As a continuation of our study of the faint galaxy luminosity function
in the Coma cluster of galaxies, we report here on the first spectros
copic observations of very faint galaxies (R less than or equal to 21.
5) in the direction of the core of this cluster. Out of these 34 galax
ies, only one may have a redshift consistent with Coma, all others are
background objects. The predicted number of Coma galaxies is 6.7 +/-
6.0, according to Bernstein et al. (1995, B95). If we add the 17 galax
ies observed by Seeker (1997), we end up with 5 galaxies belonging to
Coma, while the expected number is 16.0 +/- 11.0 according to B95. Not
ice that these two independent surveys lead to the same results. Altho
ugh the observations and predicted values agree within the error, such
results raise into question the validity of statistical subtraction o
f background objects commonly used to derive cluster luminosity functi
ons (e.g. B95). More spectroscopic observations in this cluster and ot
hers are therefore urgently needed. As a by-product, we report on the
discovery of a physical structure at a redshift z similar or equal to
0.5.