  We study adiabatic oscillations of rotating self-gravitating gaseous stars in
mathematically rigorous manner. The internal motion of the star is supposed to
be governed by the Euler-Poisson equations with rotation of constant angular
velocity under the equation of state of the ideal gas. The motion is supposed
to be adiabatic, but not to be barotropic in general. This causes a free
boundary problem to gas-vacuum interface. Existence of solutions to the
linearized equation in the Lagrange coordinates of the perturbations around a
fixed stationary solution, the eigenvalue problem with concept of quadratic
pencil of operators, and the stability problem with a new concept of stability
introduced in this article are discussed.
