44 PROCEEDINGS OF THE AMERICAN ACADEMY. 



meets the linear director once, and therefore this plane cuts out also a 

 third generator, i.e. any plane through the linear director meets Q in 

 three points, say L, M, and N, and cuts out the three generators L M, 



2. Proof of llieorem I. — We can make a quadric pass through Q 

 by making it pass through seven points of Q ; and since ^ is a double 

 cubic it counts for six in the order of the complete intersection of the 

 quadric and scroll ; if the eighth point for the determination of the quad- 

 ric be taken on any generator A, the quadric will contain A, since each 

 generator meets Q twice ; the remaining intersection will be any gener- 

 ator on which we choose to take the ninth point for the determination 

 of the quadric, and if we keep the first eight points fixed and vary the 

 ninth point continuously, the quadric will cut out in succession the differ- 

 ent generators of the scroll. A fixed number of points of C^") lie on Q 

 and the chosen generator A, and therefore every generator contains the 

 same number of points of C("), say a points. Any generator, other than 

 A, can be chosen, through which the quadric is always to pass, and 

 therefore there are a points of (7(') on A. Since the quadric meets «„ in 

 2 a points, there are 2a — 2a = 2(a — a) points of tta. on Q. Three gen- 

 erators lie in a plane through the linear director; therefore a <; -, and 



there are a — 3 a points of «„ on the linear director. 



3. Plane Curves. — Since the section by a plane cannot consist of two 

 proper conies (p. 25), a plane through a proper conic would either cut 

 out two generators or the simple director and one generator; but we 

 have seen that a plane through two generators or through the simple 

 linear director cuts out three generators and the linear director ; there- 

 fore there can be no proper conic on the scroll. 



A plane through one, and only one, generator cuts out a plane cubic, 

 a 3i, having a double point on Q and passing once through each of the 

 two points where the generator meets Q ; the generator meets the plane 

 cubic in one other point where the plane is tangent to the scroll. An 

 arbitrary plane cuts out a plane quartic, a 4i, having three double points 

 on Q, and since any plane quartic is the complete intersection of its 

 plane with the scroll, formula (1) holds for it (Theorem II). The 

 simple linear director, a l^, is cut out by a plane through three generators, 

 and every plane cubic is cut out by a plane through one generator, and 

 therefore formula (1) holds for the simple linear director and for every 

 plane cubic (Theorem III). Therefore formula (1) holds for all plane 

 curves on the scroll. 



