The Post-Newtonian Effects of General Relativity on the Equilibrium of Uniformly Rotating Bodies. III. The Deformed Figures of the Jacobi Ellipsoids
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 1:0 2972:0.2575. 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 (s , -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.