  We give an implicit equation for the accessory parameter on the torus which
is the necessary and sufficient condition to obtain the monodromy of the
conformal factor. It is shown that the perturbative series for the accessory
parameter in the coupling constant converges in a finite disk and give a
rigorous lower bound for the radius of convergence. We work out explicitly the
perturbative result to second order in the coupling for the accessory parameter
and to third order for the one-point function. Modular invariance is discussed
and exploited. At the non perturbative level it is shown that the accessory
parameter is a continuous function of the coupling in the whole physical region
and that it is analytic except at most a finite number of points. We also prove
that the accessory parameter as a function of the modulus of the torus is
continuous and real-analytic except at most for a zero measure set. Three
soluble cases in which the solution can be expressed in terms of hypergeometric
functions are explicitly treated.
