  We evidence via a computation in the reciprocal space the asymptotic
behaviour of the high order virial coefficients for a hard sphere system. These
coefficients, if their order is high enough, are those of a geometric series.
We thus are able to give an explicit expression of the equation of states of
the hard sphere system at high density when the fluid phase is no longer the
stable one; in the disordered phase this equation of states exhibits a simple
pole at the random close packing density. We can then estimate the packing
densities of the freezing point of the disordered phase and also of the melting
point of the fcc ordered phase. The results are compared with those of the
numerical simulations.
