  The relativistic Boltzmann equation for a constant differential cross section
and with periodic boundary conditions is considered. The speed of light appears
as a parameter $c>c_0$ for a properly large and positive $c_0$. A local
existence and uniqueness theorem is proved in an interval of time independent
of $c>c_0$ and conditions are given such that in the limit $c\to +\infty$ the
solutions converge, in a suitable norm, to the solutions of the
non-relativistic Boltzmann equation for hard spheres.
