The stability of gaseous masses with respect to radial as well as non-radial oscillations is considered in the framework of the post-Newtonian equations of hydrodynamics The onset of dynamical instability at a radius R determined by a formula of the type 2CM K c2 (where K is a constant) is confirmed in case the "ratio of the specific heats" = (a log p/a log p) (where the subscript S denotes that the derivative is with respect to constant entropy) is a constant. An expression for K is derived which does not involve any knowledge of the equilibrium configuration beyond the Newtonian framework; and the values of K appropriate to the polytropes are also listed. With respect to the onset of instability for non-radial oscillations, it is shown that the classical criterion of Schwarzschild based on the discriminant S(r)=dP pdp dr dr is replaced by one based on the discriminant (r) =S(r)+ (F + is;;:;) where His the internal energy (per unit volume) and F is a ratio defined by the relation p11 = p/(F - 1). An alternative form for (r), namely, (r) =S(r)[1+{' d {} } d } where F3 = 1 + (a log T/a log p),, shows that the condition for the occurrence of convective instability is unaltered in the post-Newtonian approximation.
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S. Chandrasekhar (1965) studied this question.