  Published maps of red giant stars in the halo region of M31 exhibit a giant
stellar stream to the south of this galaxy, as well as a giant "shelf" to the
northeast of M31's center. Using these maps, we find that there is a fainter
shelf of comparable size on the western side as well. By choosing appropriate
structural and orbital parameters for an accreting dwarf satellite within the
accurate M31 potential model of Geehan et al. (2006), we produce a very similar
structure in an N-body simulation. In this scenario, the tidal stream produced
at pericenter of the satellite's orbit matches the observed southern stream,
while the forward continuation of this tidal stream makes up two orbital loops,
broadened into fan-like structures by successive pericentric passages; these
loops correspond to the NE and W shelves. The tidal debris from the satellite
also reproduces a previously-observed "stream" of counterrotating PNe and a
related stream seen in red giant stars. The debris pattern in our simulation
resembles the shell systems detected around many elliptical galaxies, though
this is the first identification of a shell system in a spiral galaxy and the
first in any galaxy close enough to allow measurements of stellar velocities
and relative distances. We discuss the physics of these partial shells,
highlighting the role played by spatial and velocity caustics in the
observations. We show that kinematic surveys of the tidal debris will provide a
sensitive measurement of M31's halo potential, while quantifying the surface
density of debris in the shelves will let us reconstruct the original mass and
time of disruption of the progenitor satellite.
