On Dirichlet's Derivation of the Ellipsoid Potential

Abstract

Newton's potential of a massive homogeneous ellipsoid is derived via Dirichlet's discontinuous factor. At first we review part of Dirichlet's work in an English translation of the original German, and then continue with an extension of his method into the complex plane. With this trick it becomes possible to first calculate the potential and thereafter the force components exerted on a test mass by the ellipsoid. This is remarkable in so far as all other famous researchers prior to Dirichlet merely calculated the attraction components. Unfortunately, Dirichlet's derivation is to a large extent mathematically unacceptable which, however, can be corrected by treating the problem in the complex plane.