  It is usually believed that the Cowling approximation can give satisfactory
solutions if the stars %are highly compressible have soft equations of state
and/or if the strongly general relativistic stars are treated in the case of
rigid rotation. Since, however, there have been no systematic studies about the
accuracy of the Cowling approximation for differentially rotating compressible
stars, we investigate eigenfrequencies and eigenfunctions of the oscillation
modes in {\it rapidly} and {\it differentially} rotating compressible stars by
employing the exact method including full-perturbations and the Cowling
approximation.
  We have found that the Cowling approximation for f-mode oscillations is not a
good approximation in rapidly and differentially rotating stars, although rapid
rotation makes this approximation better for rigidly rotating stars. This
result suggests that we must be careful when we apply the Cowling approximation
to differentially rotating stars even in the framework of general relativity.
On the other hand, the approximation will work well for r-modes even if the
star is rotating differentially. Therefore, the Cowling approximation can be
used as a strong tool for the investigation of r-mode oscillations in the
general relativistic framework that it is difficult to compute including the
perturbations of gravity.
