42 PROCEEDINGS OF THE AMERICAN ACADEMY. 



are four points of the twisted quartic 4! on K, and if we take three more 

 point- of S on A". N :i will contain A"; D meets S {S twice on the curve 

 Lx, and if we take two more points of S on 1). »§ (8) will contain D ; 

 every generator will then meet S lS) once on D, once on A\ and once on 

 the curve 4^ and as we still have one point at our disposal we can make 

 5 contain a generator ; this generator, D, and K count for 7 in the 

 order of the residual, and therefore S r " cuts out one more generator. 

 The residual then consists of K, D, and two generators, and therefore 

 formula (1) holds for the twisted quartic 4, (Theorem III). Formula 

 (1) holds, therefore, for every twisted quartic. 



G. Twisted Curves in General. — When a is odd we take v = - - ; 



then r = a + 2, and, by formula (2), we have — - — points at our dis- 

 posal in the determination of £H V \ If K does not lie on S<?\ it meets 

 aS*") in 2 f — - — J = a + 1 points ; but we have seen that there are a 



points of a a on A', and consequently, if we take two more points of <SW 

 on K, »SM will contain A"; this we can always do, since the number of 



. a + 1 



points at our disposal is — - — > 2 for a > o". The residual will then 



consist of K and a curve of order r — 4 = a — 2. 



When a is even, we distinguish two kinds of curves according as is 



° ° 2 



odd or even. If is odd we take v = ; then r = a, and if 8 be the 

 number of points of a a on D, we have seen that 8 = a — 2u: now, it 

 8 > " + 1, i-e. if a < • -^— , D meets #<") in at least - + 1 = v + 1 



points," and therefore lies on S('\ so that the residual consists of 1> and 



a — 2 



a curve of order r — 2 = a — 2; if a > - . the residual either 



4 



breaks up into curves of orders less than a, or else it is a curve of order 

 i\ an a p , where p = - — a, since each generator meets $'') in v = - 



£ — 



a - i' 



points, of which a lie on ",, and p on ",, : then, since a > , 



a h 2 , . . . a + 2 . 



p < : but p is an integer and — — IS an integer, so that 



= a + 2 , . = "--' 



p < — 1, I.e. p. . and (i t , is therefore one ol the curves 



• mi -• can be made to pass through » H I points of D (Se< p 



