  Let ${\cal I}_n$ be the symmetric inverse semigroup on $X_n = \{1, 2,...,
n\}$ and let ${\cal DP}_n$ and ${\cal ODP}_n$ be its subsemigroups of partial
isometries and of order-preserving partial isometries of $X_n$, respectively.
In this paper we investigate the cycle structure of a partial isometry and
characterize the Green's relations on ${\cal DP}_n$ and ${\cal ODP}_n$. We show
that ${\cal ODP}_n$ is a $0-E-unitary$ inverse semigroup. We also investigate
the cardinalities of some equivalences on ${\cal DP}_n$ and ${\cal ODP}_n$
which lead naturally to obtaining the order of the semigroups.
