ME. A. CAYLEY’S SEVENTH MEMOIE ON QIJANTICS. 
279 
232. Tor the binary quartic {a, h, c, d, e^x, if U be the quartic itself, HU the 
Hessian, OU the cubicovai iant, I, J, the quadrinvariant and the cubinvariant, and 
□ (=1®— 27J“) the discriminant (see Fifth Memoir, Nos. 128, 134), then 
Table No. 72. 
<I>(aU+6^HU)= (1, 0, -91, -54J5(a, ,3)^OU, 
H(aU+6j0IiU)=— igb^l, 0, -91, -54JXa, /3)hU 
+ 0, -91, -54JIa, 3)M1U, 
I (aU + 6,3HU)= (I, 18J, 3PJa, 3)^ 
J(«U+63HU)= (J, P, 9IJ, -P+54PX«. 
□ (aU+6/3HU)= *(1, 0, -181, 108J, 81P, 9721J, 2916P5;a, (ifu 
= [(1, 0, -91, -54JIa, OTd. 
233. Writing for the moment 
G=(l, 0, -91, -54JI<,, (if, 
then the Hessian, cubicovariant, and discriminant of this cubic function of (a, j3) are 
respectively 
HG=- 3(1, 18J, 3PJ«, /3)^ 
(PG= 54(J, P, 9IJ, -P+54P'Xa, /3)^ 
□ G= — 108n ; 
so that the covariants of the intermediate aU-|-6|3HU are all of them expressible by 
means of the cubic function G. 
It may be noticed that G is what the left-hand side of the equation 
4(HU)^- 41 . HU . U'^-f JU^= - (cPU)^ 
(see Fifth Memoir, No. 128) becomes on writing therein a, — 6/3, for U, HU respectively, 
and throwing out the factor 4. 
234. I take the opportunity of remarking with respect to a binary quartic 
U=(a, c, d^x, yY, tliat the Hessian of the cubicovariant, to fix the numerical 
factor, say -i{b*(hU.b^OU-(B,B/PU)U, is 
= PU^- .36.1 . U. HU+12I(HU)^ 
which is 
( 1 Q T \ 2 If) 
lU-i^HUj +^(P-27P)(HU)U 
or if P— 27P — 0, that is if the quartic has a pair of equal factors, the Hessian of the 
cubicovariant is a perfect square, 
235. Coming now to the ternary cubic U = (ffl, h, c, /*, h, i, j, Ic, I'^x, y, zY, I give 
in the first place the following comparison of my notation with that of M. Akonhold. 
* The coefficient 2916 is in the Fifth Memoh’ erroneously given as — 291GJ". 
