  This paper reports on the in-plane normal modes in the self-consistent and
the cut-out power-law disks. Although the cut-out disks are remarkably stable
to bisymmetric perturbations, they are very susceptible to one-armed modes. For
this harmonic, there is no inner Lindblad resonance, thus removing a powerful
stabilising influence. A physical mechanism for the generation of the one-armed
instabilities is put forward. Incoming trailing waves are reflected as leading
waves at the inner cut-out, thus completing the feedback for the
swing-amplifier. Growing three-armed and four-armed modes occur only at very
low temperatures. The rotation curve index has a marked effect on stability.
For all azimuthal wavenumbers, any unstable modes persist to higher
temperatures and grow more vigorously if the rotation curve is rising than if
the rotation curve is falling. If the central regions or outer parts of the
disk are carved out more abruptly, any instabilities become more virulent. The
self-consistent power-law disks possess a number of unusual stability
properties. There is no natural time-scale in the self-consistent disk. If a
mode is admitted at some pattern speed and growth rate, then it must be present
at all pattern speeds and growth rates. Our analysis suggests that such a
two-dimensional continuum of non-axisymmetric modes does not occur and that the
self-consistent power-law disks admit no global non-axisymmetric normal modes
whatsoever. Without reflecting boundaries or cut-outs, there is no resonant
cavity and no possibility of unstable growing modes. The self-consistent
power-law disks certainly admit equiangular spirals as neutral modes, together
with a one-dimensional continuum of growing axisymmetric modes.
