  An iterative method is presented for solving the problem of a uniformly
rotating, self-gravitating ring without a central body in Newtonian gravity by
expanding about the thin ring limit. Using this method, a simple formula
relating mass to the integrated pressure is derived to the leading order for a
general equation of state. For polytropes with the index n=1, analytic
coefficients of the iterative approach are determined up to the third order.
Analogous coefficients are computed numerically for other polytropes. Our
solutions are compared with those generated by highly accurate numerical
methods to test their accuracy.
