The sample of 15 galaxy clusters published by Dressler and Shectman (1988) has been analyzed to search for a tendency for spiral and irregular galaxies to have greater velocity dispersion than the ellipticals and lenticulars. It is shown that this trend is highly significant, with a probability of less than 0.05 percent that it may result from statistical fluctuations in the sample of clusters. An analogous but much less significant effect appears also to be present between lenticular and elliptical galaxies. These results are interpreted as evidence that late-type galaxies are still in the process of infalling toward the virialized cluster cores. 34 refs.