THEORY OF ELECTRIC IMAGES AND ELECTRIC INVERSION.
155.] We have already shown that when a conducting sphere is under the influence of a known distribution of electricity, the distribution of electricity on the surface of the sphere can be determined by the method of spherical harmonics.
For this purpose we require to expand the potential of the influencing system in a series of solid harmonics of positive degree, having as the centre of the sphere as origin and we then find a corresponding series of solid harmonics of negative degree, which express the potential due to the electrification of the sphere.
By the use of this powerful method of analysis, Poisson determined the electrification of a sphere under the influence of a given electrical system and he also solved the more difficult problem to determine the distribution of electricity on two conducting spheres in presence of each other. These investigations have been pursued at great length by Plana and others, who have confirmed the accuracy of Poisson.
In applying this method to the most elementary case of a sphere under the influence of a single electrified point, we require to expand the potential due to the electrified point in a series of solid harmonics, and to determine a second series of solid harmonics which express the potential, due to the electrification of the sphere, in the space outside.
It does not appear that any of these mathematicians observed that this second series expresses the potential due to an imaginary electrified point, which as no physical existence as an electrified point, but which may be called an electrical image, because the action of the surface on external points is the same as that which would be produced by the imaginary electrified point if the spherical surface were removed,