  We analyze previous results on the stability of uniformly and differentially
rotating, self-gravitating, gaseous and stellar, axisymmetric systems to derive
a new stability criterion for the appearance of toroidal, m=2 Intermediate (I)
and bar modes. In the process, we demonstrate that the bar modes in stellar
systems and the m=2 I-modes in gaseous systems have many common physical
characteristics and only one substantial difference: because of the anisotropy
of the stress tensor, dynamical instability sets in at lower rotation in
stellar systems. This difference is reflected also in the new stability
criterion. The new stability parameter "alpha" is formulated first for
uniformly rotating systems and is based on the angular momentum content rather
than on the energy content of a system. For stability of stellar systems
"alpha" = 0.254-0.258, while it is in the range of 0.341-0.354 for gaseous
systems. For uniform rotation, one can write "alpha" as a function of the total
(rotational) kinetic and gravitational energies, and of a parameter which is
characteristic of the topology/connectedness and the geometric shape of a
system. "Alpha" can be extended to and calculated for a variety of
differentially rotating, gaseous and stellar, axisymmetric disk and spheroidal
models whose equilibrium structures and stability characteristics are known. We
also estimate "alpha" for gaseous toroidal models and for stellar disk systems
embedded in an inert or responsive "halo". We find that the new stability
criterion holds equally well for all these previously published axisymmetric
models.
