Convergence of Liapunov Series for Maclaurin Ellipsoids: Real Analysis

According to the classical theory of equilibrium figures, surfaces of equal density, potential and pressure concur (let us call them isobars). Isobars can be represented by means of Liapunov power series in a small parameter q, in the first approximation coinciding with the centrifugal to gravitational force ratio at the equator. In 5] the convergence radius for Liapunov series was found in case of homogeneous equilibrium figures (Maclaurin ellipsoids). Isobars were described by means of implicit functions of q. The proof was based on the analysis of these functions in a complex domain and used computer algebra tools. Here we present a much simpler proof examining the above mentioned functions in a real domain only.