36 PROCEEDINGS OF THE AMERICAN ACADEMY. 



/S^"^), and the residual then consists of D, D' , and a curve of order a — 4 ; 

 \i a> ^ ~ , a > ^5 and when « > 2 ^^ residual is an Op, where 



n = n <^ orp^ , and a^ is therefore a curve that can be 



^ 2 2 4 ^ 4 '^ 



cut out by an /S^") such that the residual will consist of curves of orders 



less than a ; finally, if a = - we take v = - + 1 ; then r = a + 4 and 



8 = - ; if then we put two more points of aSW on D and two more on 



D' , both of these double directors will lie on *?'*'>, and this we can always 

 do, since, by formula (2), we have a + 2 points at our disposal in the 

 determination of -SI") and a > 4 ; then G will meet ^SW once on D, once 



on i/, and 2 a = - times on a^, and will therefore lie on S(v) ; each gen- 



a 

 erator will meet jS^") once on D, once on Z>', and - times on Oa, and, con- 

 sequently, if we put - more points of /S^") on any generator it will lie on 



4 / 



a 



S^"') ; now we still have at our disposal a — 2>2l-l points, since 



a. > 4, and therefore we can make two generators lie on *§('); therefore 

 the residual can be made to consist of D, D', C, two generators, and a 

 curve of order r — 8 = a — 4. 



Therefore, on this scroll, we may divide all twisted curves into two 

 groups, viz. group (1), those that may be cut out by an ^('') such that the 

 residual consists of curves of orders less than a, and group (2), those 

 that may be cut out by an S^''') such that the residual is a curve of group 

 (1), with or without D. Now we have seen that formula (1) holds for 

 all plane curves and for all twisted curves of order 3 or 4 ; it therefore 

 holds for all curves of order 5 of group (1) (Theorem III), and it there- 

 fore holds for all curves of order 5 of group (2) ; it then holds for all 

 curves of order 6 of group (1), and therefore for all curves of order 6 of 

 group (2), and so on. Therefore formula (1) holds for every curve on 

 the scroll. 



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

 two-fold 2 (+ 2)-tvple linear director, and with a double generator 



S (I2, I2, 2), {Cayley^s Fifth Species, -S" (U, Ig, 4)), for this scroll is simply 

 the limiting case of the scroll just considered, where one of the double 

 directors has moved up into coincidence with the other. A plane quartic 

 has a tac-node, where it meets the two-fold director and has' a double 

 point on the double generator. A plane cubic, regarded as lying in the 



