A dual quadratic transformation associated with the Hessian conics of a pencil

Abstract

1. The invariants and covariants of a system of two conics have been much studied 2 but little has been said about those of three conies. Three conics have a symmetrical invariant Ω 123 , or in symbolical notation ( a b c ) 2 . According to Ciamberlini 3 the vanishing of this invariant signifies that the Φ conic of any two of f 1 , f 2 , f 3 is inpolar with respect to the third; and in a previous paper 4 I have derived by symbolical methods a more symmetrical result, viz., if Ω 123 vanishes, then u being any line in the plane, u 1 , u 2 , u 3 are concurrent, where u i is the polar with respect to f i of the pole of u with respect to Φ jk .