Now let this induced distribution, on the two concentric spheres, be inverted from P as centre of inversion. We obtain two non-intersecting spheres, as in Fig. 5, for the inverse geometrical system, and for the inverse electrical system an equilibrium distribution on these two spheres in presence of one another, and charged with the charges which are the inverses of the induced charges. These maintain the system of two spheres at one potential. From this inversion it is possible to proceed as shown by Maxwell in his _Electricity and Magnetism_, vol. i, § 173, to the distribution on two spheres at two different potentials; but we have shown above how the problem may be dealt with directly by the method of images. [Illustration: FIG. 8.] Again take the case of two parallel infinite planes under the influence of a point-charge between them. This system inverted from P as centre gives the equilibrium distribution on two charged insulated spheres in contact (Fig. 8); for this system is the inverse of the planes and the charges upon them. Another interesting case is that of the "electric kaleidoscope" referred to above. Here the two infinite conducting planes are inclined at an angle 360°⧸n, where n is a whole number, and are therefore bounded in one direction by the straight line which is their intersection. The image points I₁, J₁, ..., of P placed in the angle between the planes are situated as shown in Fig. 3, and are n － 1 in number. This system inverted from P as centre gives two spherical surfaces which cut one another at the same angle as do the planes. This system is one of electrical equilibrium in free space, and therefore the problem of the distribution on two intersecting spheres is solved, for the case at least in which the angle of intersection is an aliquot part of 360°. When the planes are at right angles the result is that for two perpendicularly intersecting planes, for which Fig. 9 gives a diagram. [Illustration: FIG. 9.] But the greatest achievement of the method was the determination of the distribution on a segment of a thin spherical shell with edge in one plane. The solution of this problem was communicated to M. Liouville in the letter of date September 16, 1846, referred to above, but without proof, which Thomson stated he had not time to write out owing to preparation for the commencement of his duties as Professor of Natural Philosophy at Glasgow on November 1, 1846. It was not supplied until December 1868 and January 1869; and in the meantime the problem had not been solved by any other mathematician. As a starting point for this investigation the distribution on a thin plane circular disk of radius a is required. This can be obtained by considering the disk as a limiting case of an oblate ellipsoid of revolution, charged to potential V, say. If Fig. 10 represent the disk and P the point at which the density is sought, so that CP = r, and CA = a, the density is V⧸{2π²√(a² － r²)}.