WILLIAMS. — GEOMETRY ON RULED QUARTIC SURFACES. 45 



The double cubic Q, although not a plane curve, will be considered 

 here. We have seen that it is cut out by a quadric through two gener- 

 ators, and therefore formula (1) holds for it (Theorem III). It is met 

 twice by every generator, and is, therefore, a 6 2 . A plane quartic has a 

 branch on each sheet at each of the three points where it meets Q, and 

 the number of its intersections with Q is 6, (6 2 , 4 X ) = 6 + 8 — 8 = 6. 

 A plane cubic meets Q four times, twice at the double point of the plane 

 cubic and once at each of the other two points where the plane meets 

 Q, (6 2 , 3j) = 6 + 6 — 8 = 4. The linear director does not meet Q, 

 (02,10=6 + 2-8 = 0. 



__ a 



4. Twisted Cubic, 3 V — Since a < -, every twisted cubic is a 3^ We 



o 



take v = 3 ; then r = 9, and we have 19 — 10 = 9 points at our disposal 

 in the determination of S {:i) that cuts out the twisted cubic. The number 

 of points of the twisted cubic on Q is 2 (a — a) = 4, and if we take 6 

 more points of S {?j) on Q, S {3) will contain Q ; this leaves 9 — 6 = 3 points 

 at our disposal, and, since each generator now meets S {S) twice on Q and 

 once on the twisted cubic, we can take one more point of *S (3) on each of 

 three generators and S {:i) will then contain those three generators ; the 

 residual will then consist of Q and three generators, and therefore 

 formula (1) holds for every twisted cubic (Theorem III). Since three 

 generators lie in a plane through the linear director, a twisted cubic does 

 not meet the linear director, (3 l5 l x ) = 3 + 1 — 4 = 0. 



5. Twisted Quartic 4 X . — Since a < -, every twisted quartic is a 4 X . 



o 



We take v — 3 ; then r = 8, and the number of points at our disposal in 



the determination of S l3) is 19 — 13 = 6. There are 2 (a — a) = 6 points 



of the twisted quartic on Q, and if we take four more points of S l3) on Q, 



S {3) will contain Q. Each generator will then meet S i3) twice on Q, and 



once on the twisted quartic, and, since we still have two points at our 



disposal, we can make S [3) cut out two generators. The residual will 



then consist of Q and two generators, and therefore formula (1) holds 



for every twisted quartic. 



a + 3 



6. Twisted Curves in General. — When a is odd we take v = — ^ — » 



3a + 9 ,. 



then r = a + 6, and by formula (2) we have <r points at our dis- 

 posal in the determination of £("). If 8 be the number of points of a a on 

 Q,8 = 2 (a — a) ^-^, since a ^ |. That £<"> may contain Q, it must 



