Influence of any Number of Electrified Points.
Now let us consider the sphere as divided into two parts, one of which, the spherical segment on which we have determined the electric distribution, we shall call the bowl, and the other the remainder, or unoccupied part of the sphere on which the influencing point is placed.
If any number of influencing points are placed on the remainder of the sphere, the electricity induced by these on any point of the bowl may be obtained by the summation of the densities induced by each separately.
179.] Let the whole of the remaining surface of the sphere be uniformly electrified, the surface-density being , then the density at any point of the bowl may be obtained by ordinary integration over the surface thus electrified.
We shall thus obtain the solution of the case in which the bowl is at potential zero, and electrified by the influence of the remaining portion of the spherical surface rigidly electrified with density .
Now let the whole system be insulated and placed within a sphere of diameter , and let this sphere be uniformly and rigidly electrified so that its surface-density is .
There will be no resultant force within this sphere, and therefore the distribution of electricity on the bowl will be unaltered, but the potential of all points within the sphere will be increased by a quantity where
Hence the potential at every point of the bowl will now be .
Now let us suppose that this sphere is concentric with the sphere of which the bowl forms a part, and that its radius exceeds that of the latter sphere by an infinitely small quantity.
We have now the case of the bowl maintained at potential and influenced by the remainder of the sphere rigidly electrified with superficial density .
180.] We have now only to suppose , and we get the case of the bowl maintained at potential and free from external influence.
If is the density on either surface of the bowl at a given point when the bowl is at potential zero, and is influenced by the rest of the sphere electrified to density , then, when the bowl is maintained at potential , we must increase the density on the outside of the bowl by , the density on the supposed enveloping sphere.