  Factorization of the $N$-tachyon amplitudes in two-dimensional $c=1$ quantum
gravity is studied by means of the operator product expansion of vertex
operators after the Liouville zero mode integration. Short-distance
singularities between two tachyons with opposite chiralities account for all
singularities in the $N$-tachyon amplitudes. Although the factorization is
valid, other possible short-distance singularities corresponding to other
combinations of vertex operators are absent since the residue vanishes. Apart
from the tachyon states, there are infinitely many topological states
contributing to the intermediate states. This is a more detailed account of our
short communication on the factorization.
