# Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/199

CHAPTER IX.

SPHERICAL HARMONICS.

On Singular Points at which the Potential becomes Infinite.

128.] We have already shewn that the potential due to a quantity of electricity $e$, condensed at a point whose coordinates are (a, b, c) is

 $V=\frac{e}{r}$ (1)

where r is the distance from the point (a, b, c) to the point (x, y, z), and V is the potential at the point (x, y, z).

At the point (a, b, c) the potential and all its derivatives become infinite, but at every other point they are finite and continuous, and the second derivatives of $V$ satisfy Laplace's equation.

Hence, the value of $V$, as given by equation (1), may be the actual value of the potential in the space outside a closed surface surrounding the point (a, b, c), but we cannot, except for purely mathematical purposes, suppose this form of the function to hold up to and at the point (a, b, c) itself. For the resultant force close to the point would be infinite, a condition which would necessitate a discharge through the dielectric surrounding the point, and besides this it would require an infinite expenditure of work to charge a point with a finite quantity of electricity.

We shall call a point of this kind an infinite point of degree zero. The potential and all its derivatives at such a point are infinite, but the product of the potential and the distance from the point is ultimately a finite quantity $e$ when the distance is diminished without limit. This quantity $e$ is called the charge of the infinite point.

This may be shewn thus. If $V'$ be the potential due to other electrified bodies, then near the point $V'$ is everywhere finite, and the whole potential is

$V=V'+\frac{e}{r},$

whence

$Vr+V'r+e.\,$