Preferential acceperation of alpha particles interacting with left-hand polarized ion cyclotron waves is studied. It is shown that a small positive drift velocity between alpha particles and protons can lead to alpha particle velocities well in excess of the proton bulk velocity. During the acceleration process, which is assumed to take place at heliocentric distances less than 0.3 AU, the alpha particle drift velocity should exceed the proton bulk velocity, and then the gap which exists around the alpha particle gyrofrequency should disappear. It is also shown that for proton thermal anisotropies of the order of those observed in fast solar wind, the waves either grow or are not damped excessively, so that the waves can exist and might thus lead to the observed differential speeds. However, the way in which the alpha particles exceed the proton velocity remains unexplained. ¿ American Geophysical Union 1991