  The linear stability of inviscid, incompressible, two-dimensional, plane
parallel, shear flow was considered over a century ago by Rayleigh, Kelvin, and
others. A principal result on the subject is Rayleigh's celebrated inflection
point theorem {R80}, which states that for an equilibrium flow to be unstable,
the equilibrium velocity profile must contain an inflection point. That is, if
the velocity profile is given by $U(y)$, where $y$ is the cross-stream
coordinate, then there must be a point, $y=y_I$, for which $U''(y_I)=0$. Much
later, in 1950, Fj{\o}rtoft {F50} generalized the theorem by showing that,
moreover, if there is one inflection point, then $U'''(y_I)/U'(y_I)<0$ is
required for instability (see {Bar} for further extensions). Both Rayleigh's
Theorem and Fj{\o}rtoft's subsequent generalization are necessary conditions
for instability, but they are not sufficient. That is, even though an
equilibrium profile may contain a vorticity minimum, it is not necessarily
unstable. The point of this paper is to derive, for a large class of
equilibrium velocity profiles, a condition that is necessary and sufficient for
instability.
