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 .