  We continue the study of a relationship between the instanton expansion of
the Seiberg-Witten (SW) prepotential of $D = 4$, ${\cal N }= 2$ $SU(2)$ SUSY
gauge theory and the monstrous moonshine. Extending the previous results, we
show for the cases of $N_f=2$ and $3$ that $q=e^{2\pi i\tau}$, where $\tau$ is
the complex gauge coupling, again has an expansion whose coefficients are all
integer-coefficient polynomials of the moonshine coefficients of the modular
$j$-function in terms of an appropriate expansion variable. We also demonstrate
that the new method of calculating the SW prepotential developed here is useful
by performing some explicit computations.
