The use of the complex variable z( = x + iy) and the complex potential W(= U + iV) for two-dimensional electrostatic systems is well known and the actual system in the (x, y) plane has an image system in the (U, V) plane. It does not seem to have been noticed previously that the electrostatic energy per unit length of the actual system is simply related to the area of the image domain in the (U, V) plane.

We consider a linear partial differential equation with constant coefficients in one dependent and m independent variables, the right-hand side being zero,

P being a symbolic polynomial in D1, …, Dm. If P can be decomposed into two factors, so that the equation can be written

it is evident that the sum of any solution of

and any solution of

is also a solution of (1).

Various improvements in the formula

which was discovered by Wallis in 1669, were studied by D. K. Kazarinoff in No. 40 of these Notes (December 1956).

A and B study an array of counters, placed in straight lines on a table, in separated groups (e.g. 28, 53, 1, 16). The counters are to be removed under the following conditions :–

Each “player” may take away

(1) one counter from anywhere (e.g. 27, 53, 1, 16) or (28, 20, 32, 1, 16) etc.

or (2) three counters from anywhere (e.g. 28, 53, 14) or (27, 52, 1, 15) or (28, 25, 17, 6, 2, 1, 16) etc.

or (3) two adjacent counters from one group (e.g. 28, 51, 1, 16) or (28, 53, 1, 8, 6) etc.