422 



=n(ti-j-l)-{-(27i+l)p—(n' — \)q. 



From this it ensues- that the surface f/>»+/'+i contains at least ni 

 straight lines. 



If we take here q=^l, it appears that a surface <!>"+/>+' luith an 

 n-fold and a p-fold straight line contains 2np -\- ti -{- p -\- 1 straight 

 lines resting on the multiple straight lines ^). For ?z=l, the number 

 m appears to be independent of ^; we find that a surface <f»P-l-2 

 with />-fold straight line contains m=z^p-\-1 straight lines g, resting 

 on a and ^. A plane passing through ^ and one of those straight 

 lines contains one more straight line h, wich does not rest on 

 «; on 0/'+"^ lie therefore at least 6/>-f-4 straight lines. 



8. Let us now determine the image of the straight line ^. A plane 

 7 passing through ^ contains one more curve 7"+^ which has on a but 

 not on {i, an n-fold point Q. As it intersects /? in {n-\-l) points, the 

 plane contains {n-\-\) straight lines t, which touch the surface in 

 points of ^, consequently are generators of the ruled surface 5p by 

 which /? is represented; « is therefore an (yi-|-l)fold directrix. As 

 Q comes to lie on {i for {q — t) different positions of the plane 7, t 

 will as many times coincide with (t, consequently ^ is a {q — l)fold 

 torsal line. But each of the p planes that touch at 4> in a point of 

 ïJ, contains one straight line t; consequently the multiplicity of /3 

 on 55 is equal to {p -\- q — 1) and the order of 35 ^qnaAio {n-\-p-\-q). 



The image of the straight line ^ is therefore a curve è"+/'+9 (^^1"+^ 

 j8"+9-i, mG). 



This result may be controlled by determining the number of points 

 that b has in common with a curve c apart from the principal 

 points A,B,G; we promptly find then the number p, being the 

 number of common points of ^p and y"+/'+^ 



9. Let us now determine the image of a<i. Each of the n planes 

 that touch at in a point Q of «, contains 07ie straight t of the 

 ruled surface ^ that represents u, consequently « is an n-fold directrix. 



If ^? is a y-fold directrix a plane passing through ^ contains a 

 section of order {n -\- y), and the image of « has as symbol rt"+'/ 

 {qA"", B'\ mG). 



By combining with c"+/^+9+i {qA"-^\ Bp+'/, mG) we find for the 

 determination of y, the relation {7i-\-p-\-q-\-l) (n-{-y) = ?i{n-\-l)q -\- 

 -\-{p-rq)y-\-m-\-7iq, in which it has been taken into account that a 

 plane section has nq points in common with WJ. 



1) See MiNEo ibid, page 221 or J. de Vries, Surfaces algébriques renfermant 

 un nombre fini de droite$ ('Archives Teyler, serie II, tome VIII, p. 262). 



