40 PROCEEDINGS OF THE AMERICAN ACADEMY. 



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

 which are the two generators in the plaue. 



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

 erate twisted 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 D, distinct from the point of 

 intersection of ^and D, A' and D will both lie on the quadric and we 

 shall still have two points at our disposal ; now every generator meets 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 

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

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

 D^ 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'"' meets A', D, and ^ 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 6'*"' 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 — 2a points 

 of cia on Z). K is the complete intersection of its plane and the scroll, 

 and there are therefore a points of aa on A", The number of points of 



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 2i. The double conic K is met once by each 

 generator and is therefore a 4i. The section by a plane not through D 

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

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

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

 meet on A', cuts out Z), 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 D, for there are no lines on the scroll but the generators, 

 and D and two generators meet only on K or D; conversely, through 

 every point of D pass two generators and their plane cuts out a proper 

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

 meet A' twice, for clearly the plane cannot meet D again, and the section 

 cannot have a triple point on D unless the plane contains Z> ; each of 

 the two generators in tlie plane of any conic meets the conic twice, once 



