V. On the vibrations and stability of a gravitating planet

Abstract 1. In a former paper I have considered the effect of gravitation as a factor tending towards instability, in the case of a spherical nebula of gas. The object of the present paper is to investigate the analogous problem in the case of a spherical planet, the planet being supposed composed of solid or fluid matter. The main question at issue is the following. 2. So long as gravitation is neglected there can be no doubt as to the stability of an elastic solid; any displacement increases the potential energy, and an unstressed configuration of equilibrium is therefore necessarily stable. But when gravitation is taken into account, the gravitational energy may be either increased or decreased by a displacement from equilibrium, and if a displacement can be found which effects a decrease in the gravitational potential energy of amount sufficient to outweigh the increase in the potential of the elastic forces, then the equilibrium configuration will be unstable.