

 are the potentials of three ideal distributions of gravitational 

 matter having their densities respectively equal to 1/4T of the 

 three velocity components of our Tertiary. This proposition is 

 proved in a moment, 1 in § 5 below, by expressing the velocity 

 components of our Tertiary in terms of those of our Secondary, 

 and those of our Secondary in terms of those of our Primary ; 

 and then eliminating the velocity components of Secondary, so 

 as to have those of Tertiary directly in terms of those of Primary. 



(4) Consider now, in a fixed solid or solids of no magnetic 

 susceptibility, any case of electric motion in which there is no 

 change of electrification, and therefore no incomplete electric 

 circuit, or, which is the same, any case of electric motion in 

 which the distribution of electric current agrees with the distri- 

 bution of velocity in a case of liquid motion. Let this case, 

 with velocity of liquid numerically equal to 4ir times the electric 

 current density, be our Tertiary. The velocity in our corre- 

 sponding Secondary is then the magnetic force of the electric 

 current system ; 2 and the velocity in our Primary is what Max- 

 well 3 has well called the "electro-magnetic momentum at any 

 point" of the electric current system ; and the rate of decrease 

 per unit of time, of any component of this last velocity at any 

 point, is the corresponding component of electromotive force, due 

 to electro-magnetic induction of the electric current system when 

 it experiences any change. This electromotive force, combined 

 with the electrostatic force, if there is any, constitutes the whole 

 electromotive force at any point of the system. Hence by Ohm's 

 law each component of electric current at any point is equal to 

 the electric conductivity multiplied into the sum of the corre- 

 sponding component of electrostatic force and the rate of 

 decrease per unit of time of the corresponding component of 

 velocity of liquid in our Primary. 



(5) To express all this in symbols, let («j, v v iv{), (u. 2 , v. lt 7v ? ), 

 and (7/3, v 3 , w 3 ) denote rectangular components of the velocity 

 at time t, and point (x, y, z) of our Primary, Secondary, and 

 Tertiary. We have (§ 1) — 



Eliminating u 2 , v„, zv 2 from (2) by (1), we find — 



' _ d /£*, + dz\ + dw\ 



3 dx \ dx dy dz J 



S+SKSft* <» 



But, by our assumption (§ 2) of incompressibility in the Primary — 

 dy 



d« x du x div 1 



dx dy dz ™ 



Hence (3) becomes — 



ti 3 = - v' 3 */!, v. A = - v"v v «j = - v 2 ^ . . . (5) 



where, as in Article xxvii. (November 1846) of my " Collected 

 Papers " (vol. i. ) — 



d 2 , d- , d 2 



V 2 = ,^r + -, o +-72 (6) 4 



dx A dy dz v ' 



This (5) is the promised proof of § 3. 



(6) Let now u, v, w denote the components of electric current 

 at (x, y, z) in the electric system of § 4 ; so that — 



4Ttt = « 3 = -▼"«! ; 4ttv = v z = -V*t^; 4irw = w 3 — - v 2 ^ . (7) 

 which, in virtue of (4), give — 



du , dv , div 



_ + _ + — =0 



dx dy dz 



(8) 



1 From Poisson's well-known elementary theorem, V 2 V = - ^wp. 



2 "Electrostatics and Magnetism," § 517 (Postscript) (c). 



3 "Electricity and Magnetism," §§ 585, 604. 



4 Maxwell, for o.uaternionic reasons, takes v 2 the negative of mine. 



Hence the components of electromotive force due to change of 

 current, being, (§ 5) — 



_ du 3 _ dv 3 dw 3 

 ~dP M* dt % 



are equal to — 



„du n dv M „_*d-cu . v 



at dt dt 



and therefore if ¥ denote electrostatic potential, we have, for the 

 equations of the electric motion (§ 5) — 



_^du dV\ 

 V -— - — ; v = 



dt dx) 



I / n dw dV\ 

 dt dz) 



.*dv 

 dt 



d* 

 dy 



.(10) 



where k denotes \\%ir of the specific resistance. 



(7) As y is independent of t, according to § 4, we may, 

 conveniently for a moment, put — 



« + 4?=«i v+"=»; w+ d Z=y. . .(11) 

 KdX Kdy Krtz 



and so find, as equivalents to (9) — 



^=vV); d l = vH«fi); ^ = v> 7 ) . .(12) 



The interpretation of this elimination of "V may be illustrated 

 by considering for example a finite portion of homogeneous 

 solid conductor, of any shape (a long thin wire with two ends, 

 or a short thick wire, or a solid globe, or a lump of any shape, 

 of copper or other metal homogeneous throughout) with a constant 

 flow of electricity maintained through it by electrodes from a 

 voltaic battery or other source of electric energy, and with proper 

 appliances over its whole boundary, so regulated as to keep any 

 given constant potential at every point of the boundary ; while 

 currents are caused to circulate through the interior by varying 

 currents in circuits exterior to it. There being no changing 

 electrification by our supposition of § 4, V can have no 

 contribution from electrification within our conductor ; and 

 therefore, throughout our field — 



V-¥ = o (13) 



which, with (8) and (11), gives — 



da d$ 



dx dy 



dz 



•(14) 



Between (12) and (14) we have four equations for three unknown 

 quantities. These, in the case of homogeneousness (k constant), 

 are equivalent to only three, because in this case (14) follows 

 from (12) provided (14) is satisfied initially, and proper surface 

 condition is maintained to prevent any violation of it from 

 supervening. But unless k is constant throughout our field, the 

 four equations (12) and (14) are mutually inconsistent ; from 

 which it follows that our supposition of unchangingness of 

 electrification (§ 4) is not generally true. An interesting and 

 important practical conclusion is, that when currents are induced 

 in any way, in a solid com oosed of parts having different electric 

 conductivities (pieces of copper and lead, for example, fixed 

 together in metallic contact), there must in general be changing 

 electrification over every interface between these parts. This 

 conclusion was not at first obvious to me ; but it ought to be so 

 by anyone approaching the subject with mind undisturbed by 

 mathematical formulas. 



(8) Being thus warned off heterogeneousness until we come 

 to consider changing electrification and incomplete circuits, let 

 us apply (10) to an infinite homogeneous solid. As (8) holds 

 through all space according to our supposition in § 4, and as k 

 is constant, (13) must now hold through all space, and therefore 

 "V = o, which reduces (10) to — 



• dw 



V -3 



(•! 



These equations express simply the known law of elect 

 magnetic induction. Maxwell's equations (7) of § 783 of 

 " Electricity and Magnetism," become, in this case — 



fi( 4 nC + K d Y"=v-u,S:c (11 



\ dij at 



which cannot be right, I think (? ? ?), according to any conceivable 

 hypothesis regarding electric conductivity, whether of metals, or 



