  We have developed a formalism for analyzing the stability of a fluid in the
presence of neutrinos of all flavors and in the presence of a gravitational
field. When applied to an extensive two-dimensional grid of core radii and
fluid element sizes for each of several time slices of a number of
proto-supernovae, we find no evidence for the neutron finger instability as
described by the Livermore group. We find, instead, that the rate of lepton
equilibration always exceeds that of thermal equilibration. Furthermore, we
find that thermal diffusion driven by a lepton fraction difference and lepton
diffusion driven by an entropy difference are both nonzero and that the first
of these tends to be large in magnitude. An important consequence of this is
the presence of a doubly diffusive instability, which we refer to as
``lepto-entropy fingers,'' in an extensive region below the neutrinosphere
where the lepton number, $Y_{\ell}$, is small. This instability is driven by a
mechanism very different from that giving rise to neutron fingers, and may play
an important role in enhancing the neutrino emission. Deep in the core where
the entropy is low and the lepton number higher, our analysis indicates a
region unstable to another instability which we refer to as ``lepto-entropy
semiconvection.'' These instabilities, particularly lepto-entropy fingers, may
have already been seen in some multi-dimensional core collapse simulations
described in the literature.
