  We calculate the increase in the number of modes (the Kac number) per unit
length and the change in the zero-point energy (the Casimir energy) of the
electromagnetic field resulting from the introduction of a thin perfectly
conducting cylindrical shell of elliptical cross-section. Along the way we give
a novel route to the calculation of these physical quantities. The Casimir
energy is found to be attractive with the circular case corresponding to the
energy maximum and the large eccentricity limit being the divergent energy
minimum. As a result, with only Casimir stresses present, a fixed area shell is
unstable with respect to collapse onto itself. This instability is argued to
persist at arbitrary temperature.
