  We improve the calculation of Nozawa, Itoh, and Kohyama (1998) to provide a
formula for relativistic corrections to the thermal and kinematical
Sunyaev-Zeldovich effects that is accurate to fourth order in theta_e =
kT_e/m_ec^2, T_e and m_e being the electron temperature and electron mass,
respectively. We also carry out a direct numerical integration of the Boltzmann
collision term and confirm the excellent accuracy of the analytic formula. This
formula will be useful for the analysis of the observational data of the
forthcoming experiments of the kinematical Sunyaev-Zeldovich effect for
clusters of galaxies.
