On the convergence of the theory of figures

The so-called theory of figures (TOF) uses potential theory to solve for the structure of highly distorted rotating liquid planets in hydrostatic equilibrium. An apparently divergent expansion for the gravitational potential plays a fundamental role in the traditional TOF. This questionable expansion, when integrated, leads to the standard geophysical expansion of the external gravitational potential on spherical-harmonics (via the usual J-coefficients). We show that this expansion is convergent and exact on the planet's level surfaces, provided that rotational distortion does not exceed a critical value. We examine the general properties of the Maclaurin multipole expansion and discuss conditions for its convergence on the surface of both single and nested-concentric Maclaurin spheroids.