  We consider the monodromy problem for the four-punctured sphere in which the
character of one composite monodromy is fixed, by looking at the expansion of
the accessory parameter in the modulus $x$ directly, without taking the limit
of the quantum conformal blocks for infinite central charge. The integrals
which appear in the expansion of the Volterra equation, involve products of two
hypergeometric functions to first order and up to four hypergeometric functions
to second order. It is shown that all such integrals can be computed
analytically. We give the complete analytical evaluation of the accessory
parameter to first and second order in the modulus. The results agree with the
evaluation obtained by assuming the exponentiation hypothesis of the quantum
conformal blocks in the limit of infinite central charge. Extension to higher
orders is discussed.
