  The Galaxy's stellar halo seems to be a tangle of disrupted systems that have
been tidally stretched out into streams. Each stream approximately delineates
an orbit in the Galactic force-field. In the first paper in this series we
showed that all six phase-space coordinates of each point on an orbit can be
reconstructed from the orbit's path across the sky and measurements of the
line-of-sight velocity along the orbit. In this paper we complement this
finding by showing that the orbit can also be reconstructed if we know proper
motions along the orbit rather than the radial velocities. We also show that
accurate proper motions of stream stars would enable distances to be determined
to points on the stream that are independent of any assumption about the
Galaxy's gravitational potential. Such "Galactic parallaxes" would be as
fundamental as conventional trigonometric parallaxes, but measureable to
distances ~70 times further.
