We present the first proper motion (PM) measurements for the dwarf elliptical galaxies NGC 147 and NGC 185, two satellite galaxies of M31, using multi-epoch HST imaging data with time baselines of $sim 8$ years. For each galaxy, we take an error-weighted average of measurements from HST ACS/WFC and WFC3/UVIS to determine the PMs. Our final results for the PMs are $(mu_mathrm{W}, mu_mathrm{N})_mathrm{N147} = (-0.0232, 0.0378) pm (0.0143, 0.0146),mathrm{mas},mathrm{yr}^{-1}$ for NGC 147, and $(mu_mathrm{W}, mu_mathrm{N})_mathrm{N185} = (-0.0242, 0.0058) pm (0.0141, 0.0147),mathrm{mas},mathrm{yr}^{-1}$ for NGC 185. The 2-dimensional direction of motion for NGC 147 about M31 is found to be aligned with its tidal tails. The 3-d positions and velocities of both galaxies are transformed into a common M31-centric coordinate system to study the detailed orbital histories of the combined M31+NGC 147+NGC 185 system via numerical orbit integration. We find that NGC 147 (NGC 185) had its closest passage to M31 0.3-0.5~Gyr ($gtrsim 1.6$~Gyr) within the past 6 Gyr at distances of $sim 70$ kpc (70-260 kpc). The pericentric times of NGC 147/NGC 185 correlate qualitatively well with the presence/absence of tidal tails seen around the galaxies. Our PMs show that the orbital poles of NGC 147, and also NGC 185 albeit to a lesser degree, agree within the uncertainties with the normal of the Great Plane of Andromeda (GPoA). These are the first measurements of the 3-d angular momentum vector of any satellite identified as original GPoA members. Our results strengthen the hypothesis that the GPoA may be a dynamically coherent entity. We revisit previous claims that NGC 147 and NGC 185 are binary galaxies and conclude that it is very unlikely the two galaxies were ever gravitationally bound to each other.
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