Bethe Equation of $τ^{(2)}$-model and Eigenvalues of Finite-size Transfer Matrix of Chiral Potts Model with Alternating Rapidities

We establish the Bethe equation of the $τ^{(2)}$-model in the $N$-state chiral Potts model (including the degenerate selfdual cases) with alternating vertical rapidities. The eigenvalues of a finite-size transfer matrix of the chiral Potts model are computed by use of functional relations. The significance of the "alternating superintegrable" case of the chiral Potts model is discussed, and the degeneracy of $τ^{(2)}$-model found as in the homogeneous superintegrable chiral Potts model.