  We obtain an explicit formula for the number of Lam\'e equations (modulo
scalar equivalence) with index $n$ and projective monodromy group of order
$2N$, for given $n \in \Z$ and $N \in \N$. This is done by performing the
combinatorics of the `dessins d'enfants' associated to the Belyi covers which
transform hypergeometric equations into Lam\'e equations by pull-back.
