The light profiles of elliptical (E) galaxies are known to display small systematic deviations (0.1-0.2 mag.) from the R^1/4 law. In this paper we show that the senses and amplitudes of these departures can be naturally accounted for by a simple distribution function constructed on the basis of statistical mechanics of violent relaxation. As a consequence, detailed light-profiles can be used to infer about the central potentials of E galaxies (the only free shape parameter of our model). Furthermore, the small deviations have recently been shown to correlate with luminosity, L. This observation entails a slight breaking of the generally assumed structural homology between E galaxies. Using our model we parametrize this broken homology by establishing a correlation between a suitably normalized central potential and luminosity. The non-homology means that a basic assumption in the interpretation of the Fundamental Plane (FP) breaks down, and with it the conclusions derived from it. Instead, by assuming that M/L is independent of luminosity, we derive a relation akin to the FP directly from our correlation. This implies that the FP may have a simple stellar dynamical origin. We can reproduce the observed Carlberg-Kormendy relation for the central phase-space densities (f_c) of E galaxies of identical structure, f_c = L^-2.35, but non-homology changes it to a much weaker dependence (f_c = L^-1.5 for constant M/L) which would imply that the central phase-space densities of ellipticals are comparable to those of spiral galaxies. Thus dissipationless merging is consistent with the FP, although HST observations, notably the presence of nuclear embedded disks, indicate that the assumptions behind our model break down for the nuclei of faint E galaxies in which dissipative processes seem to be important.

10 pages. To appear in ApJ, May 20, 1995