  This paper is a continuation of our previous work (solv-int/9903001). We
obtain two more functional relations for the eigenvalues of the transfer
matrices for the $sl(3)$ chiral Potts model at $q^2=-1$. This model, up to a
modification of boundary conditions, is equivalent to the three-layer
three-dimensional Zamolodchikov model. From these relations we derive the Bethe
ansatz equations.
