  Using Chandra X-ray observations of 9 nearby, X-ray luminous ellipticals with
good optical velocity dispersion measurements, we show that a tight correlation
exists between the Bondi accretion rates calculated from the X-ray data and
estimated black hole masses, and the power emerging from these systems in
relativistic jets. The jet powers, inferred from the energies and timescales
required to inflate the cavities observed in the surrounding X-ray emitting
gas, can be related to the accretion rates by a power law model. A significant
fraction (2.2^{+1.0}_{-0.7} per cent, for P_jet=10^{43} erg/s) of the energy
associated with the rest mass of material entering the accretion radius
eventually emerges in the jets. The data also hint that this fraction may rise
slightly with increasing jet power. Our results have significant implications
for studies of accretion, jet formation and galaxy formation. The tight
correlation between P_Bondi and P_jet suggests that the Bondi formulae provide
a reasonable description of the accretion process, despite the likely presence
of magnetic pressure and angular momentum in the accreting gas, and that the
accretion flows are approximately stable over timescales of a few million
years. Our results show that the black hole `engines' at the hearts of large
elliptical galaxies and groups can feed back sufficient energy to stem cooling
and star formation, leading naturally to the observed exponential cut off at
the bright end of the galaxy luminosity function.
