68 Proceedings of the Royal Irish Academy. 



The Ruled Ctibic luitli hvo Directrices. 



65. The equations of this surface can be written (cf. Sahnon, ch. 15, 

 § 520) in the form XijX = ajj/X^ = x^jd' = x^. 



The asymptotic lines are therefore given by X' = 2c9 ; or if we write 

 6 = 2cf, then X = 2ct, and the lines c are 



x^llct = xJ^c-V' = Xi\A.c-t' = xt. (33) 



The inflexions are t = 0, t = <x. These are pinch-points on the directrix 

 £Ui = Xi = 0. The other directrix is not a double line on the surface : there are 

 therefore no pinch-points on it. 



A Certain Quartic Scroll. 



66. The scroll S determined by 



Xi = Xi{fJ, - -If, Xs = Xi (fX + 2)'- 



is a quartic. It is generated by a line meeting the lines XiX, and XiOSi and a 

 conic (the section of the surface by any plane through X1X4 except «i = or 

 Xi = 0). The line p joining the points where the directrices meet the plane of 

 the conic is a double generator of the surface (Salmon, ch. 16, § 553). In the 

 present case 2^ (which is XiXi) touches the conic and is a cuspidal generator. 



Any point on >S' is X, X(/u. - 2)^^, (/x + 2)', 1, in terms of X and fA. 



The asymptotic lines are given (writing ¥ for c) by 



X= = F(m + 2)/(^ - 2). 

 Putting X = k{t + l)/{t - 1), we get fj, = t + t-\ 



Hence the asymptotic lines are 



iCi X2 x^ x^ 



mlTi) " k{t + i){t- ly " {t~i) {t + 1)* " t-{t - 1) ■ (34) 



These are rational quintics whose tangents belong to the complex 



lh% + k''pu = 0. 



The line p (xi = .1-4 = 0) is a bitangent at ^ = and t = <x>. The points 

 !^ = ± 1 are inflexions : t = 1 is the pinch-point x^XaXi on the directrix 

 XiXi : t = - 1 is the pinch-point XiXzx, on the other directrix. All the curves 

 have three-point contact at these two points. The 2, 2 relation expressing 

 that the line tO belongs to the complex is f9^ = 1, hence the four inflexions 

 are the roots of ;!' - 1 = 0, i.e. + 1, ± i. Two of these (+ 1) are the pinch- 

 points on the directrices. The other two lie on the generator ^ = 0, i.e. 

 3*2 = 4:X^, X3 = 4r4. Each pair (± 1 and ± i) satisfies the 2, 2 relation. 



