Vibrational stability of the rotating Roche model

The aim of the first part of this investigation will be to establish the explicit form of the linearized systems of differential equations governing arbitrary oscillations (of amplitudes small enough for their squares and higher powers to be negligible) of the rotating Roche model in Clairaut's coordinates (in which their radial component is identified with the total potential). By solving these equations in a closed form we shall prove that this model is incapable of performing such oscillations (for any type of symmetry) about equipotential surfaces representing the figures of equilibrium, as soon as the centrifugal force will cause their equilibrium form to depart from a sphere.In the second part of this paper we shall set up the closed forms of the Laplace equation in Clairaut (non-orthogonal) as well as Roche (orthogonal) coordinates associated with the rotating Roche model; and by a construction of their solution establish successively the explicit forms of the respective harmonic functions associated with such figures (as a generalization of Legendre functions which are similarly associated with a sphere.