11. The derivation of the virial theorem normally assumes a set of self-gravitating particles, and the derived mass is that of all the particles. It is obvious (think about it) that the motions will be the same so long as the mass is distributed in the same way as the particles, and in this case the virial mass is the mass of the whole system (particles plus any other mass distributed in a similar fashion).

In the case of an elliptical galaxy, it is clear that any dark halo extending well beyond the visible galaxy will not be included in a virial mass analysis (by Newton's theorem, a spherical shell of mass surrounding a galaxy will have no effect whatever on the gravitational field within the galaxy). At best the virial mass may give the total mass in the region where one is analysing the stars. Strictly speaking, however, the whole analysis is of doubtful validity, since the distribution of the d.m. is not believed to be similar to that of the stars.