  Taylor-Goldstein equation (TGE) governs the stability of a shear-flow of an
inviscid fluid of variable density. It is investigated here from a rigorous
geometrical point of view using a canonical class of its transformations.
Rayleigh's point of inflection criterion and Fjortoft's condition of
instability of a homogenous shear-flow have been generalized here so that only
the profile carrying the point of inflection is modified by the variation of
density. This fulfils a persistent expectation in the literature. A pair of
bounds exists such that in any unstable flow the flow-curvature (a function of
flow-layers) exceeds the upper bound at some flow-layer and falls below the
lower bound at a higher layer. This is the main result proved here. Bounds are
obtained on the growth rate and the wave numbers of unstable modes, in
fulfillment of longstanding predictions of Howard. A result of Drazin and
Howard on the boundedness of the wave numbers is generalized to TGE. The
results above hold if the local Richardson number does not exceed 1/4 anywhere
in the flow, otherwise a weakening of the conditions necessary for instability
is seen. Conditions for the propagation of neutrally stable waves and bounds on
the phase speeds of destabilizing waves are obtained. It is also shown that the
set of complex wave velocities of normal modes of an arbitrary flow is bounded.
Fundamental solutions of TGE are obtained and their smoothness is examined.
Finally sufficient conditions for instability are suggested.
