The potential for a homogeneous spheroid in a spheroidal coordinate system. I. At an exterior point

The explicit expressions of the external potential for a homogeneous spheroid in a rectangular coordinate system have been studied previously by several researchers. The author presents the simplest forms of the potential in existing expressions, as derived from the reciprocal of the distance between two points expanded with the Legendre functions of the first and second kind in a spheroidal coordinate system. The numerical comparison of the potentials in spheroidal and rectangular coordinate systems shows exactly their identity. It is now feasible to calculate the gravitational potential for an astronomical body have spheroidal shape in its spheroidal coordinate system and to determine directly the equipotential surfaces by setting the corresponding spheroidal coordinate to a constant.