46 PROCEEDINGS OP THE AMERICAN ACADEMY. 



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



3 a + 11 . _ a + 33 „ , . ,. - ^ rf/ \ Ml 

 ~ ■ — o < — of tlie points at our disposal on Q, oa") will coii- 



tain ^; if X be the number of points left at our disposal, after making 



• ^ X = 3 a + 9 a + 33 . ^ = 4 a ^ 



ol"' contain Q, A ^ — -; — , i. e. A > —- 1. J\ow each 



2 b o 



generator meets aS'') twice on Q and a times on «„, i. e. at least three 



a + 3 

 times since a > 1 ; if, then, we make aS^") pass through • — h 1 — 3 



a — 1 . 



= — - — other points on any generator, that generator will lie on S^^'h 



' . . 4fi 

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



> 2 I • — - — |, and the residual will then consist of Q, two generators, 



and a curve of order r — 6 — 2 = a — 2. 



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



Li 



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



_ 4a 

 /S"'"). The number of points of a^ on ^ is 8 = 2 (a — «) > -5-- In 



o 



3o + 8 



2 

 2)oints of Q, and therefore in addition to the 8 points we must make ^>'') 



,3a + 8 ^_a + 24 , . ,^ ,. 



pass through — — h :^ other pomts of Q ; this we can 



a + 24 

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



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

 r - 6 = a — 2. 



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

 out by an aS'") such that the residual is composed of curves of orders less 

 than a; we have also shown that formula (1) holds for every plane 

 curve and for every twisted curve of order 3 or 4 ; it holds, therefore, 

 for every twisted curve of order 5, and therefore for every twisted curve 

 of order G, and so on (Theorem III); therefore formula (1) holds for 

 every curve on the scroll. 



VIII. QuARTic Scroll, with a Double Twisted Cubic met 



TvriCE BY EACH GENERATOR, aS'(32^2). (CaYLEY's TeNTH 



Species, ^^(S^).) 



1. Let Q be the double twisted cubic. Through every point of Q 

 pass two generators. Tlie scroll differs from the Quartic Scroll ,S' {^.^, 1) 



order for /S'f'') to contain (?, it must pass through 3 v + 1 = 



