As regards simultaneous binary forms, the system of two quadratics, and of any number of quadratics, is alluded to above and has long been known.
The system of two ternary quadratics consists of 20 forms; it has been investigated by Gordan (Clebsch-Lindemann's Vorlesungen i.
Ciamberlini has found a system of 127 forms appertaining to three ternary quadratics (Batt.
2 cos w xy+y 2 = X 2 +2 cos w'XY+Y2, from which it appears that the Boolian invariants of axe+2bxy-y2 are nothing more than the full invariants of the simultaneous quadratics ax2+2bxy+y2, x 2 +2 cos coxy+y2, the word invariant including here covariant.
He successfully considers the systems of two and three simultaneous ternary quadratics.