

1863.] 447 



I. "On Skew Surfaces, otherwise Scrolls." By A. CAYLEY, 



F.R.S. Received February 3, 1863. 



(Abstract.) 



It may be convenient to mention at the outset that in the paper 

 "On the Theory of Skew Surfaces," Camb. and Dubl. Math. Journ. 

 vol. vi. pp. 171-173 (1852), I pointed out that upon any skew surface 

 of the order n there is a singular (or nodal) curve meeting each 

 generating line in (n 2) points, and that the class of the circum- 

 scribed cone, or what is the same thing, the class of the surface, is 

 equal to the order n of the surface. In the paper " On a Class of 

 Ruled Surfaces," Camb. and Dubl. Math. Journ. vol. viii. pp. 45, 46 

 (1853), Dr. Salmon considered the surface generated by a line which 

 meets three curves of the orders m, n, p respectively : such surface 

 is there shown to be of the order 2mnp', and it is noticed that 

 there are upon it a certain number of double right lines (nodal 

 generators) ; to determine the number of these, it is necessary to 

 consider the skew surface generated by a line meeting a given right 

 line and a given curve of the order m twice ; and the order of such 

 surface is found to be =%m(m l) + h, where h is the number of 

 apparent double points of the curve. The theory is somewhat 

 further developed in Dr. Salmon's memoir "On the Degree of a 

 Surface reciprocal to a given one," Trans. R. Irish Acad. vol. xxiii. 

 pp. 461-488 (read 1855), where certain minor limits are given for 

 the orders of the nodal curves on the skew surface generated by a 

 line meeting a given right line and two curves of the orders m 

 and n respectively, and on that generated by a line meeting a given 

 right line and a curve of the order m twice. And in the same 

 memoir the author considers the skew surface generated by a line, 

 the equations whereof are (a, . ^fat, l) m =Q (a 1 , . .^fct, l) n =0, where 

 a, . . a 1 , . . are any linear functions of the coordinates, and t is an 

 arbitrary parameter. And the same theories are reproduced in the 

 " Treatise on the Analytic Geometry of three Dimensions," Dubl. 

 1862. I will also, though it is less closely connected with the sub- 

 ject of the present memoir, refer to a paper by M. Chasles, " De- 

 scription des Courbes a double Courbure de tous les ordres sur les 

 surfaces reglees du troisieme et du quatrieme ordre," Comptes Ren- 

 dus, t. liii. (1861, 2 e Sem.), pp. 884-889. 



2K2 



