  We show that, for a stationary version of Hammersley's process, with Poisson
``sources'' on the positive x-axis, and Poisson ``sinks'' on the positive
y-axis, an isolated second-class particle, located at the origin at time zero,
moves asymptotically, with probability 1, along the characteristic of a
conservation equation for Hammersley's process. This allows us to show that
Hammersley's process without sinks or sources, as defined by Aldous and
Diaconis [Probab. Theory Related Fields 10 (1995) 199-213] converges locally in
distribution to a Poisson process, a result first proved in Aldous and Diaconis
(1995) by using the ergodic decomposition theorem and a construction of
Hammersley's process as a one-dimensional point process, developing as a
function of (continuous) time on the whole real line. As a corollary we get the
result that EL(t,t)/t converges to 2, as t\to\infty, where L(t,t) is the length
of a longest North-East path from (0,0) to (t,t). The proofs of these facts
need neither the ergodic decomposition theorem nor the subadditive ergodic
theorem. We also prove a version of Burke's theorem for the stationary process
with sources and sinks and briefly discuss the relation of these results with
the theory of longest increasing subsequences of random permutations.
