The fact that the value of s' does not involve the radius makes it possible to suppose the radius infinite, in which case we have the solution for a circular disk uninsulated and under the influence of a charge of electricity at a point Q in the same plane but outside the bounding circle.
Now consider the two parts of the spherical surface, the bowl B, and the remainder S of the spherical surface. Q with the charge − aV may be regarded as situated on the latter part of the surface. Any other influencing charges situated on S will give distributions on the bowl to be found as described above, and the resulting induced electrification can be found from these by summation. If S be uniformly electrified to density s, and held so electrified, the inducing distribution will be one given by integration over the whole of S, and the bowl B will be at zero potential under the influence of this electrification of S, just as if B were replaced by a shell of metal connected to the earth by a long fine wire. The densities are equal at infinitely near points on the two sides of B.
Let the bowl be a thin metal shell connected with the earth by a long thin wire and be surrounded by a concentric and complete shell of diameter f greater than that of the spherical surface, and let this shell be rigidly electrified with surface density − s. There will be no force within this shell due to its own electrification, and hence it will produce no change of the distribution in the interior. But the potential within will be − 2πfs, for the charge is − πf2s, and the capacity of the shell is ½f. The potential of the bowl will now be zero, and its electrification will just neutralise the potential − 2πfs, that is, will be exactly the free electrification required to produce potential 2πfs.
To find this electrification let the value of f be only infinitesimally greater than the diameter of the spherical surface of which B is a part; then the bowl is under the influence (1) of a uniform electrification of density − s infinitely close to its outer surface, and (2) of a uniform electrification of the same density, which may be regarded as upon the surface which has been called S above. It is obvious that by (1) a density s is produced on the outer surface of the bowl, and no other effect; by (2) an equal density at infinitely near points on the opposite sides of the bowl is produced which we have seen how to calculate. Thus the distribution on the bowl freely electrified is completely determined and the density can easily be calculated. The value will be found in Thomson's paper.
Interesting results are obtained by diminishing S more and more until the shell is a complete sphere with a circular hole in it. Tabulated results for different relative dimensions of S will be found in Thomson's paper, "Reprint of Papers," Articles V, XIV, XV. Also the reader will there find full particulars of the mathematical calculations indicated in this chapter, and an extension of the method to the case of an influencing point not on the spherical surface of which the shell forms part. Further developments of the problem have been worked out by other writers, and further information with references will be found in Maxwell's Electricity and Magnetism, loc. cit.
It is not quite clear whether Thomson discovered geometrical inversion independently or not: very likely he did. His letter to Liouville of date October 8, 1845, certainly reads as if he claimed the geometrical transformation as well as the application to electricity. Liouville, however, in his Note in which he dwells on the analytical theory of the transformation says, "La transformation dont il s'agit est bien connue, du reste, et des plus simples; c'est celle que M. Thomson lui-même a jadis employée sous le nom de principe des images." In Thomson and Tail's Natural Philosophy, § 513, the reference to the method is as follows: "Irrespectively of the special electric application, the method of images gives a remarkable kind of transformation which is often useful. It suggests for mere geometry what has been called the transformation by reciprocal radius-rectors, that is to say...." Then Maxwell, in his review of the "Reprint of Papers" (Nature, vol. vii), after referring to the fact that the solution of the problem of the spherical bowl remained undemonstrated from 1846 to 1869, says that the geometrical idea of inversion had probably been discovered and rediscovered repeatedly, but that in his opinion most of these discoveries were later than 1845, the date of Thomson's first paper.[10]
A very general method of finding the potential at any point of a region of space enclosed by a given boundary was stated by Green in his 'Essay' for the case in which the potential is known for every point of the boundary. The success of the method depends on finding a certain function, now called Green's function. When this is known the potential at any point is at once obtained by an integration over the surface. Thomson's method of images amounts to finding for the case of a region bounded by one spherical surface or more the proper value of Green's function. Green's method has been successfully employed in more complicated cases, and is now a powerful method of attack for a large range of problems in other departments of physical mathematics. Thomson only obtained a copy of Green's paper in January 1845, and probably worked out his solutions quite independently of any ideas derived from Green's general theory.