A curious and useful theorem in two-dimensional electrostatics

In 1956 metrologists Thompson and Lampard established a new theorem in two-dimensional electrostatics that gives calculable capacitances for practical systems possessing a certain symmetry. The theorem is well known among those concerned with electrical standards, but not so well known beyond. This paper brings the theorem to a wider audience with a pedagogical discussion of it, first: by treating a simple example by standard methods of electrostatics, then proving the Riemann mapping theorem on which the theorem is based. The Thompson-Lampard theorem and its generalization are then proved and its use in metrology described. Along the way, some curious results about series and products are exhibited. (C) 1999 American Association of Physics Teachers.