The $n$-th modular equation for the elliptic modular function
$j(z)$ has large coefficients even for small $n$, and those
coefficients grow rapidly as $n \to \infty$. The growth of
these coefficients was first obtained by Cohen ([5]). And,
recently Cais and Conrad ([1], \S7) considered this problem
for the Hauptmodul $j_{5}(z)$ of the principal congruence
group $\Gamma(5)$. They found that the ratio of logarithmic
heights of $n$-th modular equations for $j(z)$ and $j_{5}(z)$
converges to 60 as $n \to \infty$, and observed that 60 is
the group index $[\overline{\Gamma(1)} : \overline{\Gamma(5)}]$.
In this paper we prove that their observation is true for
Hauptmoduln of somewhat general Fuchsian groups of the first
kind with genus zero.