342 
[770 
770. 
ON THE 34 CONCOMITANTS OF THE TERNARY CUBIC. 
[From the American Journal of Mathematics, vol. IV. (1881), pp. 1—15.] 
I have, (by aid of Gundelfinger’s formulæ, afterwards referred to), calculated, and 
I give in the present paper, the expressions of the 34 concomitants of the canonical 
ternary cubic act? + by 3 + cz 3 + Qlxyz, or, what is the same thing, the 34 covariants of 
this cubic and the adjoint linear function Çx + yy+Çz: this is the chief object of 
the paper. I prefix a list of memoirs, with short remarks upon some of them ; 
and, after a few observations, proceed to the expressions for the 34 concomitants ; 
and, in conclusion, exhibit the process of calculation of these concomitants other 
than such of them as are taken to be known forms. I insert a supplemental table 
of 6 derived forms. 
The list of memoirs (not by any means a complete one) is as follows : 
Hesse, Ueber die Elimination der Variabein aus drei algebraischen Gleichungen 
vom zweiten Grade mit zwei Variabein : Crelle, t. xxvm. (1844), pp. 68—96. Although 
purporting to relate to a different subject, this is in fact the earliest, and a very 
important, memoir in regard to the general ternary cubic ; and in it is established 
the canonical form, as Hesse writes it, y 3 + y 3 + y 3 4- Çnry{y 2 y z . 
Aronhold, Zur Theorie der homogenen Functionen dritten Grades von drei 
Variabein: Grelle, t. xxxix. (1850), pp. 140—159. 
Cayley, A Third Memoir on Quantics: Phil. Trans., t. cxlvi. (1856), pp. 627—647; 
[144]. 
Aronhold, Theorie der homogenen Functionen dritten Grades von drei Variabein : 
Grelle, t. lv. (1858), pp. 97—191. 
Salmon, Lessons Introductory to the Modern Higher Algebra : 8°, Dublin, 1859. 
Cayley, A Seventh Memoir on Quantics: Phil. Trans., t. cli. (1861), pp. 277—292; 
[269]. 
Brioschi, Sur la théorie des formes cubiques à trois indéterminées : Comptes 
Rendus, t. LVi. (1863), pp. 304—307.