It is to be observed that k1, k2, &c., w,(1), w, (2), w,, &c. will be functions of x if the section of the conducting system is heterogeneous in different positions along it; but in all cases in which each conductor is uniform, and uniformly situated with reference to the others along the whole length, these coefficients will be constant, and the equations become reduced to

The most obvious general method of treatment for integrating these equations, is to find elementary solutions by assuming

q=A1u, q=Au, q=A ̧u,..... qi=Au,

where u satisfies the equation

(6),

This will reduce the differential equations (5) to a set of linear equations among the coefficients A, A, . . . . A,, giving by elimination an algebraic equation of the ith degree having i real roots, to determine x. The particular form of elementary solution of the equation (7) to be used may be chosen from among those given by Fourier, according to convenience, for satisfying the terminal conditions for the different wires.

In thinking on some applications of the preceding theory, I have been led to consider the following general question regarding the mutual influence of electrified conductors :-If, of a system of detached insulated conductors, one only be electrified with a given absolute charge of electricity, will the potential excited in any one of the others be equal to that which the communication of an equal absolute charge to this other would excite in the first? I now find that a general theorem communicated by myself to the Cambridge Mathematical Journal, and published in the Numbers for November 1842 and February 1843, but, as I afterwards (Jan. 1845) learned, first given by Green in his Essay on the Mathematical Theory of

Electricity and Magnetism (Nottingham, 1828), leads to an affirmative answer to this question.

2

The general theorem to which I refer is, that if, considering the forces due respectively to two different distributions of matter (whether real, or such as is imagined in theories of electricity and magnetism), we denote by N1, N, their normal components at any point of a closed surface, or group of closed surfaces, S, containing all parts of each distribution of matter, and by V1, V, the potentials at the same point due respectively to the two distributions, and if ds be an element of the surface S, the value of N,V,ds is the same as that of N,V,ds (each being equal to the integral SSR, R, sin 0 da dy dz extended over the whole of space external to the surface S, at any point (x, y, z) of which external space the two resultants are denoted by R1, R, respectively, and the angle between their directions by 0). To apply this with reference to the proposed question, let the first distribution of matter consist of a certain charge, q, communicated to one of a group of insulated conductors, and the inductive electrifications 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 N, V, ds will be equal to the sum of a set of terms of the form [V][N,ds], where [V] denotes the value in any of these conductors of the potential of the second distribution, and [ƒN1ds] 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, [ƒƒN1ds] is equal to 47q if q denote the absolute 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 distribution 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 [V], denote the potential in the first conductor due to the second distribution of matter, we have

૧

Hence, by the general theorem, we conclude [V]=[V1],, 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 conducting 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 quantity of electricity per unit of length in the neighbouring parts of another 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,

On the Solution of the Equations of Peristaltic Induction in symmetrical 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 telegraph cable), from the determinantal equation having sets of equal roots. Regular analytical methods are well known by which the solutions for such particular cases may be derived from the failing general solutions; but it is nevertheless interesting to investigate each particular case specially, so as to obtain its proper solution by a synthetical 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 submarine telegraph cables, in which two or more wires of equal dimen

sions are insulated in symmetrical positions within a cylindrical conducting 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 k1 = k2; w,()=w,(2); and w,(1), (2); and it will be convenient now to denote the values of the members of

these three equations by k,

1

f

and respectively; that is, to express

с

by k the galvanic resistance in each wire per unit of length, by e the electrostatical capacity of each per unit of length when the other is prevented from acquiring an absolute charge, and by f the proportion 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 e and ƒ so that

if v, and v be the potentials in the two wires in any part of the cable where they are charged with quantities of electricity respectively q, and q per unit of length. The equations of electrical conduction along the two wires then become

From these we have, by addition and subtraction,

(2).

If both wires reached to an infinite distance in each direction, the conditions to be satisfied in integrating the equations of motion would be simply that the initial distribution of electricity along each must be whatever is prescribed; that is, that

when

v1=0,(x), and v2=42(x) }

t=0

(5),

, and denoting two arbitrary functions. Hence, according to Fourier, we have, for the integrals of the equations (3),

and the solution of the problem is expressed in terms of these integrals by (4).

If now we suppose the cable to have one end at a finite distance from the part considered, for instance at the point O from which a is reckoned, and if at this end each wire is subjected to electric action so as to make its potential vary arbitrarily with the time, there will be the additional condition

to be fulfilled. In the other conditions, (5), only positive values of a have now to be considered, but they must be fulfilled in such a way as not to interfere with the prescribed values of the potentials at the ends of the wires; which may be done according to the principle of images, by still supposing the wires to extend indefinitely in both directions, and in the beginning to be symmetrically electrified with contrary electricities on the two sides of O. To express the new condition (7), a form of integral, investigated in a communication to the Royal Society (Proceedings,' May 10, 1855, p. 385), may be used; and we thus have for the integrals of equations (3),

Lastly, instead of the cable extending indefinitely on one side of