  It is shown that the parameters contained in any two complete solutions of
the Hamilton-Jacobi equation, corresponding to a given Hamiltonian, are related
by means of a time-independent canonical transformation and that, in some
cases, a generating function of this transformation is given by the envelope of
a family of surfaces defined by the difference of the two complete solutions.
Conversely, in those cases, one of the complete solutions is given by the
envelope of a family of surfaces defined by the sum of the other complete
solution and the generating function of the canonical transformation. Some
applications of these results to geometrical optics are also given.
