The complete systems of the quintic and sextic were first obtained by Gordan in 1868 (Journ.
Now, when C = o, clearly (see ante) R 2 j = 6 2 p where p = S +2 B a; and Gordan then proves the relation 6R 4 .f = B65Ã¯¿½5B64p - 4A2p5, which is Bring's form of quintic at which we can always arrive, by linear transformation, whenever the invariant C vanishes.
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.
three quintic forms f; (f, i) 1; (i 2, T)4 two sextic forms H; (H, 1)1 one septic form (i, T)2 one nonic form T.
Certain convariants of the quintic involve the same determinant factors as appeared in the system of the quartic; these are f, H, i, T and j, and are of special importance.
Hermite expresses the quintic in a forme-type in which the constants are invariants and the variables linear covariants.
Sylvester showed that the quintic might, in general, be expressed as the sum of three fifth powers, viz.
a little further progress has been made by Cayley, who established the two generating functions for the quintic 1 -a3s 11 -a8.1 a12.
His first notable work was a proof of the impossibility of solving the quintic equation by radicals.
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