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