  The axial mode oscillations are examined for relativistic rotating stars with
uniform angular velocity. Using the slow rotation formalism and the Cowling
approximation, we have derived the equations governing the r-mode oscillations
up to the second order with respect to the rotation. In the lowest order, the
allowed range of the frequencies is determined, but corresponding spatial
function is arbitrary. The spatial function can be decomposed in non-barotropic
region by a set of functions associated with the differential equation of the
second-order corrections. The equation however becomes singular in barotropic
region, and a single function can be selected to describe the spatial
perturbation of the lowest order. The frame dragging effect among the
relativistic effects may be significant, as it results in rather broad spectrum
of the r-mode frequency unlike in the Newtonian first-order calculation.
