  The stability of an initially one-dimensional electron hole to perturbations
varying sinusoidally transverse to its trapping direction is analysed in
detail. It is shown that the expected low-frequency eigenmode of the linearized
Vlasov-Poisson system consists of a shift-mode, proportional to the gradient of
the equilibrium potential. The resulting dispersion relation is that the total
jetting force exerted by a perturbed hole on the particles balances the
electric restoring tension of the hole. The tension is quantitatively small and
can often be ignored. The particle force is expressed as integrals of
equilibrium parameters over the hole and is shown at low frequency to be
exactly equal to what has recently been found (by different analysis) to
express `kinematic' hole momentum conservation. The mechanism of instability
has nothing to do with the previously hypothesized transverse electron
focusing. The unmagnetized growth rate $\gamma(k)$ is found numerically and is
in excellent agreement with recent kinematic estimates. Magnetic field
stabilization of the transverse mode is also evaluated. The resulting stability
boundary for Maxwellian holes is in reasonable agreement with previously
published criteria based on particle simulation. It arises from a change of
trapped force sign across the resonance between bounce and cyclotron
frequencies.
