  We present a brief numerical study of the Darwin-Fowler method applied to the
analysis of the energy partition of essembles of bosons and fermions. We
analyze the assertion of the existence of a "strong maximum" made in the
original paper of Darwin and Fowler and other studies and show that although
the presumed saddle point along the real axis of the grand canonical parameters
may exist it cannot, in general, be characterized as "strong", in the sense of
having much larger magnitude than the other points along the path of
integration. We show that in some cases the saddle point is not even present
and the various approximations of the method can be interpreted as a tricky
reformulation of usual thermodynamic relations. The close connection of the
method with the formalism of the Laplace transform may produce wrong results if
the internal energy of the components of the ensemble is not large enough.
Therefore, although useful in many applications the Darwin-Fowler method may
not be suitable, in general, for a detailed microscopic analysis of the nuclear
structure in connection with the Shell Model approach, as it is usually done in
studies of the pre-equilibrium stage of nuclear reactions.
