We present a study of the variation of the relative abundance of helium to hydrogen in the solar wind as a function of solar wind speed and heliographic latitude over the previous solar cycle. The average values of A(sub He), the ratio of helium to hydrogen number densities, are calculated in 25 speed intervals over 27-day Carrington rotations using Faraday Cup observations from the Wind spacecraft between 1995 and 2005. The higher speed and time resolution of this study compared to an earlier work with the Wind observations has led to the discovery of three new aspects of A(sub He), modulation during solar minimum from mid-1995 to mid-1997. First, we find that for solar wind speeds between 350 and 415 km/s, A(sub He), varies with a clear six-month periodicity, with a minimum value at the heliographic equatorial plane and a typical gradient of 0.01 per degree in latitude. For the slow wind this is a 30% effect. We suggest that the latitudinal gradient may be due to an additional dependence of coronal proton flux on coronal field strength or the stability of coronal loops. Second, once the gradient is subtracted, we find that A(sub He), is a remarkably linear function of solar wind speed. Finally, we identify a vanishing speed, at which A(sub He), is zero, is 259 km/s and note that this speed corresponds to the minimum solar wind speed observed at one AU. The vanishing speed may be related to previous theoretical work in which enhancements of coronal helium lead to stagnation of the escaping proton flux. During solar maximum the A(sub He), dependences on speed and latitude disappear, and we interpret this as evidence of two source regions for slow solar wind in the ecliptic plane, one being the solar minimum streamer belt and the other likely being active regions.