For the cubic (ab) 2 axbx is a covariant because each symbol a, b occurs three times; we can first of all find its real expression as a simultaneous covariant of two cubics, and then, by supposing the two cubics to merge into identity, find the expression of the quadratic covariant, of the single cubic, commonly known as the Hessian.
The system of the quadratic and cubic, consisting of 15 forms, and that of two cubics, consisting of 26 forms, were obtained by Salmon and Clebsch; that of the cubic and quartic we owe to Sigmund Gundelfinger (Programm Stuttgart, 186 9, 1 -43); that of the quadratic and quintic to Winter (Programm Darmstadt, 1880); that of the quadratic and sextic to von Gall (Programm Lemgo, 3873); that of two quartics to Gordan (Math.
This author questioned the possibility of solving cubics by pure algebra, and biquadratics by geometry.
Similarly as regards cubics, or curves of any other order: a cubic depends on 9 constants, and the elementary problems are to find the number of the cubics (9 p), (8p, 1l), &c., which pass through 9 points, pass through 8 points and touch 1 line, &c.; but it is in the investigation convenient to seek for the characteristics of the systems of cubics (8p), &c., which satisfy 8 instead of 9 conditions.
The elementary problems in regard to cubics are solved very completely by S.
He determines in every case the characteristics (µ, v) of the corresponding systems of cubics (4p), (3 p, il), &c. The same problems, or most of them, and also the elementary problems in regard to quartics are solved by Zeuthen, who in the elaborate memoir " Almindelige Egenskaber, &c.," Danish Academy, t.