The optical counterpart of the new extreme ultraviolet source EUVE J1016-053 (=RE 1016-053) is known to show intermittent sharp Balmer and He I emission and traces of an M dwarf associated with a white dwarf optical spectrum. We present extensive optical spectroscopy showing the emission lines vary in velocity and intensity on a period of 0(d).78929 +/- 0(d).00003. The phase of the emission-line strengths lags that of the velocities by 0.237 +/- 0.013 cycle, consistent with the quarter-cycle offset expected if emission arises from the side of the secondary facing the hot white dwarf. EUVE J1016-053 is another example of an extreme ultraviolet illumination-effect binary, which only recently emerged from a common-envelope phase of binary evolution. Based on spectroscopic measurements and new BVR photometry, we find that the M dwarf secondary's contribution to the combined light is smaller than previously reported, probably because of the difficulty of avoiding contamination from a third star 3''.2 distant. The velocity of the weak H II absorption is in antiphase to the emission, suggesting that it originates in the white dwarf photosphere. The mass function implied by the emission-line motion is f(M) = 0.28 +/- 0.08 M.. Assuming the He II absorption does follow the white dwarf, the mass ratio M(WD)/M(dM) is 1.8 +/- 0.5, and the gravitational redshift is 45 +/- 14 km s(-1). The modulation of the emission lines suggests the inclination i > 40 degrees, but plausible masses demand the inclination be well above this value.
