  The superfluid phase transition of the general vortex gas, in which the
circulations may be any non-zero integer, is studied. When the net circulation
of the system is not zero the absence of a superfluid phase is shown. When the
net circulation of the vortices vanishes, the presence of off-diagonal long
range order is demonstrated and the existence of an order parameter is
proposed. The transition temperature for the general vortex gas is shown to be
the Kosterlitz---Thouless temperature. An upper bound for the average vortex
number density is established for the general vortex gas and an exact
expression is derived for the Kosterlitz---Thouless ensemble.
