Components of the proper motion velocities along RA and Dec for M31 (red dots) and M33 (orange dots) sampling 1000 combinations of initial conditions. For each galaxy we circle in black the velocities for which we find a pericentre passage of M33 around M31 in the past (for the standard mass model). The blue stars mark the values of the velocity components of M31 and M33 selected for showing the results of a test-particle simulation in the next figure. For this case the MW, M31 and M33 mass at the present time are 1.35, 2.00, and 0.36 × 10¹² M_⊙ respectively.