  In this paper, we develop an iterative scheme to enable the explicit
calculation of an arbitrary post-Newtonian order for a relativistic body that
reduces to the Maclaurin spheroid in the appropriate limit. This scheme allows
for an analysis of the structure of the solution in the vicinity of bifurcation
points along the Maclaurin sequence. The post-Newtonian expansion is solved
explicitly to the fourth order and its accuracy and convergence are studied by
comparing it to highly accurate numerical results.
