  We consider the small separation asymptotic expansions of the Casimir
interaction energy and the Casimir interaction force between two parallel
cylinders. The leading order terms and the next-to-leading order terms are
computed analytically. Four combinations of boundary conditions are considered,
which are Dirichlet-Dirichlet (DD), Neumann-Neumann (NN), Dirichlet-Neumann
(DN) and Neumann-Dirichlet (ND). For the case where one cylinder is inside
another cylinder, the computations are shown in detail. In this case, we
restrict our attention to the situation where the cylinders are strictly
eccentric and the distance between the cylinders $d$ is much smaller than the
distance between the centers of the cylinders. The computations for the case
where the two cylinders are exterior to each other can be done in the same way
and we only present the results, which turn up to be similar to the results for
the case where one cylinder is inside another except for some changes of signs.
In all the scenarios we consider, the leading order terms are of order
$d^{-7/2}$ and they agree completely with the proximity force approximations.
The results for the next-to-leading order terms are new. In the limiting case
where the radius of the larger cylinder approaches infinity, the well-known
results for the cylinder-plate configuration with DD or NN boundary conditions
are recovered.
