XIV. On toroidal functions

Abstract

The following investigation was originally undertaken as the foundation for certain researches on the theory of vortex rings, with especial reference to a theory of gravitation propounded by the author in the Proceedings of the Cambridge Philosophical Society (vol. iii., p. 276). As the results seemed interesting in themselves, and as they also serve as a basis for other investigations, more particularly in electricity and conduction of heat, I have thought it advisable to publish it as a separate paper, especially as I cannot hope to find leisure for some time to complete my original purpose. The word “tore” is used as a name for an anchor-ring, here restricted to a circular section, and by “toroidal functions” are understood functions which satisfy Laplace’s equation and which are suitable for conditions given over the surfaces of tores.