333 



XIX. " On Curves of the Third Order." By the Rev. GEORGE 

 SALMON, of Trinity College, Dublin. Communicated by 

 ARTHUR CAYLEY, Esq. Received May 20, 1858. 



(Abstract.) 



The author remarks that his paper was intended as supplementary 

 to Mr. Cayley's Memoir " On Curves of the Third Order" (Philoso- 

 phical Transactions, 1857, p. 415). He establishes in the place of 

 Mr. Cayley's equation, p. 442, a fundamental identical equation, 

 which is as follows, viz. if substituting in the cubic U, x+\x', 

 y+\y' t z + \z for x, y, e, the result is 



U + 3AS + 3\ 2 P+\ 3 U' ; 



so that S and P are the polar conic and polar line of (x, y', *'), with 

 respect to the cubic, viz. 



38-^+^+^; 3p-e; + ,e;+^, 



dx dy dz dx dy dz 



and if making the same substitution in the Hessian H, the result is 



so that I> and n are the polar conic and polar line of the Hessian 

 then the identical equation in question is 



3(Sn-SP) = H'U-HU'. 



And it follows that when (x' y y' y z') is a point on the cubic, the 

 equation U=0 of the cubic may be written in the form 



Sn-sp=o, 



an equation which is the basis of the subsequent investigations of the 

 paper. The author refers to a communication to him by Mr. Cayley, 

 of an investigation of the equation of the conic passing through five 

 consecutive points of the cubic, in the case where the equation of 

 the cubic is presented in the canonical form ff 3 +y 3 + * 3 + 6foy*=0, 

 and he shows that by the help of the above mentioned identity, the 

 investigation can be effected with equal facility when the equation 

 of the cubic is presented in the general form ; and he establishes 

 various geometrical theorems in relation to the conic in question. 

 Finally, the author considers an entirely new question in the theory 

 of cubics, viz. the determination of the points of a cubic, through 

 which it is possible to draw an infinity of cubics having a nine-point 



