128 
[443 
443. 
NOTE ON THE SOLUTION OF THE QUARTIC EQUATION 
a 77 + 6/3// = 0. 
[From the Mathematische Annalen, vol. l. (1869), pp. 54, 55.] 
If £7 denote the quartic function (a, b, c, d, eQx, y) 4 , H its Hessian 
= (ac — b 2 , 2 (ad — be), ae + 2bd — 3c 2 , 2 (be - cd), ce — d 2 \x, y)\ 
a and yS constants, then we may find the linear factors of the function a£7+6/3// 
(or what is the same thing solve the equation a U + 6/3/7 = 0) by a formula almost 
identical with that given by me (Fifth Memoir on Quantics, Phil. Trans, vol. cxlviii. 
(1858), see p. 446, [156]) in regard to the original quartic function 77. 
In fact (reproducing the investigation) if /, J are the two invariants, M = , 
<I> the cubicovariant 
= (- a-d + 3abc — 2b 3 , &c$x, yf, 
then the identical equation JU 3 —/77 2 /7+4/7 3 =—<J> 2 , maybe written(1, 0, —71/, M\IH, JU) 3 
= —I/ 3< F 2 , whence if u) 1 , co 2 , w 3 are the roots of the equation (1, 0, —71/, M\w, 1) 3 = 0, 
or what is the same thing w 3 - M (w — 1) = 0 ; then the functions 
IH-CO.JU, IH-coJU, IH-CO.JU 
are each of them a square: writing 
(g) 2 — £+) (IH — w x JU) — X'\ 
(co 3 — (Oj) (IH — a>.,JU) = F 2 , 
(ft)! — (Wo) (IH — ft);. JU) = Z• ,