34 PROCEEDINGS OP THE AMERICAN ACADEMY. 



points at our disposal in determining the qaadric. Every generator then 

 :- the quadric once on the twisted cubic and once on J), and cannot 

 meet it again without lying on it; therefore, the residual consists of D 

 and three generators. Since formula (1 ) holds for D&nd the generators, 

 it holds for every twisted cubic on the scroll (Theorem III). 



5. Twisted Quartic 4j. — Since a< . every twisted quartic is a 



1, ; 8 = a — 2 a = 2; i. e. a plane through h or 1/ meets tlie twisted 

 quartic twice on that line. If tlie curve is a "quartic of the lirst kind "' 

 it may have a double point on D, D' , or G. but in any case it has two 

 distinct or consecutive points on one of the double directors, Bay />. and 

 -hire we can pass a quadric through a M quartic of the first kind " and 

 any arbitrary point, we may take this arbitrary point on D, and D will 

 then lie on the quadric ; if the quartic has no double point on G tlie 

 residual will then consist of D and G, and if the quartic- has a double 

 point on G the residual will consist of D and two generators, since 

 each generator will then meet the quadric once on 1) and once on 

 the quartic. If the curve is a "quartic of the second kind," it is more 

 convenient to cut it out by a cubic surface, i. e. we take v = 3 ; the re- 

 sidual then is of order 8, and by formula (2) we have six points at our 

 disposal in the determination ot .s' 11 ; since this quartic has do double 

 point, it meets both D and D' in two distinct or consecutive point-, and 

 if we put two more points of «S on each of these double directors they 

 will both lie on S 3 ' ; then G will meet S once on D, once on //, and 

 twice on the quartic, and will, therefore lie on >'•"' ; each generator meets 

 8 once on D, once on />', and once on the quartic, and, since we -till 

 have two points at our disposal, we can make two generators lie on S ; 

 the residual will then consist of D,D\ G, and two generators. Since 

 formula (1) holds for />. I >' <>. and the generators, it hold- for every 

 twisted quartic on the Bcroll (Theorem [II). 



6. Twitted Curvet in General — When a is odd we take v = — - — , 



a 

 and the oiil.r of the residual is r = a + 2. We have -ecu that a < - 



or l .- <i . .and that there- i> the same number "1 point-. -i\ 8 



points, on each double director, where 8 =a— 2a, i.e. 1 < 8< " — -• 



It has also been Bhown, p. 25, that we need not consider whether ",, has 



a \- 1 

 multiple points on 1) and l> or not. By formula (2), we have 



point- at our disposal, in ih< del ( i initiation of > ■ . and therefore we can 



