  In this paper, basic observational properties of elliptical galaxies such as
the integrated spectra, the chemical abundances and the enhancement of
$\alpha$-elements inferred from broad-band colors and line strength indices
$Mg_2$ and $< Fe >$ (and their gradients), the color-magnitude relation, the UV
fluxes, and the mass-luminosity ratios, are examined in the light of current
theoretical interpretations, and attention is called on several points of
internal contradiction. Indeed existing models for the formation and evolution
of elliptical galaxies are not able to simultaneously account for all of the
above observational features. We suggest that the new initial mass function
(IMF) by Padoan et al. (1997), which depends on the temperature, density, and
velocity dispersion of the medium in which stars are formed, may constitute a
break-through in the difficulties in question. Models of elliptical galaxies
incorporating the new IMF (varying with time and position inside a galaxy) are
presented and discussed at some extent. As a result of the changing IMF, the
enhancement of $\alpha$-elements and tilt of the Fundamental Plane are easily
explained leaving unaltered the interpretation of the remaining properties
under examination. Finally, some implications concerning the relative
proportions of visible stars, collapsed remnants (baryonic dark matter), and
gas left over by the star forming process are examined.
