Bipolar and Toroidal Harmonics
Proceedings of the Edinburgh Mathematical Society
Most of the solutions of Laplace's equation in common use in mathematical physics have been expressed in the integral form given by Whittaker,* viz. f 2?r I /(xcos t + y sin t + iz, t) dt. Jo The solutions are well known for which f, regarded as a function of its first argument, is a power, a circular function, a Legendre function of the first or second kind, or a Bessel function. Thus this general form of solution provides a means of classifying known potential functions and of suggesting new
... of suggesting new ones. It is therefore not without interest to express outstanding solutions of Laplace's equation in this form. In this paper it will be shown that bipolar harmonics of integral order or potential functions for two spheres are obtained by taking / to be a certain rational function of its first argument. The corresponding forms for toroidal harmonics are deduced. In each case the zonal and sectorial harmonics take a simple form, while the tesseral harmonics are somewhat more complicated, If TS, z, 9 are cylindrical coordinates, we may define a set of orthogonal curvilinear coordinates u, v, w, by means of the relations The surfaces corresponding to constant values of u are a set of coaxial spheres having z -0 for their common radical plane. By suitably choosing the position of the origin and the value of the constant a it is possible to ensure that any two given spheres are included in this set. * Mathtm. Analen, 1903, 57, p. 333.