WILLIAMS. — GEOMETRY ON RULED QUARTIC SURFACES. 33 



double director. Every plane quartic is the complete intersection of its 

 plane and the scroll, and therefore formula (1) holds for it (Theorem II). 

 A plane cubic is cut out by a plane through a single generator, and D 

 and D' are cut out by planes through two generators, and therefore, by 

 Theorem III, formula (1) holds for every plane cubic and for D and D'. 

 We can cut out G by a plane through D, and, since formula (1) holds for 

 D, by corollary to Theorem III, it holds for G. Each conic is cutout by 

 a plane through G, and, since formula (1) holds for G, it holds for each 

 conic. Therefore formula (1) holds for every plane curve on the scroll. 



Either double director and G lie on both sheets of the scroll through 

 them, respectively, and G, therefore, intersects either double director 

 twice, once on each sheet, (2 , 2 X ) = 2. A plane quartic has a branch 

 on each sheet, at each of the three points where it meets D, D', and G, 

 and it therefore intersects each of these lines twice, once on each sheet, 

 as the formula shows, (4 X , 2 X ) = 2 and (4 l5 2 ) = 2. A plane cubic 

 meets G twice, since it has a branch on each sheet where it crosses G, 

 (3 1? 2 ) = 2, but it meets each double director once, since it has a 

 branch on one sheet only, where it crosses either double director, 

 (3j, 2 X ) = 8 + 2 — 4 = 1. The plane of a conic passes through G and 

 is tangent to the scroll at two points along G, one on each sheet ; these 

 points of tangency are the two points of intersection of the conic and G, 

 (2 r 2 ) = 2, and the conic has a branch on each sheet; one point of 

 tangency lies on the finite segment of G, between D and //, and the 

 other lies on the infinite segment, so that, as we turn the plane about G 

 in one direction, these two points both move toward the intersection of 

 G and D and coincide at this intersection, forming a pinch point, when 

 the conic becomes D, i. e. a line on each sheet; if we turn the plane in 

 the other direction, or continue to turn it in the same direction after it 

 cuts out D, the two points of tangency will both move toward I)' and 

 will coincide at the pinch point, the intersection of G and I/, when the 

 conic becomes D '. 



It is easy to see, by the aid of formula (1), how the other plane curves 

 intersect. 



4. Twisted Cubic, S v — We have seen that a < - for all twisted 



curves, and, consequently, every twisted cubic is a 3 X . Also, if 8 be the 

 number of points of the curve on D or D ', 8 = a — 2 a = 1 for the 



. / a + 1 \ , 



cubic. The twisted cubic is cut out by a quadnc [v = — ^ — 1, and we 



can make the quadric contain D, since, by formula (2), we have two 



VOL. XXXVI. — o 



