## Chapter 6POTENTIAL THEORY

‘But all that moveth doth Mutation love.’

SPENSER, *The Faerie Queene*, Bk. 7

### 6·01. 1/*r* as a solution of ∇^{2}*ϕ* = 0.

Let *x*_{i} be the coordinates of a point *P* and *r* its distance from the origin. Then

Now put *k* = *i* and apply the summation convention; since

so that 1/*r* is a solution of Laplace’s equation, ∇^{2}*ϕ* = 0, except at the origin.

It follows at once that if *ξ*_{i} are the coordinates of another point *Q*, and

(summed), then ...