  The purpose of the present work is to apply the method recently developed in
reference [chain_m] to the spin-1 Ising chain, showing how to obtain analytical
$\beta$-expansions of thermodynamical functions through this formalism. In this
method, we do not solve any transfer matrix-like equations. A comparison
between the $\beta$-expansions of the specific heat and the magnetic
susceptibility for the $s=1/2$ and $s=1$ one-dimensional Ising models is
presented. We show that those expansions have poorer convergence when the
auxiliary function of the model has singularities.
