  We have recently suggested that the combination of the scalar virial theorem
($M_s \sim R_e \sigma^2$) and the $L=L'_0 \sigma^\beta(t)$ law, with L'_0 and
$\beta$ changing from galaxy to galaxy (and with time), can provide a new set
of equations valid for investigating the evolution of early-type galaxies
(ETGs) (Donofrio & Chiosi, 2022). These equations are able to account for the
tilt of the Fundamental Plane (FP) and to explain the observed distributions of
ETGs in all its projections. In this paper we analyze the advantages offered by
those equations, derive the $\beta$ and $L'_0$ parameters for real and
simulated galaxies, and demonstrate that, according to the value of $\beta$,
galaxies can move only along some permitted directions in the FP projections.
Then, we show that simple galaxy models that grow in mass by infall of gas and
form stars with a star formation rate depending on the stellar velocity
dispersion nicely reproduce the observed distributions of ETGs in the FP
projections and yield $\beta$s that agree with the measured ones. We derive the
mutual relationships among the stellar mass, effective radius, velocity
dispersion, and luminosity of ETGs as a function of $\beta$ and calculate the
coefficients of the FP. Then, using the simple infall models, we show that the
star formation history of ETGs is compatible with the $\sigma$-dependent star
formation rate, and that both positive and negative values of $\beta$ are
possible in a standard theory of galaxy evolution. The parameter $\beta(t)$
offers a new view of the evolution of ETGs. In brief, i) it gives a coherent
interpretation of the FP and of the motions of galaxies in its projections; ii)
it is the fingerprint of their evolution; iii) it measures the degree of
virialization of ETGs; iv) and finally it allows us to infer their evolution in
the near past.
