16 PROCEEDINGS OF THE AMERICAN ACADEMY. 



Q 111 



pass through 3 v + 1 = points of Q, and therefore, if we take 



- 1 1 . _ a + 33 , , . „ , _ _, . ... 



— 8 < — of tlie points at our disposal on Q, oW will con- 



_ u 



(J; if A be the number of points left at our disposal, after making 



n * = 3 a + 9 « + 33 = 1 a 



aS<*'' contain Q, A > — — , i. e. A > — — 1. JSow each 



2 o 



venerator meets *S(') twice on Q and a times on '/.,. i. e. at Least three 



- « « + 3 

 times since a > 1 ; if, then, we make o(") pass through — h 1 — 3 



= — - — other points on any generator, that generator will lie on . s . 



2 1 a 



We can, therefore, make at least two generators lie on S( l '\ since — — 1 



> 2 ( — — ), and the residual will then consist of Q, two generators, 



and a curve of order r — G — 2 = a — 2. 



n 

 When a is even we take v = - 4- 1 ; then r = a -f 4, and, by form- 



ula (2), we have a + 2 points at our disposal in the determination of 



I a 



$''). The number of points of o tt on Q is 8 = 2 (a — a) > — . In 



o 



order for &<'') to contain (?, it must pass through 8v-J- 1 = 



22 



] k »ints of <?, and therefore in addition to the 8 points we must make 5 



. 3a + 8 . _a + 24 . „ _ .. 



pass tlirougb — o < other points ol \' : this we can 



2 a I 2 1 



always do, since the number of points at our disposal is a + 2 >- 



for a > 4. The residual will then consist of (? and a curve of order 

 r - 6 = a — 2. 



We have shown, then, that every twisted curve of order a can be cut 

 out 1>\ an SM such that the residual is composed of curves of orders less 

 than "; we have also shown that formula (1) holds for every pi 

 curve and for every twisted curve of order S or 1; it holds, therefore, 

 for every twisted curve of order •">, and therefore ti>r every twisted curve 

 of order <'.. and so on (Theorem III); therefore formula (l ) holds for 

 v curve on the scroll. 



YIIJ. Qi kRTic Scroll, with a Doubli Twisted Cubic kei 



Tumi BT BACB GENERATOR, S (8 .-'). (CaYLEY'8 TENTB 



Sri- n>. 5(8 |.) 



I. l.i t Q be the donble twisted cubic Through every point of Q 

 two generators. The Bcroll diffei from the Quartic Scroll S 



