Geometry on the Cubic Scroll of the First Kind. 33 



Hence in every case where no further multiple points 

 occur on the curve, we have 



r==p(m' — f — kO-f q(m' — g' — g^ — h' — h^ + k'-f-lj. 



Consider the number of points in which S meets any 

 generator not multiple on it, which is m'. Of these ra' points 

 the f'-tuple line x , y passes through f and the k'-tuple line 

 z , s passes through k', hence m' — f — k' is the number of 

 points on any generator belonging to the curve (p,q), i.e., 

 m' — f — k' = q. If k' = 0, the line z , s is met by S in m' 

 points; of this number g' -|- g^ + h' + h^ are cut out by the 

 pinch point generators of the residual intersection; the 

 remaining m' — g' — g;^ — li' — h^ belong to the curve and 

 hence m' — g' — g^ — h' — h^ = p — q when k' = ; but we 

 have seen that S may be so taken for every curve that we 

 shall have k' = 0; therefore in every case 



i" = pq + q(p — q + i); 



and in general if it be further imposed that S and 2 have 

 ordinary contact at H points and stationary contact at 

 p points, — 



r = q(2p — q-f 1) — 2H —3(3 . 



(This demonstration is in no way invalidated if g-' -^ g^' , 

 h' -j- hp , 1' -j- ^0 ' ^" "^ ^o'l occur, making many genera- 

 tors multiple instead of simply the two pinch point genera- 

 tors. But we have seen that every (p,q) can have its residue 

 of generators entirely made up of the two given, and, since 

 it is not essential to this proof that m' be a minimum, the 

 demonstration seems preferable as given.) 



PI is not taken to include the double points which may 

 occur on the double director from the intersection 'there of 



3 — Archiv for Math, og Xaturv. B. XXI. Nr. 3. 

 Trvkt deu lôûe Juni 1809. 



