The de Vaucouleurs' profile (or de Vaucouleurs' law, or R to the 1/4 law) is a relation between the surface brightness of a point on an elliptical galaxy and that point's distance from the galaxy's center (i.e., a surface brightness profile).
ln ΣR = ln Σ0 - k R1/4
This is an empirical relation and unfortunately it has limited use for determining other information, but is a relation that any theory of elliptical galaxies had best reproduce. I believe some models of galaxy mergers appear to produce galaxies with such a relation, consistent with the notion that elliptical galaxies are often the result of mergers.
Other such empirical relations regarding galaxy surface brightness have been devised. The de Vaucouleurs' profile is an example of a Sérsic profile, i.e., a surface brightness profile using an exponent of R1/n for some small number n.