34 PROCEEDINGS OF THE AMERICAN ACADEMY. 



points at our disposal in determining the quadric. Every generator then 

 meets the quadric once on the twisted cubic and once on D, and cannot 

 meet it again without lying on it ; therefore, the residual consists of D 

 and tliree generators. Since formula (1) holds for Z>and the generators, 

 it holds for every twisted cubic on the scroll (Theorem III). 



5. Twisted Quartic 4i. — Since a < ^r, every twisted quartic is a 



4i; S = a — 2tt = 2;i. e. a plane through D ov 1/ meets the twisted 

 quartic twice on that line. If the curve is a " quartic of the first kind " 

 it may have a double point on D, D', or G, but in any case it has two 

 distinct or consecutive points on one of the double directors, say D, and 

 since we can pass a quadric through a " quartic of the first kind " and 

 any arbitrary point, we may take this arbitrary point on D, and D will 

 then lie on the quadric ; if the quartic has no double point on G the 

 residual will then consist of D and G, and if the quartic has a double 

 point on G the residual will consist of D and two generators, since 

 each generator will then meet the quadric once on D and once on 

 the quartic. If the curve is a " quartic of the second kind," it is more 

 convenient to cut it out by a cubic surface, i. e. we take v = 3 ; the re- 

 sidual then is of order 8, and by formula (2) we have six points at our 

 disposal in the determination of ^S'^' ; since this quartic has no double 

 point, it meets both D and D' in two distinct or consecutive points, and 

 if we put two more points of <S''' on each of these double directors they 

 will both lie on &^^ ; then G will meet S-^^ once on D, once on D\ and 

 twice on the quartic, and will, therefore lie on ^S'^' ; each generator meets 

 S once on D, once on D' , and once on the quartic, and, since we still 

 have two points at our disposal, we can make two generators lie on *S'"' ; 

 the residual will then consist of Z), I>\ G, and two generators. Since 

 formula (1) holds for D, U, G, and the generators, it holds for every 



twisted quartic on the scroll (Theorem III). 



« + 1 



6. Twisted Curves in General. — When a is odd we take v = — - — , 



and the order of the residual is r = a + 2. We have seen that a < s 

 or 1 ^ tt < — - — , and that there is the same number of points, say 8 



points, on each double director, where 8 = a — 2 a, i. e. 1 < 8 < a — 2. 

 It has also been shown, p. 25, that we need not consider whether a^ has 



multiple points on D and D' or not. By formula (2), we have — - — 

 points at our disposal, in the determination of ^S^"), and therefore we can 



