  We use a restricted sample of elliptical galaxies, whose kinematical
parameters inside the semimajor axis were calculated correcting the effect of
the integration of the light along the line of sight, in order to analyze a
possible relationship between the mass of a Supermassive Black Hole (SMBH) and
the kinetic energy of random motions in the host galaxy. We find $M_{BH}
\propto (M_{G} \sigma^{2})^\alpha$ with $0.87 \leq \alpha \leq 1$ depending on
the different fitting methods and samples used. This result could be
interpreted as a new fundamental relationship or as a new way to explain the
old $M_{BH} - \sigma$ law. In fact, the relations of the velocity dispersion
both with the mass of the SMBH ($M_{BH} \propto \sigma^{4.12}$) and with the
mass of the host galaxy ($M_{G} \propto \sigma^{2.16}$) induce us to infer an
almost direct proportionality: $M_{BH} \propto M_{G} \sigma^{2}$. A similar
relationship is found for the total kinetic energy involving the rotation
velocity too.
