  We present Faint Object Camera (FOC) ultraviolet images of the central
14x14'' of Messier 31 and Messier 32. The hot stellar population detected in
the composite UV spectra of these nearby galaxies is partially resolved into
individual stars, and their individual colors and apparent magnitudes are
measured. We detect 433 stars in M 31 and 138 stars in M 32, down to detection
limits of m_F275W = 25.5 mag and m_F175W = 24.5 mag. We investigate the
luminosity functions of the sources, their spatial distribution, their
color-magnitude diagrams, and their total integrated far-UV flux. Although M 32
has a weaker UV upturn than M 31, the luminosity functions and color-magnitude
diagrams of M 31 and M 32 are surprisingly similar, and are inconsistent with a
majority contribution from any of the following: PAGB stars more massive than
0.56 Msun, main sequence stars, or blue stragglers. Both the the luminosity
functions and color-magnitude diagrams are consistent with a dominant
population of stars that have evolved from the extreme horizontal branch (EHB)
along tracks with masses between 0.47 and 0.53 Msun. These stars are well below
the detection limits of our images while on the zero-age EHB, but become
detectable while in the more luminous (but shorter) AGB-Manque' and post-early
asymptotic giant branch (PEAGB) phases. The FOC observations require that only
a only a very small fraction of the main sequence population (2% in M 31 and
0.5% in M 32) in these two galaxies evolve though the EHB and post-EHB phases,
with the remainder evolving through bright PAGB evolution that is so rapid that
few if any stars are expected in the small field of view covered by the FOC.
