Prof. Thomson on Peristaltic Induction of Electric Currents. 139 



distribution of matter consist of a certain charge, q, communicated to 

 one of a group of insulated conductors, and the inductive electrifica- 

 tions of the others, not one of which has any absolute charge ; let 

 the second distribution of matter consist of the electrifications of the 

 same group of conductors when an equal quantity q is given to a 

 second of them, and all the others are destitute of absolute charges ; 

 and let surfaceS be the group of the surfaces of the different conductors. 

 Since the potential is constant through each separate conductor, the 

 integral //"Ni Vj ds will be equal to the sum of a set of terms of the 



form \y iW^JT^ xds], where [V2] denotes the value in any of these 

 conductors of the potential of the second distribution, and \_ff'^xds] 

 an integral including the whole surface of the same conductor, but 

 no part of that of any of the others. Now by a well-known theorem, 

 first given by Green, [ //"Njf/s] is equal to Aitq if q denote the abso- 

 lute quantity of matter within the surface of the integral (as is the case 

 for the first group of conductors), and vanishes if there be no distri- 

 bution of matter, or (as is the case with each of the other conductors) 

 if there be equal quantities of positive and negative matter within the 

 surface over which the integral is extended. Hence if [Vj]j denote 

 the potential in the first conductor due to the second distribution of 

 matter, we have 



Similarly, we have 



Hence, by the general theorem, we conclude [V2]i = [V,].„ and so 

 demonstrate the affirmative answer to the question stated above. 



I think it unnecessary to enter on details suited to the particular 

 case of lateral electrostatic influence between neighbouring parts of a 

 number of wires insulated from one another under a common con- 

 ducting sheath, when uniform or varying electric currents are sent 

 through by them ; for which a particular demonstration in geometry 

 of two dimensions, analogous to the demonstration of Green's theorem 

 to which I have referred as involving the consideration of a triple 

 integral for space of three dimensions, may be readily given ; but, as 

 a particular case of the general theorem I have now demonstrated, it 

 is obviously true that the potential in one wire due to a certain quan- 

 tity of electricity per unit of length in the neighbouring parts of an- 

 other under the sr,:iie sheath, is equal to the potential in this other, 

 due to an equal electrification of the first. 



Hence the following relations must necessarily subsist among the 

 coefficients of mutual peristaltic induction in the general equations 

 given above, 



ctW = <'); vt,^^^=isW; ^,p = ^.p) ■ &c. 



On the Solution of the Equations of Peristaltic Induction in symme- 

 trical Hijslems of Submarine Tdeyraph Wires. 

 The general method wliich lias just been indicated for resolving 

 the equations of electrical motion in any number of linear conductors 



