  We combine morphological classifications from deep HST imaging of a sample of
3 clusters at z=0.31 and a further 9 clusters at z=0.37-0.56 with existing
spectroscopic observations of their E and S0 populations, to study the relative
spectral properties of these two galaxy types. We have also used similar data
for the Coma cluster as a present day bench-mark with which to compare our data
at higher redshift. The data span the range where strong evolution is claimed
in the proportion of S0 galaxies within rich clusters. Techniques have been
recently developed to analyse the strengths of absorption lines in the spectra
of local, passive galaxies to separate the effects of age & metallicity and
hence date the ages of the most recent, substantial star formation episode in
these galaxies. We have applied these techniques to the co-added spectra of the
distant cluster galaxies to determine the relative ages of the E and S0
populations. We find no statistically significant difference between the
luminosity-weighted ages of the E and S0 galaxies in these clusters. The
precision of this measurement would indicate that any recent star formation in
the cluster S0s at z=0.31 could have involved bursts of no more than ~11% (by
mass) in strength in no more than half of these galaxies. Our results, in
conjunction with other work, suggest that the progenitors of S0 galaxies in
rich clusters are mostly early-type spirals who, through interactions with the
cluster environment, have had their star formation truncated. This indicates a
relatively unspectacular origin for the missing S0 population at high redshift.
