The complete systems of the quintic and sextic were first obtained by Gordan in 1868 (Journ.
-, reduce s x2ax1 -x10x2 to the form j Oz ON 2 1 1 j 2 i The Binary Quintic.-The complete system consists of 23 forms, of which the simplest are f =a:; the Hessian H = (f, f') 2 = (ab) 2axbz; the quadratic covariant i= (f, f) 4 = (ab) 4axbx; and the nonic co variant T = (f, (f', f") 2) 1 = (f, H) 1 = (aH) azHi = (ab) 2 (ca) axbycy; the remaining 19 are expressible as transvectants of compounds of these four.
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.
The process is not applicable with complete success to quintic and higher ordered binary forms. This arises from the circumstance that the simple syzygies between the ground forms are not all independent, but are connected by second syzygies, and these again by third syzygies, and so on; this introduces new difficulties which have not been completely overcome.
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.