WILLIAMS. — GEOMETRY ON RULED QUARTIC SURFACES. 29 



in which «„ is met by a plane through T. For a = 3, we saw that T 

 could be made to lie on /S'<^', the quadric that cuts out the twisted cubic. 



Now for « > 3, V = — - — < T when a is odd and v = - < t when a is 



even. It follows at once, therefore, from what was said on p. 24, that 

 we can always make J" lie on »S'(''). Therefore when a is odd the re- 

 sidual can be made to consist of Tand a curve of order /• — 3 = a — 1, 

 and when a is even the residual can be made to consist of T and a curve 

 of order r — 3 = a — 3. Therefore, by Theorem III, if formula (1) 

 holds for every curve of order less than a, it holds for every curve of 

 order a ; but we have proved that it holds for all plane curves and for 

 all twisted curves of orders 3 and 4 ; it therefore holds for every curve 

 of order 5 and therefore for every curve of order 6 and so on, and it 

 therefore holds for every curve on the scroll. 



7. The above proof is also applicable to the Quartic Scroll, with a 

 two-fold 3 (+1) -tuple linear director, S (I3, 1, 4). (Cayley''s Sixth Spe- 

 cies.) This scroll is, in fact, the limiting case of the scroll just consid- 

 ered, where the simple director has moved up into coincidence with the 

 triple director. Cayley denotes this symbolically by drawing a bar over 

 the two I's. 



The triple linear director on this scroll is torsal along one of the three 

 sheets through it, i. e. the tangent plane to this sheet, along this director, 

 is the same for every point of this director; the generator, lying on this 

 sheet, that is cut out by this tangent plane, coincides with the simple 

 linear director, and is regarded as intersecting the triple linear director 

 at the point where the other two generators cut out by this tangent 

 plane, one on each sheet, intersect. 



III. Quartic Scroll, with a Triple Linear Director, 

 S {\s,2,2). (Cayley's Ninth Species, S (l^).) 



1. The triple director, T, is scrolar on each of the three sheets that 

 pass through it, and this scroll differs from the Quartic Scroll ^(l,.,, 1, 4) 

 in not having a simple linear director, in consequence of which we have, 

 on this scroll, three generators through each point of T that do not lie in 

 the same plane. The plane of any two of these three generators meets 

 the scroll otherwise in a conic that passes through their point of inter- 

 section on 7\ making up the triple point of the complete intersection. 

 TlKircfore, there are three conies through each point of T, one in each 

 of the tliree planes that contain two generators through the point. 



