( 623 ) 



The enne C lias in /^ llireo sucressivo |)<>iiils in common with 

 the phme V. The cni-ve 6' is sitnated on llie sni-hice O, so tlie tln'ee 

 common points mnst lie on the section of () with V. This section 

 consists of the curve d and of the common tangent / in P of C 

 and (/, counting doul)le. In the following w-ay can be proxed that 

 of the three common points only two lie on d. 



Let us first take instead of a general curve a twisted cubic 6'j. 

 Let F be the origin ; the property being projective the plane at 

 infinity can be chosen in such a way, that the curve is i-epresented 

 by the equations 



,,; = t\ y =z t\ z ^t. 



If now i]i point t = the curxes C, Jind (I„ had three common 

 points then in the origin also the radii of curvature of the two 

 curves Avould be the same. Let R be the radii of curvature of 

 Cj iu P and /• the radius of curvature of d^ in P, we then easily 

 find : 



E = = --^— — — . 



1 



So for ^ =: we find 7^ = -. 



The surface 0^ is enveloped by the plane 



,r — 3tjt-\-3z f' — t' = 0. 

 So the curve (/., is enveloped by the line 



— 3^ + 3^^ — ^1=0 

 Consequently the equation of d^ is 



3 



2 

 so the radius of curvature r = -. The curves C3 and d.^ have in 



o 



the origin not the same radius of curvature, so they have not three 

 or more points in common. 



Let j? be the orthogonal projection of C^ on V (,/; = 0), then C\ 

 has with jj three successive points in common in /\ for all jioints 

 which ('3 has in common with T"^ must lie on the section p of V 

 with the projecting cylinder of C\. The projection p has for e(piation 



and so also the radius of curvature in the origin P = -, showing 



again that the curves p and C\ have in the origin three points in 

 common. 



