30 PROCEEDINGS OF THE AMERICAN ACADEMY. 



The proof of Theorem lis the same as for the Quartic Scroll S (I3, 1, 4), 



_ a 

 p. 2G. Siuce two generators lie in a plane, « < k* 



2. Fla7ie Curves. — As before, each plane curve is met once by an 

 arbitrary generator, i. e. a = 1 for any plane curve. There is no Ij on 

 this scroll. Each conic is a 2i, the triple director 7' is a 3i, and the other 

 plane curves, 3i and 4i, are the same as for the Quartic Scroll S (1 3, 1, 4). 

 Every plane curve is either the complete intersection of the scroll by its 

 plane or else the residual is composed of generators, and therefore, by 

 Theorems II and III, formula (1) holds for every plane curve. 



Two conies do not intersect ; even if they pass thi-ough the same point 

 they lie on different sheets, and cannot be regarded as intersecting on the 

 scroll ; the formula gives (2i, 2i) = 2 + 2 — 4 = 0; the line of inter- 

 section of the planes of the two conies meets the scroll in the four points 

 where the two generators in the plane of either conic meet the other 

 conic. In the plane of a conic each of the two generators that lie in that 

 plane meets the conic on T and at one other point where the plane is 

 tangent to the scroll ; therefore the plane of every conic is a double 

 tangent plane to the scroll. T meets each conic once, (S^, 2i) = 3 + 2 

 — 4=1. A conic meets a plane cubic once, (2i, 3i) = 1, and meets a 



plane quartic twice, (2i, 4i) = 2 + 4 — 4 = 2. 



a 



3. Twisted Curves. — Since a < -, we have for the twisted cubic 



a = 1 and t = 2, where t is the number of points of intersection of the 

 curve and an arbitrary plane through T that lie on T. By the same 

 reasoning as that employed on page 28, we see that T can be made to lie 

 on the quadric that cuts out the twisted cubic, and that formula (1) holds 

 for every twisted cubic. 



When a is odd we take v = • Then, since two generators lie in 



a plane and a is an integer, a ^ — - — and t > — ;r — ; but by formula 



a A- \ 

 (2) we have — - — = v points at our disposal in the determination of 



S^") and therefore we can make T lie on S'^^) ; the residual will then con- 

 sist of T and a curve of order r — 3 = « + 2 — 3 = a— 1. 



iTTL • , ft — a ., _ a ^. a a 



When a is even we take v = -; a :p - and t > -. If a < -, t > -, 



2 ^2 2 22 



i. e. T > v, and it follows from what was said on page 24 that T can be 



made to lie on Si") ; the residual will then consist of T and a curve of 



order a — 3. If a = - = v, every generator meets S^"^ in v points, which 



