-, 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.

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 = B65ï¿½5B64p - 4A2p5, which is Bring's form of **quintic** at which we can always arrive, by linear transformation, whenever the invariant C vanishes.

The complete systems of the **quintic** and sextic were first obtained by Gordan in 1868 (Journ.

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.

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.