126 



Similarly, we have 



Hence, by the general theorem, we conclude [V 2 ], = [V,] 2 , 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 same 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, 



/>=,(>; 7 1 W=r 3 ( 1 ); vr t W=v 9 W; &c. 



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



trical systems of Submarine Telegraph Wires. 

 The general method which has just been indicated for resolving 

 the equations of electrical motion in any number of linear conductors 

 subject to mutual peristaltic influence, fails when these conductors 

 are symmetrically arranged within a symmetrical conducting sheath 

 (and therefore actually in the case of any ordinary multiple wire tele- 

 graph cable), from the determinantal equation having sets of equal 

 roots. Regular analytical methods are well known by which the solu- 

 tions for such particular cases may be derived from the failing general 

 solutions ; but it is nevertheless interesting to investigate each par- 

 ticular case specially, so as to obtain its proper solution by a synthe- 

 tical process, the simplest possible for the one case considered alone. 

 In the present communication, the problem of peristaltic induction 

 is thus treated for some of the most common cases of actual sub- 

 marine telegraph cables, in which two or more wires of equal dimen- 



