310 



This expression is shown to give known particular integrals, such 

 as (I 2/-cos0+r 2 )"*, and 



r n (sin 0)~ n (sin ^ sin 6 J ( tan ^ j cos t*0. 



It appears probable, therefore, that the generalization of the result 

 obtained for the limited value of n is legitimate ; but the author does 

 not profess to demonstrate this conclusion, believing that the principle 

 of the "permanence of equivalent forms " is not at present established 

 in such a sense as to amount to a demonstration. 



VII. " A Memoir on Curves of the Third Order." By ARTHUR 

 CAYLEY, Esq., F.R.S. Received Oct. 30, 1856. 



(Abstract.) 



A curve of the third order, or cubic curve, is the locus represented 

 by an equation such as ~U=(* ~J[x, y, ^) 3 =0 ; and it appears by my 

 " Third Memoir on Quantics," that it is proper to consider, in con- 

 nexion 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, 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 , T), of PU, 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 

 geometrical definition of the Pippian was readily found ; the curve 

 is in fact Steiner's curve R mentioned in the memoir " Allgemeine 

 Eigenschaften der algebraischen Curven," Crelle, t. xlvii. 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 " Memoire sur les Courbes du Troisieme Ordre," Liouville, 

 t. ix. p. 285, and " Nouvelles Remarques sur les Courbes du 

 Troisieme Ordre," Liouv. t. x. p. 102. As regards the Quippian, I 



