140 Royal Society : — "■ i 



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 M'ell 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- 

 sions are insulated in symmetrical positions within a cylindrical con- 

 ducting sheath of circular section. 



Case I. — Two-wire Cable. 



In the general equations (according to the notation of the first part 

 of this communication) we have k^^/t.^; ■m^^^^^^^^^-'); andvrJ^^^vTi'-'^^: 

 and it will be convenient now to denote the values of the members of 



1 f 



these three equations by k, -, and ^ respectively ; that is, to express 



c c 



by /c the galvanic resistance in each wire per unit of length, by c the 

 electrostatical capacity of each per unit of length when the other is 

 prevented from acquiring an absolute charge, and by /the propor- 

 tion in which this exceeds the electrostatical capacity of each when 

 the other has a charge equal to its own ; or in other words, to assume 

 c and /so that 



1 / T 



; : '"■ 



if ?j, and v^ be the potentials in the two wires in any part of the 

 cable where they are charged with quantities of electricity respectively 

 qi and q^ per unit of length. The equations of electrical conduction 

 along the two wires then become 



dv^ 1 ffd^-v^ ^'^'"■A f 



di ~ Ac V rf^ '^Ix^Jj 



From these we have, by addition and subtraction, 



^ = i±f^.and'^'"=L=^^ . . . (3), 

 dt kc dx"^ dt kc dx'^ 



where S' and w are such that 



v,=S'-|-w, v^=d—w (4). 



'. If both wires reached to au infinite distance in each direction, the 



(2). 



