Equilibrium of inverse-distance forces in three-dimensions

Abstract

This paper is about the field of force in three dimensions due to particles that repel according to the inverse-distance law, the analog of the field in the complex plane used by C. F. Gauss in studying the zeros of the derivative of a polynomial. In particular it deals with the analog in three dimensions of Jensen's theorem and related theorems in the plane, shows that the positions of equilibrium in the field of force when the total mass is zero are invariant under spherical transformation, and gives a geometric construction to find the force due to an axially symmetric distribution of matter.