The post-Newtonian effects of general relativity on the equilibrium of uniformly rotating dodies. III. The deformed figures of the Jacobi ellipsoids

Abstract

The effects of general relativity, in the post-Newtonian approximation, on the Jacobian figures of equilibrium of uniformly rotating homogeneous masses are determined, It is shown, for example, that the post-Newtonian figure is obtained by a deformation of the Jacobi ellipsoid by a suitable Lagrangian displacement cubic in the coordinates. The solution of the post-Newtonian equations exhibits an indeterminacy at the point of bifurcation M₂, where the Jacobian sequence branches off from the Maclaurin sequence, and a singularity at a point J₄, where the axes of the Jacobi ellipsoid are in the ratios 1:02972 :0,2575. The indeterminacy in the solution at M₂ arises from the fact that at this point the Maclaurin spheroid is neutral to an infinitesimal deformation proportional to (X₁, -X₂, 0); and the singularity at J₄ arises from the fact that at this point the Jacobi ellipsoid is unstable to the deformation induced by the effects of general relativity.