  The index of a meromorphic function $g$ on a compact Riemann surface is an
invariant of $g$, which is defined as the number of negative eigenvalues of the
differential operator $L:=-{\Delta}-|dG|^2$, where ${\Delta}$ is the Laplacian
with respect to a conformal metric $ds^2$ on the Riemann surface, $G \colon M
\to S^2$ is the holomorphic map corresponding to $g$. We consider the
meromorphic function $w$ on the Riemann surface $M_a= \{(z,w)
\in\widehat{\mathbb{C}}^2 \mid w^2=z(z-a)(z+\frac{1}{a})\}(a \geqslant 1 )$
homeomorphic to a torus, and we determine the index of $tw$ for all $a$ in the
range $1 \leqslant a \leqslant a_0$ (where $a_0$ can be numerically evaluated)
and all $t>0$.
