Schwarzschild's criterion states that a fluid layer in hydrostatic equilibrium in a gravitational field will become unstable if the rate of change of pressure with density exceeds the corresponding adiabatic derivative. The validity of the criterion is discussed. The initial-value problem for the hydrodynamic perturbation equations for adiabatic motion is reduced to a time-independent problem. The latter is solved by a systematic classification of the possible motions. The method uses the theory of symmetric, semibounded operators. The distinction between thermodynamic and general gravitational equilibrium follows naturally. In particular, it is proved that the density in a gravitating body cannot increase outwards.