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. [Illustration: FIG. 6.] The inverse of a circle is another circle, and therefore the inverse of a sphere is another sphere, and the inverse of a straight line is a circle passing through the centre of inversion, and of an infinite plane a sphere passing through the centre of inversion. Obviously the inverse of a sphere concentric with the sphere of inversion is a concentric sphere. The line P'Q' is of course not the inverse of the line PQ, which has for its inverse the circle passing through the three points O, P', Q', as indicated in Fig. 6. The following results are easily proved. A locus and its inverse cut any line OP at the same angle. To a system of point-charges q₁, q₂, ... at points P₁, P₂, ... on one side of the surface of the sphere of inversion there is a system of charges aq₁⧸f₁, aq₂⧸f₂, ... on the other side of the spherical surface [OP₁ = f₁, OP₂ = f₂]. This inverse system, as we shall call it, produces the same potential at any point of the sphere of inversion, as does the direct system from which it is derived. If V, V' be the potentials produced by the whole direct system at Q, and by the whole inverse system at Q', V'⧸V = r⧸a = a⧸r', where OQ = r, OQ' = r'. Thus if V is constant over any surface S', V' is not a constant over the inverse surface S', unless r is a constant, that is, unless the surface S' is a sphere concentric with the sphere of inversion, in which case the inverse surface is concentric with it and is an equipotential surface of the inverse distribution.