From the invariant a2 -2a 1 a 3 -2aoa4 of the quartic the diminishing process yields ai-2a 0 a 21 the leading coefficient of the Hessian of the cubic, and the increasing process leads to a3 -2a 2 a 4 +2a i a 5 which only requires the additional term-2aoa 6 to become a seminvariant of the sextic. A more important advantage, springing from the new form of S2, arises from the fact that if x"-aix n- +a2x n-2.
This arose from the study by Felix Klein and Sophus Lie of a new theory of groups of substitutions; it was shown that there exists an invariant theory connected with every group of linear substitutions.
F(a ' a ' a, ...a) =r A F(ao, a1, a2,ï¿½ï¿½ï¿½an), 0 1 2 n the function F(ao, al, a2,...an) is then said to be an invariant of the quantic gud linear transformation.
This expression of R shows that, as will afterwards appear, the resultant is a simultaneous invariant of the two forms.
Remark.-The invariant C is a numerical multiple of the resultant of the covariants i and j, and if C = o, p is the common factor of i and j.