  X-ray observations of M87 in the Virgo cluster and NGC 4874 in Coma reveal
that the gas temperature beyond about 50 kpc from these cD galaxies is
comparable to the virial temperature of the cluster, 3 or 9 keV respectively,
but within the optical galaxy the temperature drops to the galactic virial
temperature, about 1 keV. We show that these steep thermal (and density)
gradients follow naturally from the usual cooling inflow assumptions without
recourse to thermal conductivity. To avoid unobserved central mass
concentrations, most of the gas must radiatively cool (``dropout'') before it
flows to the galactic core, i.e. the gas must be multiphase. Recent XMM spectra
of M87 indicate single phase flow at every radius with no apparent radiative
cooling to low temperatures. However, the X-ray spectral evidence for
multiphase cooling beginning at lower temperatures near 1 keV (within about 10
kpc) may be less apparent and may have escaped detection. Finally, we show that
the standard decomposition method used by X-ray observers to determine the mass
flow dM/dt(r) may fail rather badly due to small deviations from perfectly
steady state flow. When applied to our slightly non-steady computed flows, this
decomposition (incorrectly) gives the usual result, dM/dt proportional to r,
and overestimates the mass dropout dM/dt(r) at large radii.
