10 PROCEEDINGS <>F THE AMERICAN ACADEMY. 



conic having a double point on K, i. e. two lines that meet in this point, 

 which arc the two generators in the plane. 



2. Proof of Theorem 1. — The double curve on this scroll is a degen- 

 erate t \vi>t«'l cubic, consisting of the double conic K and the double 

 director D. We can pass a quadric through nine arbitrary points, and if 

 we take five of these on K and two on A>, distinct from the point of 

 intersection of A' and A*. A" and D will both lie on the quadric and we 

 shall still have two points at our disposal; now every generator meet- K 

 and D. and if we take one more point of the quadric on any generator A, 

 it will lie on the quadric ; the quadric will then intersect the scroll in 

 A' counted twice, D counted twice, and the generator A, and will, there- 

 fore, cut out one more generator. Making the quadric always contain A', 

 I), and A, we have one point at our disposal, and by varying this point 

 continuously we make the quadric cut out, in succession, all the gener- 

 ators of the scroll. C (a) meets A". D, and .1 in a definite number of points, 

 and as it meets every quadric in 2 a points it meets each generator in the 

 same number of points, say a points. It is evident that A also meets 

 C"" in a points, for any other generator may be chosen as the one through 

 which the quadric is always to pass, and then A will meet C" in the same 

 number of points as the other generators, i. e. in a points. A plane 

 through D cuts out two generators, and there are therefore a — 2 a points 

 of a a on D. A' is the complete intersection of its plane and the scroll, 

 and there are therefore a points of a a on K. The number of points of 



a a on D cannot be less than zero, and therefore a < =. 



3. Plane Curves. — The double director D meets every generator 

 once and is therefore a 2,. The double conic K is met once h\ each 

 generator and is therefore a \ x . The section by a plane not through h 

 or K has three double points on the double curve, one on D and two 

 on A'. We have seen (p. 25), that the section cannot consist of two 

 proper conies, and we know that a plane through two generators thai 

 meet On A", cuts out A>, for each generator meets D\ therefore, if a 

 plane cuts out a simple conic, it cuts out also two generators that meet 

 in a point on l>. for there arc no lines on the scroll but the generators, 

 and D and two generators meet only on A' or /■>; conversely, through 

 every point of l> pass two generators and their plain- cuts out a proper 

 conic; consequently, there is a system of conies that do not meet /), but 



meet A' twice, for clearly the plane cannot meet / ; again, and the .section 



cannot have a triple point on £> unless the plane contain- />-. each of 

 the two generators in the plane of an) come meets the conic twice, once 



