We study the distribution of projected ellipticity n(epsilon) for galaxies in a sample of 20 rich (Richness >= 2) nearby (z < 0.1) clusters of galaxies. We find no evidence of differences in n(epsilon), although the nearest cluster in the sample (the Coma Cluster) is the largest outlier (P(same) < 0.05). We then study n(epsilon) within the clusters, and find that epsilon increases with projected cluster-centric radius R (hereafter the epsilon-R relation). This trend is preserved at fixed magnitude, showing that this relation exists over and above the trend of more luminous galaxies to be both rounder and more common in the centres of clusters. The epsilon-R relation is particularly strong in the subsample of intrinsically flattened galaxies (epsilon > 0.4), therefore it is not a consequence of the increasing fraction of round slow rotator galaxies near cluster centers. Furthermore, the epsilon-R relation persists for just smooth flattened galaxies and for galaxies with de Vaucouleurs-like light profiles, suggesting that the variation of the spiral fraction with radius is not the underlying cause of the trend. We interpret our findings in light of the classification of early type galaxies (ETGs) as fast and slow rotators. We conclude that the observed trend of decreasing epsilon towards the centres of clusters is evidence for physical effects in clusters causing fast rotator ETGs to have a lower average intrinsic ellipticity near the centres of rich clusters.
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