  We demonstrate that for the sine-Gordon theory at the free fermion point, the
2-point correlation functions of the fields $\exp (i\al \Phi )$ for $0< \al <
1$ can be parameterized in terms of a solution to a sinh-Gordon-like equation.
This result is derived by summing over intermediate multiparticle states and
using the form factors to express this as a Fredholm determinant. The proof of
the differential equations relies on a $\Zmath_2$ graded multiplication law
satisfied by the integral operators of the Fredholm determinant. Using this
methodology, we give a new proof of the differential equations which govern the
spin and disorder field correlators in the Ising model.
