THERE are two opposing interpretations of the small dispersion σ1∼0.35 mag in the absolute magnitudes M1 of the brightest members of rich clusters of galaxies measured by Sandage1. In one view, advanced, among others, by Peebles2,3, it is assumed that there is a universal cluster luminosity function, and that the value of σ1 and of the variation of M1 with cluster richness can be explained as the statistical result of drawing the brightest member from a set of clusters obeying this function. There is therefore nothing physically remarkable about the brightest member, and its luminosity is determined by the same mechanism as that which determines the luminosity of the fainter members. The opposing view, which has been advanced by Peach4, Sandage1, and Tremaine and Richstone5, claims that the small dispersion is due to special conditions governing the formation or evolution of the brightest member, and is therefore entirely or in part independent of the luminosity function of the other cluster members. On this view one evokes either special conditions at the epoch of galaxy formation1, or a mechanism, such as galactic accretion6, which governs the size and luminosity of the most massive galaxy in such a way as to give a small dispersion in the metric magnitude. The weakness of the statistical model, according to its opponents, is its inability to account for σ1 without postulating a luminosity function steeper than those directly observed. The opposing view is open to the criticism that it is either ad hoc or that the mechanisms of galaxy evolution are too uncertain to lead to reliable conclusions. Here, we discuss the two interpretations in the light of new results and suggest that the case against the statistical model may not be as strong as sometimes suggested.