36 PROCEEDINGS OF THE AMERICAN ACADEMY. 



5 i, ami the residual then consists of JJ, JJ', ami a curve of order a — 4 ; 



a — A _ a , (i .... 



if a > - — , a > — , and wlii'ii a ;> - the residual is an " p , where 

 14 4 



a a a _ a — 4 , . . . , 



p = — a < ^ — - or p p — , and a D is therefore a curve that can lie 



2 2 4 4 



cut out hy an 5M such that the residual will consist of curves of orders 



a a 



less than '/ ; finally, if a = - we take v = - + 1 ; then r — a -f 1 and 



8 = ; if then we put two more points of S^ on D and two more on 



I)', both of these douhle directors will lie on S (l '\ and this we can always 

 do, since, hy formula (2), we have a -f 2 points at our disposal in the 



determination of S^ and a > 4 ; then C will meet <SW once on D, once 



o 



on JJ', and 2a = - times on a a , and will therefore lie on S( v ); each gen- 

 erator will meet <SW once on Z), once on Z)', and - times on a a , and, con- 



4 



sequently, if we put - more points of *SW on any generator it will lie on 



51") ; now we still have at our disposal a — 2 > 2 ( - J points, since 



a > 1. and therefore we can make two generators lie on <Sv"); therefore 

 the residual can he made to consist of U, JJ', G, 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 <SW such that the 

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

 thai may he cut out by an »$(") 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 '■'> or I ; it therefore 

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

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

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

 group (2), and bo on. Therefore formula (1) holds for even curve on 

 the scroll. 



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

 two-fold 2 | • 'l)-tnjilr linear director, and with a double generator 



S I . I . •_' i, ( ( 'at/ley's Fifth Species, S \ . I . I > >. for this Bcroll is Bimplj 

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

 director- baa moved up into coincidence with the other, A plane quartic 

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

 point on the double g< aerator. A plane cubic, iv^ml.-d as lying in the 



