  This is the third in a series of papers in which we set up and discuss the
functional relations for the ``split rapidity line'' correlation function in
the N - state chiral Potts model. The order parameters of the model can be
obtained from this function. Here we consider the case N = 3 and write the
equations explicitly in terms of the hyperelliptic functions parametrization.
We also present four-term low-temperature series expansions, which we hope will
cast light on the analyticity properties needed to solve the relations. The
problem remains unsolved, but we hope that this will prove to be a step in the
right direction.
