770] 
ON THE 34 CONCOMITANTS OF THE TERNARY CUBIC. 
343 
Hermite, Extrait d’une lettre ä M. Brioschi: Crelle, t. lxiii. (1864), pp. 30—32, 
followed by a note by Brioschi, pp. 32—33. 
The skew covariant of the ninth order, which is y 3 — z 3 . z 3 — x 3 .a? — y 3 for the canoni 
cal form a? + y 3 + z 3 + Qlxyz, and the corresponding contravariant y 3 — . £ 3 — £ 3 . f 3 — y 3 , 
alluded to p. 116 of Salmon’s Lessons, were obtained, the covariant by Brioschi and 
the contravariant by Hermite, in the last-mentioned papers. 
Clebsch and Gordan, Ueber die Theorie der ternären cubischen Formen: Math. 
Annalen, t. i. (1869), pp. 56—89. 
The establishment of the complete system of the 34 covariants, contravariants 
and Zwischen for men, or, as I have here called them, the 34 concomitants, was first 
effected by Gordan in the next following memoir: 
Gordan, Ueber die ternären Formen dritten Grades: Math. Annalen, t. I. (1869), 
pp. 90—128. 
And the theory is further considered: 
Gundelfinger, Zur Theorie der ternären cubischen Formen: Math. Annalen, t. VI. 
(1871), pp. 144—163. The author speaks of the 34 forms as being “theils mit den 
von Gordan gewählten identisch, theils möglichst einfache Combinationen derselben.” 
They are, in fact, the 34 forms given in the present paper for the canonical form 
of the cubic, and the meaning of the adopted combinations of Gordan’s forms will 
presently clearly appear. 
There is an advantage in using the form ax 3 + by 3 + cz 3 + Qlxyz rather than the 
Hessian form x 3 + y 3 + 2 s + Qlxyz, employed in my Third and Seventh Memoirs on 
Quantics : for the form ax 3 + by 3 + cz 3 + Qlxyz is what the general cubic 
(a, b, c, f, g, h, i, j, k, l) (x, y, z) s 
becomes by no other change than the reduction to zero of certain of its coefficients; 
and thus any concomitant of the canonical form consists of terms which are leading 
terms of the same concomitant of the general form. 
The concomitants are functions of the coefficients (a, b, ..., 1), of (f, y, £), and of 
(x, y, z): the dimensions in regard to the three sets respectively may be distinguished 
as the degree, class, and order; and we have thus to consider the deg-class-order of 
a concomitant. 
Two or more concomitants of the same deg-class-order may be linearly combined 
together: viz., the linear combination is the sum of the concomitants each multiplied 
by a mere number. The question thus arises as to the selection of a representative 
concomitant. As already mentioned, I follow Gundelfinger, viz., my 34 concomitants 
of the canonical form correspond each to each (with only the difference of a 
numerical factor of the entire concomitant) to his 34 concomitants of the general 
form. The principle underlying the selection would, in regard to the general form, 
have to be explained altogether differently; but this principle exhibits itself in a 
very remarkable manner in regard to the canonical form ax 3 + by 3 + cz 3 + Qlxyz.