  We investigate the chemical properties of M32, the bulges of M31 and the
Milky Way, and the dwarf spheroidal galaxies NGC 205, NGC 185, Sagittarius, and
Fornax using previously published oxygen abundances for their planetary
nebulae. Our principal result is that the mean stellar oxygen abundances for
all of these galaxies correlate very well with their mean velocity dispersions.
This implies that the balance between energy input from type II supernovae and
the gravitational potential energy controls how far chemical evolution proceeds
in bulges, ellipticals, and dwarf spheroidals. It appears that chemical
evolution ceases once supernovae have injected sufficient energy that a
galactic wind develops. All of the galaxies follow a single relation between
oxygen abundance and luminosity, but the dwarf spheroidals have systematically
higher [O/Fe] ratios than the other galaxies. Consequently, dynamically hot
galaxies do not share a common star formation history nor need they share a
common chemical evolution, despite attaining similar mean stellar oxygen
abundances when forming similar stellar masses. The oxygen abundances support
previous indications that the stars in higher luminosity ellipticals and bulges
were formed on a shorter time scale than their counterparts in less luminous
systems. We calibrate the Mg$_2$ index for dynamically hot galaxies as a
function of observed age and oxygen abundance using the mean oxygen abundances
from the planetary nebulae and the best age estimates available for the stars
in M32 and the bulges of M31 and the Milky Way.
