WILLIAMS. — GEOMETRY ON RULED QUARTIC SURFACES. 47 



in not having a simple linear director, in consequence of which a plane 

 through two generators cuts out a proper conic. 



2. Proof of. Theorem I. — By passing a quadric surface through Q 

 and any chosen generator, Theorem I is proved in exactly the same way 

 as it is for the Quartic Scroll S (3. 2 2 , 1), p. 44., and if a be the number 

 of points of C (tt > on each generator, we see, as before, that the number of 



points of a a on Q is 2 (a — a) . Since two generators lie in a plane, 



_ a 



a <2\ 



3. Plane Curves. — Since there are no lines on the scroll except the 



generators, a plane through two generators cuts out a proper conic, and 

 since the section has three double points on Q, the conic passes through 

 each of the two points in which the generators meet Q, different from 

 their point of intersection. Each conic is a 2 l9 and regarding the conic 

 and generators as lying on the scroll, the conic is met once only by 

 each generator in its plane, since at the point where the conic and gen- 

 erator cross Q they lie on different sheets. The two points where 

 the two generators meet the conic, not on Q, are points of tangency 

 of the plane, and therefore through every point of Q there is a double 

 tangent plane to the scroll. By Theorem III, formula (1) holds for 

 every conic. There is no l x on the scroll, and the other plane curves, 

 the cubic, 3 l , and the quartic, 4 1} are the same as those of the same order 

 on the Quartic Scroll S (3 2 2 , 1) and therefore formula (1) holds for all 

 curves on the scroll. Since Q is a double twisted cubic met twice by 

 each generator, it is a 6», and since it is cut out by a quadric through 

 two generators, formula (1) holds for it (Theorem III). We have seen 

 that Q meets each conic twice, and the formula gives 



(G 2 , 2 X ) = 6 + 4-8 = 2. 



4. Twisted Curves. — It is proved in exactly the same way as for the 

 Quartic Scroll S (3 2 2 , 1) that formula (1) holds for every twisted curve 

 on the scroll ; therefore, it holds for every curve on the scroll. 



IX. Quartic Developable. 



1. This surface is formed by the tangents to a twisted cubic, E\ E is 

 then a double curve on the surface, and from one point of view, this sur- 

 face is the limiting case of the Quartic Scroll S(3 2 2 , 1) where the two 

 points in which any generator meets the twisted cubic have become con- 

 secutive ; the twisted cubic then becomes the " edge of regression " and 

 the three double points of the section by a plane become cusps. 



