  We introduce a new parametrisation for the Fermi sea of the $c = 1$ matrix
model. This leads to a simple derivation of the scattering matrix, and a
calculation of boundary corrections in the corresponding $1+1$--dimensional
string theory. The new parametrisation involves relativistic chiral fields,
rather than the non-relativistic fields of the usual formulations. The
calculation of the boundary corrections, following recent work of Polchinski,
allows us to place restrictions on the boundary conditions in the matrix model.
We provide a consistent set of boundary conditions, but believe that they need
to be supplemented by some more subtle relationship between the space-time and
matrix model. Inspired by these boundary conditions, some thoughts on the black
hole in $c=1$ string theory are presented.
