XXIII. Theorems on the attraction of ellipsoids for certain laws of force other than the inverse square

Abstract

Abstract 1. To find the potential at an internal point P of a thin ellipsoidal shell bounded by similar concentric ellipsoids, the law of force being the inverse kth power of the distance. Let (x, y, z) be the coordinates of P; (l, m, n) the direction cosines of a radius vector drawn from P to any point Q on the surface; and let PQ = r. Let (a, b, c) be the semi-axes of the ellipsoids, and (α, β, γ) their squared reciprocals. Then from the equation of the ellipsoid (αl2 + βm2 + γn2) r2 + 2 (αlx + βmy + γnz)r - (1 - αx2 - βy2 - γz2) = 0.