  We present chemical abundances for planetary nebulae in M32, NGC 185, and NGC
205 based upon spectroscopy obtained at the Canada-France-Hawaii Telescope
using the Multi-Object Spectrograph. From these and similar data compiled from
the literature for other Local Group galaxies, we consider the origin and
evolution of the stellar progenitors of bright planetary nebulae in galaxies
where star formation ceased long ago. The ratio of neon to oxygen abundances in
bright planetary nebulae is either identical to that measured in the
interstellar medium of star-forming dwarf galaxies or at most changed by a few
percent, indicating that neither abundance is significantly altered as a result
of the evolution of their stellar progenitors. Several planetary nebulae appear
to have dredged up oxygen, but these are the exception, not the rule. The
progenitors of bright planetary nebulae typically enhance their original helium
abundances by less than 50%. In contrast, nitrogen enhancements can reach
factors of 100. However, nitrogen often shows little or no enhancement,
suggesting that nitrogen enrichment is a random process. The helium, oxygen,
and neon abundances argue that the typical bright planetary nebulae in all of
the galaxies considered here are the progeny of stars with initial masses of
approximately 1.5 Msun or less, based upon the nucleosynthesis predictions of
current theoretical models. These models, however, are unable to explain the
nitrogen enrichment or its scatter. Similar conclusions hold for the bright
planetary nebulae in galaxies with ongoing star formation. Thus, though
composition varies significantly, there is unity in the sense that the
progenitors of typical bright planetary nebulae appear to have undergone
similar physical processes. (Abridged)
