136 Royal Society : — - -^iuriiivui .i^/i 



in four, six, or more circular passages pfivallel to its length, will cor- 

 respond to an ordinary telegraph cable containing the same number 

 of copper wires separated from one another only by gutta-percha : 

 and the hydraulic motion will follow- rigorously the same laws as the 

 electrical conduction, and will be expressed by identical language in 

 mathematics, provided the lateral dimensions of the bores are so 

 small in comparison with their lengths, or the viscosity of the liquid 

 so great, that the motions are not sensibly affected by inertia, and 

 are consecpiently dependent altogether on hydrostatic pressure and 

 fluid friction. The electrical induction now alluded to depends on 

 the electrostatic forces determined by Coulomb ; hut it would be in 

 one respect a real, and in all respects an apparent, contradiction of 

 terms, to speak of electrostatic induction of electric currents, and I 

 therefore venture to introduce the term peristaltic to characterize 

 that kind of induction by which currents are excited in elongated 

 conductors through the variation of electrostatic potential in the sur- 

 rounding matter. On the other hand, as any inductive excitation of 

 electric motion might be called electro-dynamic induction, it will be 

 convenient to distinguish the kind of electro-dynamic induction first 

 discovered by Faraday, by a distinctive name ; and as the term elec- 

 tro-magnetic, which has been so applied, appears correctly character- 

 istic, I shall call electro-magnetic iiuhiction that kind of action by 

 which electric currents are excited, or inequalities of electric potential 

 sustained, in a conductor of electricity, by variations of magnetic or 

 electro-magnetic potential, or by absolute or relative motion of the 

 conductor itself across lines of magnetic or electro-magnetic force. 



The most general problem of peristaltic induction is to determine 

 the motion of electricity in any number of long conducting wires, 

 insulated from one another within an uninsulated tube of conducting 

 material, when subjected each to any prescribed electrical action at 

 its extremities ; without supposing any other condition regarding the 

 sections and relative dispositions of the conductors than — (1), that 

 their lateral dimensions and mutual distances are so small in propor- 

 tion to their lengths, that the effects of peristaltic induction are para- 

 mount over those of electro-magnetic induction ; and (2), that the 

 section of the entire system of conductors, if not uniform in all parts, 

 varies so gradually as to be sensibly uniform through every part of 

 the length not a very large multiple of the largest lateral dimension. 

 In the present communication I shall only give the general equations 

 of motion by which the physical conditions to be satisfied are ex- 

 pressed for every case ; and I shall confine the investigation of solu- 

 tions to certain cases of uniform and symmetrical arrangement, such 

 as are commonly used in the submarine telegraph cable. 



At any time t, let q-^, q^, q^ &c. be the quantities of electricity 

 with which the different wires are charged, per unit of length of each, 

 at a distance x from one extremitj^, O, of the conducting system ; 

 and let v,, v.p v^, &c. be the electrostatical potentials in the same 

 parts of those conductors. Let OTj^D, cTj^'^', •nr,(3), &c., W", ro'jt^', 

 cr.,(^>, &c., 'Z3'.j('\ '^■J'^K "^-.p^ &c. be coefficients, such that the electro- 

 statical potentials (wj, v,^, &c.), due to stated charges (</j, q^, &c.) 

 of the different wires, are expressed by the equations 



