  We study the perturbation modes of rotating superfluid ellipsoidal figures of
equilibrium in the framework of the two-fluid superfluid hydrodynamics and
Newtonian gravity. Our calculations focus on linear perturbations of background
equilibria in which the two fluids move together, the total density is uniform,
and the densities of the two components are proportional to one another, with
ratios that are independent of position. The motions of the two fluids are
coupled by mutual friction, as formulated by Khalatnikov. We show that there
are two general classes of modes for small perturbations: one class in which
the two fluids move together and the other in which there is relative motion
between them. The former are identical to the modes found for a single fluid,
except that the rate of viscous dissipation, when computed in the secular (or
``low Reynolds number'') approximation under the assumption of a constant
kinematic viscosity, is diminished by a factor $f_N$, the fraction of the total
mass in the normal fluid. The relative modes are completely new, and are
studied in detail for a range of values for the phenomenological mutual
friction coefficients, relative densities of the superfluid and normal
components, and, for Roche ellipsoids, binary mass ratios. We find that there
are no new secular instabilities connected with the relative motions of the two
fluid components. Moreover, although the new modes are subject to viscous
dissipation (a consequence of viscosity of the normal matter), they do not emit
gravitational radiation at all.
