  As a continuation of our study of the faint galaxy luminosity function in the
Coma cluster of galaxies, we report here on the first spectroscopic
observations of very faint galaxies (R $\le$ 21.5) in the direction of the core
of this cluster. Out of these 34 galaxies, only one may have a redshift
consistent with Coma, all others are background objects. The predicted number
of Coma galaxies is 6.7 $\pm$ 6.0, according to Bernstein et al. (1995, B95).
If we add the 17 galaxies observed by Secker (1997), we end up with 5 galaxies
belonging to Coma, while the expected number is 16.0 $\pm$ 11.0 according to
B95. Notice that these two independent surveys lead to the same results.
Although the observations and predicted values agree within the error, such
results raise into question the validity of statistical subtraction of
background objects commonly used to derive cluster luminosity functions (e.g.
B95). More spectroscopic observations in this cluster and others are therefore
urgently needed. As a by-product, we report on the discovery of a physical
structure at a redshift z$\simeq$0.5.
