WILLIAMS. — GEOMETRY ON RULED QUAKTIC SURFACES. 35 



make both D and iX lie on *S(-) if 8 + —^ ^ "^ h 1, i. e. if 



4 2 



^ _ a + 5 _ Za—6 _3a-5 



8 > or a < ^ . lliererore for a ^ the residual 



8 o 8 



can be made to consist of D, U , and a curve of order r — 4 = a — 2. 



When a > r , at least one point of «„ lies on D, and since we have 



8 



— points at our disposal, we can make D lie on ^(''), and the residual 



will then consist of D and a curve of order r — 2 = a, say Wp, where p 

 is the number of points of this curve on each generator; now formula (1) 

 holds for Z), and if it holds for a^ it will hold for «„ (Theorem III), 

 so we need only consider the curve Op ; each generator meets .SW in 



« + 1 . T\ ■ 1 • , 



V = — ;^ — points, once on x', a times on a^-, and p times on ap, so that 



2 



a+1 , a + l 3 (7 — 5 a+1 



P = -y--l-a<-^--l 8— °'^< 8 ' 



3 Q 5 



i. e. p < ^ for a > 3, and Op is therefore a curve like that consid- 



8 



ered above, that can be cut out by an S^") such that the residual consists 

 of curves of orders less than a. 



When a is even, it is convenient to separate the curves into two divi- 

 sions, according as - is odd or even. If -- is odd we take v = - : then 

 * 2 2 2 



r =: a, a < -, and 8 must be even since 8 = a — 2 a ; if 8 > — h 1, i- e. 



a — 2 

 if a < — - — , both D and D' lie on *§■(") (or can be made to lie on *5('')), 



and the residual consists of D, I/, and a curve of order ?- — 4 = a — 4 ; 



a — 2 

 if a > — - — the residual is a curve of order a, say an ap, where 



a a « — 2 a -f 2 , a + 2 . 



p = - — a<- — , or p < — — — ; but — - — is an integer and p 



2 2 4 4 4 



-«+2 ,. _a — 2, 



IS an integer, and thererore p :< — 1, i. e. p :< — - — ; therefore, 



Op is a curve like that just considered, that can be cut out by an <S'('') 



such that the residual will consist of curves of orders less than a. If - 



is even we first take v = , , for which r ^^ a\ 8 is even, and if 8 > ^ + 2, 



_ a — 4 

 i. e. if a < — - — , both D and U lie on S^"^ (or can be made to lie on 



