  Ionization cooling in a straight beamline reduces the transverse emittance of
a beam, and has little effect on the longitudinal emittance (generally, in
fact, it increases the longitudinal emittance). Once the beamline bends, the
introduction of dispersion creates a coupling between the transverse and
longitudinal planes. If this coupling is handled properly, one can achieve
cooling in all three phase space planes. This is usually done by placing a
wedge-shaped absorber in a region where there is dispersion. I will demonstrate
using an eigenvalue analysis that there are other configurations of dispersion
and absorber shape that will achieve ionization cooling in all phase space
planes. In particular, I will show that a one can even achieve cooling in all
phase planes with a parallel-faced absorber in a dispersion-free region. I will
use perturbation theory to approximate the change in the cooling rates due to
longitudinal-transverse coupling. I will then describe how the cooling of
longitudinal oscillations can be understood via the projection of the
"longitudinal" eigenmodes into the transverse plane.
