We discuss the physics of bending instabilities in inhomogeneous stell
ar systems and propose a simple criterion for stability. We derive the
bending modes of a family of thin disk models and show that long-wave
length modes in thin systems with realistic density profiles are not a
lways stabilized by gravity, in contrast to an infinite sheet. We also
present the results of N-body experiments with finite-thickness disks
that confirm instability to bending even when the velocity anisotropy
is much less extreme than the critical value for instability in an in
finite slab. We suggest that the most important mechanism that stabili
zes bending in realistic models is the out-of-phase response of stars
that encounter the bend with a frequency greater than their free verti
cal oscillation frequency kappa(z). This mechanism successfully accoun
ts for our N-body results, and for the behavior of the infinite slab a
t short wavelengths, where gravity can be neglected. It further predic
ts stability at all wavelengths to longitudinal bending modes for pres
sure-supported systems in which the ratio of orbital oscillation frequ
encies is less extreme than about 2:1, that is, for which the isodensi
ty contours are rounder than about 1:3. (Oblate systems can be flatter
if the velocities are azimuthally biased.) The latter number is in ag
reement with the behavior of bending modes in N-body models and in uni
form spheroids, as well as with the absence of elliptical galaxies fla
tter than about E7.