- ComputerN -body experiments are desribed which test the validities of the original Toomre's (1964) criterion and of a generalized criterion for local stability of Jeans-type perturba- tions in a self-gravitating, infinitesimally thin, and practically collisionless disk of stars. The fact that the nonaxisymmetric perturbations in the differentially rotating system are more un- stable than the axisymmetric ones is taken into account in this generalized criterion. It is shown that for differentially rotat- ing disks, the generalized criterion works as well as Toomre's ordinary criterion does for rigidly rotating ones. A modest discrepancy is observed between the analytical stability criteria and the numerical results. We tentatively at- tribute this to the shortcomings of the asymptotic density wave theory and possibly additional ones introduced by approxima- tions in the local numerical code employed here. In addition, the linear stability theory of small oscillations of a disk of stars is reexamined by using the method of particle orbit theory. This representation gives new insight into the problem of gravitating disk stability. Certain applications of the theory and the N -body simulations to actual disk-shaped spiral galaxies are explored as well.