  We present several different statistical methods to determine the transverse
velocity vector of M31. The underlying assumptions are that the M31 satellites
on average follow the motion of M31 through space, and that the galaxies in the
outer parts of the Local Group on average follow the motion of the Local Group
barycenter through space. We apply the methods to the line-of-sight velocities
of 17 M31 satellites, to the proper motions of the 2 satellites M33 and IC 10,
and to the line-of-sight velocities of 5 galaxies near the Local Group turn
around radius, respectively. This yields 4 independent but mutually consistent
determinations of the heliocentric M31 transverse velocities in the West and
North directions, with weighted averages <v_W> = -78+/-41 km/s and <v_N> =
-38+/-34 km/s. The Galactocentric tangential velocity of M31 is 42 km/s, with
1-sigma confidence interval V_tan <= 56 km/s. The implied M31-Milky Way orbit
is bound if the total Local Group mass M exceeds 1.72^{+0.26}_{-0.25}x10^{12}
solar masses. If indeed bound, then the timing argument combined with the known
age of the Universe implies that M = 5.58^{+0.85}_{-0.72}x10^{12} solar masses.
This is on the high end of the allowed mass range suggested by cosmologically
motivated models for the individual structure and dynamics of M31 and the Milky
Way, respectively. It is therefore possible that the timing mass is an
overestimate of the true mass, especially if one takes into account recent
results from the Millennium Simulation that show that there is also a
theoretical uncertainty of 41 percent (Gaussian dispersion) in timing mass
estimates. The M31 transverse velocity implies that M33 is in a tightly bound
orbit around M31. This may have led to some tidal deformation of M33. It will
be worthwhile to search for observational evidence of this.
