  The results obtained in our previous paper are now extended to the case of
stationary axially symmetric dyonic black boles within the theory of two
electromagnetic potentials. We slightly enlarge the classical Ernst formalism
by introducing, with the aid of the $t$- and $\varphi$-components of the dual
potential $B_\mu$, the magnetic potential $\Phi_m$ which, similar to the known
electric potential $\Phi_e$, also takes constant value on the black hole
horizon. We analyze in detail the case of the dyonic Kerr-Newman black hole and
show how the Komar mass must be evaluated correctly in this stationary dyonic
model. In particular, we rigorously prove the validity of the standard
Tomimatsu mass formula and point out that attempts to "improve" it made in
recent years are explained by misunderstanding of the auxiliary role that
singular potentials play in the description of magnetic charges. Our approach
is symmetrical with respect to electric and magnetic charges and, like in the
static case considered earlier, Dirac strings of all kind are excluded from the
physical picture of the stationary black hole dyonic spacetimes.
