  We investigate the properties of relativistic $r$-modes of slowly rotating
neutron stars by using a relativistic version of the Cowling approximation. In
our formalism, we take into account the influence of the Coriolis like force on
the stellar oscillations, but ignore the effects of the centrifugal like force.
For three neutron star models, we calculated the fundamental $r$-modes with
$l'=m=2$ and 3. We found that the oscillation frequency $\bar\sigma$ of the
fundamental $r$-mode is in a good approximation given by $\bar\sigma\approx
\kappa_0 \Omega$, where $\bar\sigma$ is defined in the corotating frame at the
spatial infinity, and $\Omega$ is the angular frequency of rotation of the
star. The proportional coefficient $\kappa_0$ is only weakly dependent on
$\Omega$, but it strongly depends on the relativistic parameter $GM/c^2R$,
where $M$ and $R$ are the mass and the radius of the star. All the fundamental
$r$-modes with $l'=m$ computed in this study are discrete modes with distinct
regular eigenfunctions, and they all fall in the continuous part of the
frequency spectrum associated with Kojima's equation (Kojima 1998). These
relativistic $r$-modes are obtained by including the effects of rotation higher
than the first order of $\Omega$ so that the buoyant force plays a role, the
situation of which is quite similar to that for the Newtonian $r$-modes.
