WILLIAMS. — GEOMETRY ON RULED QUARTIC SURFACES. 43 



just considered, that can be cut out by such an aS^") that the residual will 



consist of D and a curve of order a — 2. It - is even and ^ — -, the 



residual is of order a ; then 8 must be even, being equal to a — 2 a, and, 



if S = _ _}- 2, i.e. if a ^ — - — , D lies on /S^"), and the residual consists 



of D and a curve of order « — 2. If a > — - — , then a > . . When 



4 4 



a > - the residual either breaks up into curves of orders less than a, or 



4 



a . 

 else it is a curve of order a, say an Op, where p = v — a < t , i-e. 



d 4 



p ^ , and consequently a^, is a curve like that just considered, that 



can be cut out by an -S^") such that the residual will consist of D and a 



curve of order a — 2. Finally, when a = - , we take v = - + 1 ; then 



4 2 



7- = o + 4, and, by formula (2), we have « + 2 points at our disposal 

 in the determination of -S'^''). If K does not lie on <S'(''), it meets <S(^) in 

 « + 2 points ; now K has a points of a^ on it, and if we make jS^(') pass 

 through three more points of K^ not on «„) 'S'^") will contain K\ this we 

 can always do and still have at least three more points at our disposal, 



since a + 2 > 6 for a > 4. Since a — - , 8 = - , and if we make S^"^ 



4 2 



pass through two more points of Z), not on aa. , S^"^ will cut out D. The 



residual will then consist of K, D, and a curve of order r — 4 — 2 = a — 2. 



The twisted curves on this scroll can therefore be divided into two 



groups, viz., group (1), those curves of order a, each of which can be cut 



out by such an S^"') that the residual will consist of curves of orders less 



than a, and group (2), those curves of order a, each of which can be cut 



out by such an aS^") that the residual will be a curve of order a and of 



group (1). Therefore, by the same reasoning as that employed for 



Quartic Scroll -S* (l^, 1^, 2), p. 36, Formula (1) holds for all curves on 



the scroll. 



VII. Quartic Sckoll, with a Double Twistk.d Cubic mf.t 



TWICE BY EACH GENERATOR AND WITH A SiMPLE LiNEAR DIREC- 

 TOR, S{3^^,1). (Cayley's Eighth Species, ^(1, 3-).) 



1. Let Q represent the twisted cubic, which is the double curve on 

 the scroll. Through each point of Q pass two generators ; the plane of 

 these two generators contains the linear director, since each generator 



