Fig. 5.
In the same way these two induced distributions on S2 may now be regarded as reacting on the distribution on S1 as would point-charges − a2q1 ⁄ f and a2a1q2 ⁄ ff', situated at J1 and J2 respectively, and would give two induced distributions on S1 corresponding to their images in S1.
Thus by partial influences in unending succession the equilibrium state of the two spheres could be approximated to as nearly as may be desired. An infinite trail of electric images within each of the two spheres is thus obtained, and the final state of each conductor can be calculated by summation of the effects of each set of images.
If the final potentials, V1, V2, say, of the spheres are given the process is somewhat simpler. Let first the charges be supposed to exist uniformly distributed over each sphere, and to be of amount a1V1, a2V2 in the two cases. The uniform distribution on S1 will raise the potential of S2 above V2, and to bring the potential down to V2 in presence of this distribution we must place an induced distribution over S2, represented as regards the external field by the image-charge − a2a1V1 ⁄ f (at the image of C1 in S2) where f is the distance between the centres. The charge a2V2 on S2 will similarly have an action on S1 to be compensated in the same way by an image-charge − a1a2V2 ⁄ f at the image of C2 in S1. Now these two image-charges will react on the spheres S1 and S2 respectively, and will have to be balanced by induced distributions represented by second image-charges, to be found in the manner just exemplified. These will again react on the spheres and will have to be compensated as before, and so on indefinitely. The charges diminish in amount, and their positions approximate more and more, according to definite laws, and the final state is to be found by summation as before.
The force of repulsion is to be found by summing the forces between all the different pairs of charges which can be formed by taking one charge of each system at its proper point: or it can be obtained by calculating the energy of the system.
The method of successive influences was given originally by Murphy, but the mode of representing the effects of the successive induced charges by image-charges is due to Thomson. Quite another solution of this problem is, however, possible by Thomson's method of electrical inversion.
A similar process to that just explained for two charged and mutually influencing spheres will give the distribution on two concentric conducting spheres, under the influence of a point-charge q at P between the inner surface of the outer and the outer surface of the inner, as shown in Fig. [7]. There the influence of q at P, and of the induced distributions on one another, is represented by two series of images, one within the inner sphere and one outside the outer. These charges and positions can be calculated from the result for a single sphere and point-charge.
Thomson's method of electrical inversion, referred to above, enabled the solutions of unsolved problems to be inferred from known solutions of simpler cases of distribution. We give here a brief account of the method, and some of its results. First we have to recall the meaning of geometrical inversion. In Fig. [6] the distances OP, OP', OQ, OQ' fulfil the relation OP.OP' = OQ.OQ' = a2. Thus P' is (see p. [37]) the inverse of the point P with respect to a sphere of radius a and centre O (indicated by the dotted line in Fig. [6]), and similarly Q' is the inverse of Q with respect to the same sphere and centre. O is called the centre of inversion, and the sphere of radius a is called the sphere of inversion. Thus the sphere of Figs. [1] and [4] is the sphere of inversion for the points P and P', which are inverse points of one another. For any system of points P, Q, ..., another system P', Q', ... of inverse points can be found, and if the first system form a definite locus, the second will form a derived locus, which is called the inverse of the former. Also if P', Q', ... be regarded as the direct system, P, Q, ... will be the corresponding inverse system with regard to the same sphere and centre. P' is the image of P, and P is the image of P', and so on, with regard to the same sphere and centre of inversion.
