  For the analysis of the r-mode oscillation of hot young neutron stars, it is
necessary to consider the effect of it differential rotation, because viscosity
is not strong enough for differentially rotating young neutron stars to be lead
to uniformly rotating configurations on a very short time scale after their
birth. In this paper, we have developed a numerical scheme to solve r-mode
oscillations of differentially rotating polytropic inviscid stars. This is the
extended version of the method which was applied to compute r-mode oscillations
of uniformly rotating Newtonian polytropic stars. By using this new method, we
have succeeded in obtaining eigenvalues and eigenfunctions of r-mode
oscillations of differentially rotating polytropic stars. Our numerical results
show that as the degree of differential rotation is increased, it becomes more
difficult to solve r-mode oscillations for slightly deformed configurations
from sphere compared to solving r-mode oscillations of considerably deformed
stars. One reason for it seems that for slightly deformed stars corotation
points appear near the surface region if the degree of differential rotation is
strong enough. This is similar to the situation that the perturbational
approach of r-mode oscillations for it slowly rotating stars in general
relativity results in a singular eigenvalue problem.
