146. 
A MEMOIR ON CURVES OF THE THIRD ORDER. 
[From the Philosophical Transactions of the Royal Society of London, vol. cxlvii. for the 
year 1857, pp. 415—446. Received October 30,—Read December 11, 1856.] 
A curve of the third order, or cubic curve, is the locus represented by an 
equation such as TJ = ( *fx, y, zf = 0 ; and it appears by my “ Third Memoir on 
Quantics,” [144], that it is proper to consider, in connexion with the curve of the third 
order TJ = 0, and its Hessian HU = 0 (which is also a curve of the third order), two 
curves of the third class, viz. the curves represented by the equations PU — 0 and QU = 0. 
These equations, I say, represent curves of the third class; in fact, PU and QU are 
contravariants of U, and therefore, when the variables x, y, z of U are considered as 
point coordinates, the variables £, y, £ of P U and Q U 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. [The curve PU = 0 is now usually called the Cayleyan.] 
A geometrical definition of the Pippian was readily found ; the curve is in fact Steiner’s 
curve R 0 mentioned in the memoir “Allgemeine Eigenschaften der algebraischen Curven,” 
Crelle, t. xlvii. [1854] 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 “Mémoire sur les courbes du troisième ordre,” Liouville, t. ix. [1844] pp. 
285—293 [26] and “Nouvelles remarques sur les courbes du troisième ordre,” Liouville, 
t. x. [1845] pp. 102—109 [27]. 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 investi 
gation are contained in the present memoir. Some of these results are due to Mr 
Salmon, with whom I was in correspondence on the subject. 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