  Given the recently deduced relationship between X-ray temperatures and
stellar velocity dispersions (the "T-sigma relation") in an optically complete
sample of elliptical galaxies (Davis & White 1996), we demonstrate that L>L_*
ellipticals contain substantial amounts of dark matter in general. We present
constraints on the dark matter scale length and on the dark-to-luminous mass
ratio within the optical half-light radius and within the entire galaxy. For
example, we find that minimum values of dark matter core radii scale as r_dm >
4(L_V/3L_*)^{3/4}h^{-1}_80 kpc and that the minimum dark matter mass fraction
is >~20% within one optical effective radius r_e and is >~39-85% within 6r_e,
depending on the stellar density profile and observed value of beta_spec. We
also confirm the prediction of Davis & White (1996) that the dark matter is
characterized by velocity dispersions that are greater than those of the
luminous stars: sigma_dm^2 ~ 1.4-2 sigma_*^2. The T-sigma relation implies a
nearly constant mass-to-light ratio within six half-light radii: M/L_V ~ 25h_80
M_sun/L_V_sun. This conflicts with the simplest extension of CDM theories of
large scale structure formation to galactic scales; we consider a couple of
modifications which can better account for the observed T-sigma relation.
