WILLIAMS. — GEOMETRY ON RULED QUARTIC SURFACES. 39 



every generator meets (?("> in the same number of points, say a points. 

 Two generators lie in a plane through D ; therefore, 8 — a — 2 a, and 



_ a 

 a <2' 



2. Plane Curves. — D and D' are both 2/s. There is no system of 

 conies on the scroll, for there is no double generator, and a plane through 

 two generators cuts out D or D'.* A plane through one and only one 

 generator cuts out a plane cubic, a 3 1? that meets each double director 

 once and has no double point, since the section has only three double 

 points which are the points where the generator in the plane meets the 

 cubic, one on each of the double directors, and one where the plane is 

 tangent to the scroll. An arbitrary plane cuts out a plane quartic, a 4 1? 

 having two and only two double points, one on each double director. 

 Every plane curve is, therefore, either the complete intersection of the 

 plane and scroll, or else the residual consists entirely of generators, and 

 consequently, by Theorems II and III, formula (1) holds for every plane 

 curve on the scroll. 



3. Twisted Curves. — It may be shown in exactly the same way as 

 for the Quartic Scroll, S (1 2 , 1 2 , 2), pp. 33-3G, that formula (1) holds 

 for every twisted curve on the scroll. It will be observed that, in the 

 proof referred to, the double generator is shown to be a part of the 

 residual ; now, there is no double generator on this scroll, but disregard- 

 ing the double generator the residual is still composed of curves of orders 

 less than a, and the conclusion follows as before, without change. 



4. The proof just employed is also applicable to the Quartic Scroll 

 ivit/i a two-fold 2 (+ 2) -tuple linear director and without a double gener- 

 ator, S (I.,, 1 2 , 4), (Cayleys Fourth Species), which is the limiting case 

 of the scroll S (1 2 1 2 4), just considered, where one of the double linear 

 directors has moved up into coincidence with the other. 



VI. Quartic Scroll, with a Double Conic and a Doup.ee 

 Linear Director meeting it, S (1 2 , 2 2 , 2). (Cayley's Seventh 

 Species, S (1, 2, 2).) 



1. For convenience, let D represent the double linear director and let 

 A' represent the double conic. Any plane through D meets K in one 

 more point, besides the intersection of K and D, and this is a double 

 point on the section by the plane. The plane therefore cuts out D and a 



* The section cannot consist of two proper conies. (See p. 25.) 



