WILLIAMS. — GEOMETRY ON RULED QUARTIC SURFACES. 35 



make both D and U lie on £W if 8 + °—*— % ^— + 1, i.e. if 



4 2 



__ a + 5 = 3 a — 5 _ 3 a — 5 



3 tt - - or a < 5 . .therefore for a < — — the residual 



■^ 8 o 8 



can be made to consist of Z), ZX, and a curve of order r — 4 = a — 2. 



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



8 



■ — — — points at our disposal, we can make D lie on <S("), and the residual 



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

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

 holds for D, and if it holds for a p it will hold for a a (Theorem III), 

 so we need only consider the curve a p ; each generator meets SW in 

 a + 1 



v = 



2 



points, once on D, a times on a a , and p times on a p , so that 



a+1 , a+1 3a-5 a+ 1 



p == — 1 - a < -j- - 1 r -orp< -g-, 



3 a 5 



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



8 



ered above, that can be cut out by an »SW 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 

 2i Z u 



a . . — a . 



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

 ^2 * 



a- 2 



T 



and the residual consists of D, D' ', and a curve of order r — 4 = a — 4 ; 



if a > the residual i9 a curve of order a, say an a p , where 



4 



a a a — 2 " + 2,<7 + 2. . , 



pz=z a< _, or p < —j- ; but — — is an integer and p 



"■ +2 _ a — 2 



is an integer, and therefore p < — 1, i. e. p < -—r — ; tnererore, 



a p is a curve like that just considered, that can be cut out by an £W 



a 

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



is even we first take v — r, for which r — a; 8 is even, and if 8 > -- + 2, 



i. e . if = a ~ 4 both i) and D' lie on 5(") (or can be made to lie on 

 ^ 4 



ft — Zt 



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



a 



