  The hydrodynamic stability of accretion disks is considered. The particular
question is whether the combined action of a (stable) vertical density
stratification and a (stable) radial differential rotation gives rise to a new
instability for nonaxisymmetric modes of disturbances. The existence of such an
instability is not suggested by the well-known Solberg-Hoiland criterion. It is
also not suggested by a local analysis for disturbances in general
stratifications of entropy and angular momentum which is presented in our
Section 2 confirming the results of the Solberg-Hoiland criterion also for
nonaxisymmetric modes within the frame of ideal hydrodynamics but only in the
frame of a short-wave approximation for small m. As a necessary condition for
stability we find that only conservative external forces are allowed to
influence the stable disk. As magnetic forces are never conservative, linear
disk instabilities should only exist in the magnetohydrodynamical regime which
indeed contains the magnetorotational instability as a much-promising
candidate. To overcome some of the used approximations in a numerical
approach,the equations of the compressible adiabatic hydrodynamics are
integrated imposing initial nonaxisymmetric velocity perturbations with m=1 to
m=200.
  Only solutions with decaying kinetic energy are found. The system always
settles in a vertical equilibrium stratification according to pressure balance
with the gravitational potential of the central object. keywords: accretion
disks -- hydrodynamic instabilities -- turbulence
