32 PROCEEDINGS OP THE AMERICAN ACADEMY. 



present«case the twisted cubic is degenerate, consisting of D, D , and G. 

 Let us pass a quadric through eight points, three on D, three on D' , one 

 on G, and one on any generator A^ the last two points not being on 

 D OT D' ; then D, D\ G, and A will all lie on the quadric and count for 

 7 lines in the intersection of the scroll and quadric, and therefore the 

 quadric cuts out one more generator; the quadric passes through eight 

 fixed points and we can make it pass through an arbitrary ninth point, so 

 if we vary this ninth point continuously the quadric will cut out, in suc- 

 cession, each generator of the scroll. Now C'"' meets the quadric in 2 a 

 points, of which a fixed number lie on D, I/, G, and -4, and therefore 

 the same number of points of C'"', say a points, lie on each generator. 

 It is evident that there must be a points on A^ for if any other generator 

 be chosen, through which the quadric is always to pass, then there is the 

 same number of points, a, on A, as on each of the other generators. 

 Since we can pass a plane through D and two generators, there are 

 a — 2a points of fla on D, and, similarly, there are a — 2 a points of 

 Qa on D' . A plane through D and G meets the scroll in these two lines 

 only, and there are, therefore, 2 a points of aa on G, as is otherwise evi- 

 dent from the fact that G counts for two generators. Since a twisted 



curve of order a cannot have a points on any line, 2a<aora<- for 



every twisted curve on the scroll. 



3. Plane Curves. — Each double director is met once by any gener- 

 ator and is therefore a 2^. No generator can meet G\ for, suppose a 

 generator A does meet it; then A meets either D or D' in a point differ- 

 ent from that in which G meets it, and therefore the plane through G 

 and A contains also D or ly, making the order of the complete intersec- 

 tion of the plane and scroll as great as 5, which is impossible. G is 

 therefore a 2o. Then any plane through G, that does not contain D or D', 

 meets the scroll in a proper conic that does not meet either double 

 director, since the section has only a double point on each double director ; 

 since each generator meets the plane once and does not meet G, each 

 conic is a 2i, and the conies and D and U are the only curves on the 



scroll for which a ^=^ -. A plane through one and only one generator 



cuts out a plane cubic, a 3i, having a double point on G and passing once 

 through the two points where the generator in the plane meets D and D' ; 

 the generator meets the cubic again where the plane is tangent to the 

 scroll. A plane that does not contain a line of the scroll cuts out a plane 

 quartic, a 4i, having three double points, one on G and one on each 



