On the Stability of Spheres of Simple Mechanical Fluid Held Together by Newtonian Gravitation.

A sphere of simple mechanical fluid held together by Newtonian forces of gravitation will be in stable equilibrium if its total potential energy is a minimum. Using the prin- ciples of the calculus of variations, it is shown that any equilibrium distribution of fluid would, in general, automatically satisfy all tests for a minimum except that of Jacobi as to the location of conjugate points. The circumstances which lead to failure of the Jacobi test and the consequences which arise from such failure are discussed and illus- trated