While it seems possible to present a fairly complete uni?ed theory of undistorted polytropes
as attempted in the previous chapter the theory of distorted polytropes is much more extended
and - phisticated so that I present merely a brief overview of the theories that seem to me
most interesting and important. Basically the methods proposed to study the hydrostatic
equilibrium of a distorted self-gravitating mass can be divided into two major groups
(Blinnikov 1975): (i) Analytic or semia- lytic methods using a small parameter connected with
the distortion of the polytrope. (ii) More or less accurate numerical methods. Lyapunov and
later Carleman (see Jardetzky 1958 p. 13) have demonstrated that a sphere is a unique solution
to the problem of hydrostatic equilibrium for a ?uid mass at rest in tridimensional space. The
problem complicates enormously if the sphere is rotating rigidly or di?erentially in space
round an axis and or if it is distorted magnetically or tidally. Even for the simplest case of
a uniformly rotating ?uid body with constant density not all possible solutions have been found
(Zharkov and Trubitsyn 1978 p. 222). The sphere becomes an oblate ?gure and we have no a
priori knowledge of its strati?cation boundary shape planes of symmetry transfer of angular
momentum in di?erentially rotating bodies etc.
