We consider the linear stability of exact, temporally periodic solutions of the Euler
equations of incompressible, inviscid flow in an ellipsoidal domain. The problem of
linear stability is reduced, without approximation, to a hierarchy of finite-dimensional
Floquet problems governing fluid-dynamical perturbations of differing spatial scales
and symmetries. We study two of these Floquet problems in detail, emphasizing
parameter regimes of special physical significance. One of these regimes includes
periodic flows differing only slightly from steady flows. Another includes long-period
flows representing the nonlinear outcome of an instability of steady flows. In both
cases much of the parameter space corresponds to instability, excepting a region
adjacent to the spherical configuration. In the second case, even if the ellipsoid
departs only moderately from a sphere, there are filamentary regions of instability in
the parameter space. We relate this and other features of our results to properties of
reversible and Hamiltonian systems, and compare our results with related studies of
periodic flows.