« PreviousContinue »
sions are insulated in symmetrical positions within a cylindrical conducting sheath of circular section.
these three equations by k, s, and respectively ; that is, to express
CASE I.-Two-wire Cable. In the general equations (according to the notation of the first part of this communication) we have k,=kg; w,(=,(2); and w, =w 2): and it will be convenient now to denote the values of the members of
e by k 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 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 c and f so that
92 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 9 and q, per unit of length. The equations of electrical conduction along the two wires then become
1 (do dt kc | dx?
(2). dv, 1 d'v,
+ kc dx?
dt kc dx2 where og and w are such that v=9+w, v.=9-w
(4). 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 0,=0.(r), and v,=02(x)
+ f dat
Q, and o, 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 x 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
v,=4,(t), and v,=42
to be fulfilled. In the other conditions, (5), only positive values of x 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 0. 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
the end o, let it be actually limited at a point E. If the ends of the
if X, and X. denote two arbitrary functions, and a the length OE.
Either of these requirements may be fulfilled in an obvious way by the method of successive images, and we so obtain the following respective solutions :
where F, F, E, E denote for brevity the following functions :
Each of the functions F and E is clearly the difference between two periodical functions of (&-x) and (8 + x); and each of the functions F and E is a periodical function of x simply. The expressions for these four functions, obtained by the ordinary formulæ for the expression of periodical functions in trigonometrical series, are as follows:
Either (11) or (12) may be used to obtain explicit expressions for
so obtained, (11) converge very rapidly and (12) very slowly when t
The solutions obtained by using (12), in (10) and (10)', are the
method given by Professor Stokes* to express the effects of the variations arbitrarily applied at the free ends of the wires.
CASE II.-Three-wire Cable. The equations of mutual influence between the wires may be clearly put under the forms
cv,=9. +f(92 +93), cv,=9z+f(93+1), cv,=9z+f(+92); and the equations of electrical motion along them are then as follows:
d 9: dʻ93
dt dx2 dx?
kc dqi_ dq
If we assume
and require that w, +w,+w,=0, we find by addition and subtraction, among the equations of conduction,
do = (1+20)
where for w may be substituted either W., W.,, or wg.
CASE III.-Four-wire Cable. The equations of mutual influence being
co,=4.+f(92 +9) +993, and other four symmetrical with this; and the equations of motion,
* See Cambridge Phil. Trans. vol. viii. p. 533, “On the Critical Values of the sums of Periodic Series."