42 PEOCEEDINGS OF THE AMERICAN ACADEMY. 



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

 points of ^S*^'* on A' -S'^^ will contain K; D meets »S'*'^' twice on the curve 

 4i, and if we take two more points of ^'^' on D, S^^^ will contain D ; 

 every generator will then meet aS'''* once on D, once on A', and once on 

 the curve 4i, and as we still have one i)oint at our disposal we can make 

 aS'''** contain a generator ; this genei-ator, D, and K count for 7 in the 

 order of the residual, and therefore aS''^' 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. 



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



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

 posal in the determination of *S'(''). If K does not lie on *S'('''), it meets 

 ^(i-) in 2 ( — - — j = a + 1 points ; but we have seen that there are a 



points of «„ on A", and consequently, if we take two more points of »S'('') 

 on K, *S'('') will contain K; this we can always do, since the number of 



points at our disposal is — - — > 2 for a > 8. The residual will then 

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



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

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

 number of points of «a on D^ we have seen that 8 = a — 2 a : now, if 

 8 ^ ^- + 1, i.e. if a ^ , D meets »S<'') in at least - + 1 = v + 1 



points,* and therefore lies on >SM, so that the residual consists of D and 



« — 2 

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



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



a, say an «p, where p = - — a, since each generator meets jS'") in v = - 



a ~ 2 

 points, of which a lie on «„ and p on cfp ; then, suice a > — - — , 



« + 2, . . -,fl + 2. . ^ , ^ 



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



= «+2 . ^ a — 2 , ., „ p, 



p <; — -j 1, I.e. p <; , and Up is therefore one ot the curves 



* Or S^"^ can be made to pass through v + 1 points of D. (See p. 25.) 



