  This paper is devoted to the presentation of the lateral Casimir force
between two sinusoidally corrugated eccentric cylinders. Despite that applying
scattering matrix method explains the problem exactly, procedure of applying
this method is somehow complicated specially at nonzero temperature. Using the
proximity force approximation (PFA) helps to achieve the lateral Casimir force
in a truly explicit manner. We assume the cylinders to be slightly eccentric
with similar radiuses and separations much smaller than corrugations' wave
length for the validity of PFA. For such short distances the effect of finite
conductivity would be non negligible. In addition to the effect of finite
conductivity, we investigate thermal corrections of the lateral Casimir force
to reduce the inaccuracy of the result obtained by PFA. Assuming the Casimir
force density between two parallel plates, the normal Casimir force between two
cylinders is obtained. With the aid of additive summation of the Casimir energy
between cylinders without corrugation, we obtain the lateral Casimir force
between corrugated cylinders.
