  We revisit a little known theorem due to Beltrami, through which the
integration of the geodesic equations of a curved manifold is accomplished by a
method which, even if inspired by the Hamilton-Jacobi method, is purely
geometric. The application of this theorem to the Schwarzschild and Kerr
metrics leads straightforwardly to the general solution of their geodesic
equations. This way of dealing with the problem is, in our opinion, very much
in keeping with the geometric spirit of general relativity. In fact, thanks to
this theorem we can integrate the geodesic equations by a geometrical method
and then verify that the classical conservation laws follow from these
equations.
