On Dirichlet's Derivation of the Ellipsoid Potential

TL;DR

This paper revisits Dirichlet's method for deriving the potential of a homogeneous ellipsoid, extending it into the complex plane to address mathematical issues and improve the derivation process.

Contribution

It extends Dirichlet's original approach into the complex plane, enabling the calculation of both potential and force components for ellipsoids.

Findings

Dirichlet's method can be extended into the complex plane.

The extension allows for a mathematically acceptable derivation.

Potential and force components can be calculated using the extended method.

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.