[ 277 ] 
XIV. A Seventh Memoir on Quantics. Bij Aethur Cayley, Esq., F.R.S. 
Eeceived February 28, — Eead March 14, 1861. 
The present memoir relates chiefly to the theory of ternary cubics. Since the date of 
my Third Memoir on Quantics, M. Aronhold has published the continuation of his 
researches on ternary cubics, in the memoir “ Theorie der homogenen Functionen 
dritten Grades von drei Veranderlichen,” Crelle, t. Iv. pp. 97-191 (1858). He there 
considers two derived contravariants, linear functions of the fundamental ones, and 
which occupy therein the position which the fundamental contravariants PU, QU do in 
my Third Memoir ; in the notation of the present memoir these derived contravariants 
are 
YU= 3T.PU-4S.QU, 
ZU = -48ShPU+T. QU; 
and for the canonical form ^Ixyz, they acquire respectively the factor (1 + 8/^)^ 
viz. in this case 
YU = (l + 8Pf{ I 31,0 
ZU=(l + 8«=)’{(l+2i’)(?+,»+0)+18n,0- 
The derived contravariants have with the covariants U, HU, even a more intimate con- 
nexion than have the contravariants PU, QU ; and the advantage of the employment of 
Ytr, ZU fully appears by M. Aroxiiold’s memoir. 
But the conclusion is, not that the contravariants PU, QU are to be rejected, but that 
the system is to be completed by the addition thereto of two derived covariants, linear 
functions of U, HU ; these derived covariants, suggested to me by M. Aronhold’s 
memoir, are in the present memoir called CU, DU ; their values are 
CU= -T.U+ 24S.hu 
DU= 8S^U-3T.HU: 
and for the canonical form they acquire respectively, not indeed 
(1-1-8/®)^ but the simple power (1 + 8Z^), as a factor, viz. in this case 
CU=(l-l-87’){ (-l + 4/^)(a’H,y^+2*)+ mxyz) 
DU=(l-f 8Z=’){^^( 5 + 4/*)(^^-f/+2^)+3(l-lO7>^0} ; 
it was in fact by means of this condition as to the factor (1 + 8/^), that the foregoing 
expressions for CU, DU were obtained *. 
* M. Aronhold, in a letter dated Berlin, 17 .June 1861, has pointed out to me that the covariants 
CIJ, DU are in his notation Ps^, Pxy, and that they belong to the forms called Conjugate Forms, § 27 of 
his memoir. But the explicit development of the properties of these covariants is not on this account the 
less interesting. Added 20 Sept. 1861. — A. C. 
