The potential for a homogeneous spheroid in a spheroidal coordinate system. II. At an interior point

Abstract

For pt.I see ibid., vol.21, p.4245-50 (1988). The internal potential for a homogeneous spheroid in a rectangular coordinate system may be obtained with the substitution of the root of the equation for confocal spheroids being zero in the expression of the external potential. In a spheroidal coordinate system, the integration of the internal potential would be composed of two parts: inner and outer volumes with respect to the interior point of a spheroid, which generates the explicit forms of the internal potential and the expressions on spheroidal surface, where the external and internal potentials should merge, for both prolate and oblate spheroids. The reductions for the potential at the centre of the spheroid to a sphere and the numerical results compared between the rectangular and spheroidal coordinate systems have certainly confirmed the correctness of the formulae of the internal potentials for both spheroids.