  We give a perturbative analysis for an infinitesimally thin cylindrical shell
composed of counter rotating collisionless particles, originally devised by
Apostolatos and Thorne. They found a static solution of the shell and concluded
by C-energy argument that it is stable. Recently, the present authors and Ida
reanalyzed this system by evaluating the C-energy on the future null infinity
and found that the system has an instability, though it was not shown how the
system is unstable. In this paper, it is shown in the framework of the linear
perturbation theory that, if the constituent particles move slowly, the static
shell is unstable in the sense that the perturbation of its circumferential
radius oscillates with exponentially growing amplitude, whereas if the speed of
the constituent particle exceeds a critical value, the shell just expands or
contracts exponentially with time.
