  It was recently established that a function which is harmonic on an infinite
cylinder and vanishes on the boundary necessarily extends to an entire harmonic
function. This paper considers harmonic functions on an annular cylinder which
vanish on both the inner and outer cylindrical boundary components. Such
functions are shown to extend harmonically to the whole of space apart from the
common axis of symmetry. One of the ingredients in the proof is a new estimate
for the zeros of cross product Bessel functions.
