  The probability that a particle, crossing the shock along a given direction,
be reflected backwards along another direction, was shown to be the key element
in determining the spectrum of non--thermal particles accelerated via the Fermi
mechanism around a plane--parallel shock in the test--particle limit. Here an
explicit equation for this probability distribution is given, for both the
upstream and downstream sections. Though analytically intractable, this
equation is solved numerically, allowing the determination of the spectrum in
full generality, without limitation to shock speed or scattering properties. A
number of cases is then computed, making contact with previous numerical work,
in all regimes: Newtonian, trans--relativistic, and fully relativistic.
