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.