27 S ME. A. CAYLEY’S SEVENTH MEMOIE ON QUANTICS. 
The formuhe of my Third Memoir and those of M. Aeonhold are by this means 
brought into harmony and made parts of a whole ; instead of the two intermediates 
aU + 6/3HU, 6aPU+i3QU, 
in Tables 68 and 69 of my Third Memoir, or of the intermediates 
aU+6/3HU, -2aYU+2/3ZU, 
of ]M. Akoniiold’s theory, we have the four intermediates 
aU+6/3HU, -2«YU+2/3ZU, 2aCU-2i3DU, 6aPU+/3QU, 
in Tables 74, 75, 76, and 77 of the present memoir. These four Tables embrace the 
former results, and the new ones which relate to the covariants CU, DU ; and they are 
what is most important in the present memoir. I have, however, excluded from the 
Tables, and I do not in the memoir consider (otherwise than incidentally) the covariant 
of the sixth order ©U, or the contravariant (reciprocant) FU. 
I have given in the memoir a comparison of my notation with that of M. Aeonhold. 
A short part of the memoir relates to the binary cubic and the binary quartic, viz. each 
of these qualities has a covariant of its own order, forming with it an intermediate 
ciU-P/iW, the covariants whereof contain qualities in (a, (3), the coefficients of which are 
invariants of tlie original quantic. The formulae which relate to these, cases are in fact 
given in my Fifth Memoir, but they are reproduced here in order to show the relations 
between the quantics in («, (3) contained in the formulae. As regards the binary quartic, 
tliese results are required for the discussion of the like question in regard to the ternary 
cubic, viz. that of finding the relations between the different quantics in (a, j3) contained 
in the formulae relating to the ternary cubic. Some of these relations have been 
obtained by M. Heemite in the memoir “ Sur les formes cubiques a trois indeterminees,” 
(Liouville, t. hi. pp. 37-40 (1858), and in that “ Sur la Pesolution des equations du qua- 
trieme degre,” Comptes Pendus, xlvi. p. 715 (1858), and by M. Aeonhold in his memoir 
already referred to ; and in particular I reproduce and demonstrate some of the results 
in the last-mentioned memoir of M. Heemite. But the relations in question are in the 
present memoir exhibited in a more complete and systematic form. 
The paragraphs and Tables of the present memoir are numbered consecutively with 
those of my former memoirs on Quantics. 
231. For the binary cubic (a, 5, c, yY, if U be the cubic itself, HU the Hessian, 
(PU the cubicovariant, and □ the discriminant (see Fifth Memoir, Nos. 115, 118), then 
Covaiiant and other Tables, No. 71. 
H(«U-fj30U)=(«^-/3^n)HU, 
<P(«U-f/3OT)= _iB^(«^_/32n).U 
-fR(a^-/3^n).ou, 
□ («u-fpou)= («^-i3^n)^n, 
so that the quantics in (a, j3) all of them depend on cd — (S^O. 
