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 62. 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, (62, 4i)=6 + 8 — 8 = G. 

 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, (62, 3i) = 6 + 6 — 8 = 4. The linear director does not meet Q, 

 (62,10 = 6 + 2-8 = 0. 



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4. Twisted Cubic, 3i. — Since a < „, every twisted cubic is a 3i. We 



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

 in the determination of /S'''^* that cuts out the twisted cubic. The number 

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

 more points of aS*^' on Q, /S'^' will contain Q ; this leaves 9 — 6 = 3 points 

 at our disposal, and, since each generator now meets ;S^'"' twice on Q and 

 once on the twisted cubic, we can take one more point of >S''^' on each of 

 three generators and *S'"' 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^, li) = 3 + 1 — 4 = 0. 



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



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

 the determination of S^"'' is 19 — 13 = 6. There are 2 ((t — a) = 6 points 

 of the twisted quartic on Q, and if we take four more points of /S''"^* on Q, 

 <S''^' will contain Q. Each generator will then meet ^'^' twice on Q, and 

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

 disposal, we can make »S''' 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 = 



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then r = rt + 6, and by formula (2) we have — ~ — points at our dis- 



posal in the determination of S'^"). If 8 be the number of points of a„ on 



^, 8 = 2 (a — «■) > -3-j since a < -. That aS^") may contain Q, it must 



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