Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/210

 Then, by Green’s theorem, the potential energy of $$E$$ on the electrified surface is equal to that of the electrified surface on $$E$$, or

where the first integration is to be extended over every element $$dS$$ of the surface of the sphere, and the summation $$\sum$$ is to be extended to every part $$dE$$ of which the electrified system $$E$$ is composed.

But the same potential function $$V_i$$ may be produced by means of a combination of $$2^i$$ electrified points in the manner already described. Let us therefore find the potential energy of $$E$$ on such a compound point.

If $$M_0$$ is the charge of a single point of degree zero, then $$M_{0}V$$ is the potential energy of $$V$$ on that point.

If there are two such points, a positive and a negative one, at the positive and negative ends of a line $$h_i$$, then the potential energy of $$E$$ on the double point will be

and when $$M_0$$ increases and $$h_1$$ diminishes indefinitely, but so that

the value of the potential energy will be for a point of the first degree

Similarly for a point of degree $$i$$ the potential energy with respect to $$E$$ will be

This is the value of the potential energy of $$E$$ upon the singular point of degree $$i$$. That of the singular point on $$E$$ is $$\sum V_{i}dE$$ and, by Green’s theorem, these are equal. Hence, by equation (50),

If $$\sigma=CY_{i}$$ where $$C$$ is a constant quantity, then, by equations (49) and (14),

Hence, if $$V$$ is any potential function whatever which satisfies Laplace’s equation within the spherical surface of radius $$a$$, then the