  For a large class of repulsive interaction models, the Mayer cluster
integrals can be transformed into a tridiagonal symmetric matrix, whose
elements converge to a constant with a 1/n^2 correction. We find exact
expressions, in terms of these correction terms, for the two critical exponents
describing the density near the two singular termination points of the fluid
phase. We apply the method to the hard-spheres model and find that the
metastable fluid phase terminates at rho_t=0.751(5). The density near the
transition is given by (rho_t-rho)~(z_t-z)^sigma', where the critical exponent
is predicted to be sigma'=0.0877(25). The termination density is close to the
observed glass transition, and thus the above critical behavior is expected to
characterize the onset of glassy behavior in hard spheres.
