  In this paper we prove a conjectured modular equation of Farkas and Kra,
which involving a half sum of certain modular form of weight $1$ for congruence
subgroup $\Gamma_1(k)$ with any prime $k$. We prove that their conjectured
identity holds for all odd integer $k\ge 2$. A new modular equation of Farkas
and Kra type is also established.
