An introduction to the mathematical theory of attraction ...

234 Electric Images.
123. Finite Series of Images and continuous
Inverse Systems.-The relation between systems of images
and the surface distributions to which they are equivalent
may usually be based on the following general theorem:If a closed surface S be one of equilibrium for a distribution of mass of which part is external to S and part internal,
a surface distribution on S, whose density at any point is
equal to the resultant force at that point divided by 47,
produces at all points external to S a potential equal to that
of the internal mass, and at all internal points a potential
whose difference from a constant is equal to the potential of
the external mass.
This theorem is easily proved by supposing a distribution
on S whose potential at all points of this surface is equal to
that of the internal mass.
Numerous problems of considerable difficulty have been
solved by Thomson, Clerk Maxwell, and other mathematicians,
by the use of images finite in number, or by the inversion
of one continuous system into another. Some of the most
important of these problems are given in the following
Examples:EXAMPLES.
1. An insulated conductor formed of the larger segments of two spheres
cutting at an angle -, where n is any integer, is charged to potential; find
the distribution of mass on the surface of the conductor, and the potential in
external space.
Adopting the method and the notation of Ex. 5, Art. 119, we find
i'2m = 2m (a +,3), j'2n = - 2n (a + j3),
i 2m+l = 2a + 2nzs (a + /3), j2nm+i = - 23 - 2m (a + /3).
In this case a + 3 = -, and therefore, whether n = 2m or 2ms + 1, we have
in -j' = 2n (a + 3) = 27r,
and the point ',, coincides with J'%.
In the inverse spherical system, Il and Ji being the centres of the spheres
(A) and (B), we have the images I2, 13, &c., I, in the sphere (A), and the
images J2, J3, &c., J,, in the sphere (B), and mJ coincides with I,. The charges
to be placed at these points, the distribution of mass on the conductor, and the