Note on the attraction of an ellipsoid in a spherical universe

TL;DR

This paper extends a classical Newtonian theorem about ellipsoids to spherical geometry, providing a stronger statement and a shorter proof by adapting Chasles' method.

Contribution

It generalizes a classical ellipsoid attraction theorem from Euclidean to spherical geometry with a novel, simplified proof.

Findings

The theorem applies to confocal ellipsoids with homeoidal mass distributions in spherical space.

The proof is shorter and more elegant than previous methods.

The result is a stronger statement than existing Euclidean counterparts.

Abstract

A classical theorem states that two confocal ellipsoids, each of them endowed with a surface distribution of mass which is called homeoidal, exert the same Newtonian force on an exterior point if they have the same total mass. We extend this theorem to the spherical geometry by adapting a forgotten proof by Chasles of the classical theorem. Compared to the existing results, this is a slightly stronger statement with a much shorter proof.