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XX. A Memoir on Curves of the Third Order. By A. Cayley, Esy., F.R.S. 
Eeceived October 30, — Bead December 11, 1856. 
A CUKVE of the third order, or cubic curve, is the locus represented by an equation such 
as U=(-X-X^; Hi zy=0 ; and it appears by my “Third Memoir on Quantics,” that it is 
proper to consider, in connexion with the curve of the third order U=0, and its 
Hessian HU=0 (which is also a curve of the third order), two curves of the third class, 
\iz. the cui’ves represented by the equations PU = 0 and QU=0. These equations, I 
say, represent curves of the thu’d class ; in fact, PU and QU are contravariants of U, 
and therefore, when the variables s, y, z of U are considered as point coordinates, the 
variables fj, ^ of PU and QU must be considered as line coordinates, and the curves will 
be curves of the third class. I propose (in analogy with the form of the word Hessian) 
to call the two curves in question the Pippian and Quippian respectively. A geome- 
trical definition of the Pippian was readily found ; the curve is in fact Steinee’s curve 
Ko mentioned in the memoir “Allgemeine Eigenschaften der algebraischen Curven,” 
Crelle, t. xl\ii. pp. 1-6, in the particular case of a basis-curve of the third order ; and 
I also found that the Pippian might be considered as occurring implicitly in my “ Me- 
moire sur les Courbes du Troisieme Ordre,” Liouville, t. ix. p. 285, and “ Nouvelles Pe- 
marques sur les Com’bes du Troisieme Ordre,” lAouville, t. x. p. 102. As regards the 
Quippian, I have not succeeded in obtaining a satisfactory geometrical definition ; but 
the search after it led to a variety of theorems, relating chiefly to the first-mentioned 
curve, and the results of the investigation are contained in the present memoir. Some 
of these results are due to Mr. Salmon, with whom I was in correspondence on the sub- 
ject. The character of the results makes it difficult to develope them in a systematic 
order ; but the results are given in such connexion one with another as I have been able 
to present them in. Considering the object of the memoir to be the establishment of a 
distinct geometrical theory of the Pippian, the leading results will be found summed up 
in the nine different definitions or modes of generation of the Pippian, given in the con- 
cluding number. In the course of the memoir I give some further developments relating 
to the theory in the memoirs in Liouville above referred to, showing its relation to the 
Pippian, and the analogy with theorems of Hesse in relation to the Hessian. 
Article No. 1. — Definitions., &c. 
1. It may be convenient to premise as follows: — Considering, in connexion with a 
curve of the third order or cubic, a pointy we have — 
{a) Th .0 first or conic polar of the point. 
{h) The second or line polar of the point. 
The meaning of these terms is well known, and they require no explanation. 
