  A two-loop (cylinder) amplitude of the 2d pure gravity theory is obtained in
the proper-time gauge ($g_{00}=1$, $g_{01}=g_{10}=0$) in the continuum
formulation. The constraint $T_{01}=0$ is solved and used to reduce the problem
of field theory to that of quantum mechanics. This reduction can also be proved
by using a conformal Ward identity. The amplitude depends on the lengths $l_1,
l_2$ of the boundaries, the proper time $T$ and a non-negative integer $m$
associated with winding modes around the boundaries.
