  We apply the method of direct perturbation theory for the Foldy-Wouthuysen
(FW) transformation upon the Dirac-Pauli Hamiltonian subject to external
electromagnetic fields. The exact FW transformations exist and agree with those
obtained by Eriksen's method for two special cases. In the weak-field limit of
static and homogeneous electromagnetic fields, by mathematical induction on the
orders of $1/c$ in the power series, we rigorously prove the long-held
speculation: the FW transformed Dirac-Pauli Hamiltonian is in full agreement
with the classical counterpart, which is the sum of the orbital Hamiltonian for
the Lorentz force equation and the spin Hamiltonian for the
Thomas-Bargmann-Michel-Telegdi equation.
