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We present a study of the internal kinematics of two globular clusters, M10 (NGC 6254) and M71 (NGC 6838), using individual radial velocity (RV) measurements obtained from observations using the Hydra multi-object spectrograph on the WIYN 3.5 m telescope. We measured 120 RVs for stars in M10, of which 107 were determined to be cluster members. In M71, we measured 82 RVs and determined 78 of those measurements belonged to cluster members. Using the cluster members, we determine a mean RV of $75.9 \pm 4.0$ (s.d.) km s$^{-1}$ and $-22.9 \pm 2.2$ (s.d.) km s$^{-1}$ for M10 and M71, respectively. We combined the Hydra RV measurements with literature samples and performed a line-of-sight rotational analysis on both clusters. Our analysis has not revealed a statistically significant rotation in either of these clusters with the exception of the inner region (10\arcsec - 117\arcsec) of M10 for which we find hints of a marginally significant rotation with amplitude V$_{rot} = 1.14 \pm 0.18$ km s$^{-1}$. For M10, we calculate a central velocity dispersion of $σ_{0} = 5.44 \pm 0.61$ km s$^{-1}$, which gives a ratio of the amplitude of rotation to the central velocity dispersion $V_{rot}/σ_{0} = 0.21 \pm 0.04$. We also explored the rotation of the multiple stellar populations identified in M10 and M71 and found rotation (or lack thereof) in each population consistent with each other and the cluster global rotation signatures.