28 PROCEEDINGS OP THE AMERICAN ACADEMY. 



S'^\ that cuts out the cubic. Since a < - we have a = 1 , and every 



twisted cubic is a 3i ; therefore t = 3 — 1 = 2, i. e. the twisted cubic 

 meets T in two points, which must be distinct, since a twisted cubic 

 cannot have a double poiut. Therefore the quadric that cuts out the 

 twisted cubic meets T in two points on this curve, and since we can 

 make the quadric pass through any two points we please that are not on 

 the curve, we can make it pass through another poiut of T, and it will 

 theu contain T. The residual intersection, which is of order 5, then 

 consists of T, which counts for three lines, and two generators, since 

 there are no conies on the scroll, and, moreover, each generator meets 

 the quadric once on 5" and once on the twisted cubic, and cannot meet it 

 again without lying on it ; and if a conic or the simple director formed 

 part of the residual, an infinite number of generators would lie on the 

 quadric, which is impossible. Since formula (1) holds for 2^ aud the 

 generators, by Theorem III it holds for every twisted cubic. 



A plane through the simple director cuts out three generators and 

 meets the twisted cubic three times, once on each generator, and there- 

 fore the twisted cubic does not meet the simple director. 



5. Twisted Quartic, 4j. — We have a = 4, v = - = 2, and r r= 4, 



where r is the order of the residual ; a = 1, and every twisted quartic is 

 a 4j. Hence t = 3, and if the quartic has no double point on T there 

 must be three distinct points of the curve on T, i. e. T meets the quadric 

 that cuts out the quartic three times, and therefore lies on it. If the 

 quartic has a double point on 7", it is a " quartic of the first kind," and 

 we can pass a quadric through it and through any Arbitrary point not on 

 the curve ; the (piadric already meets T twice on the quartic curve, and 

 if we make it pass through another point of T, T will lie entirely on it. 

 In any case, the twisted quartic can be cut out by a quadric such that 

 the residual will consist of T and one generator, and, since formula (1) 

 holds for T and all generators, it holds for every twisted quartic ; if T 

 lies on the quadric the simple director cannot form part of the residual, 

 since each generator already meets the quadric once on Tand once on 

 the twisted quartic. The twisted quartic has a point on each generator, 

 and therefore meets the simple director once, (4i, l^) ^^ 1. 



G. Twisted Curves, in general. — When a is odd we take v = — - — , 



a 

 whence r = a -(- 2 ; and when a is even we take v ■= -, whence r = a. 



_ a _ 2 a 



We saw that « < q and t > — where r is the number of points on T, 



