( 624 ) 



1 2 



(_)ii( of llic two values ^t* = ^ i^iid r =: - ensues tluit tlie seclioii 



(I, lies near 7^ on tlie convex side of the projection />. 



§ 3. Tlie hitter can also he proved as follows for any cnr\c 

 Avilli the aid of Descriptive (leometi'v. 



h' we take the osculating plane T^ in any ])oint of the curve C^, 

 to be the vei-tical plane of projection and the normal plane of the 

 curve to he the horizontal one (plane of projection), then the vertical 

 projection j/' is a curve cutting the axis per|»endicnlarly in a common 

 point /^ and the horizontal projection y>' is a cur\'e having in F a 

 cusp with the axis for cuspidal tangent. Let us now constrnct 

 the vei'tical trace d" of (K The vertical traces of the generating 

 lines of (^ lie on the laugents to //', liius all on the convex side 

 of />", \\ hilst those traces are also situated on the same side of the 

 perpendicular on the axis in P, where also ^/' and p' are lying. So 

 the curves </" and />" turu the concave side to the same side, whilst 

 </" ]iear P lies outside p". So the cnr\es </" and //' have an even 

 jHimher of [)oiuts in common. 



Thus if r' and d" had the three points of C', lying in I" in common, 

 then also //' and t/" would have three points near P in common. 

 So accordiug to what was jtroved above //' and d" had four points 

 in counnon. That fourth common point of />" and d" would, lying 

 on d", also lie on (^ and would l)e the })rojection of a ])oint (2 of 

 tlie curve (' lying as near to it as one desires. So at the limit the 

 projecting line of ^^ would be a tangent of ''A Thus neai' /''a tangent 

 of () might be [lerpendicular to the osculating plane l^, which is 

 iin|)ossible. So ^'and d" have no three ]ioiiits in common. 



§ 4. We can prove moreover as follows synthetically that the 

 twisted cubic T., has not three points in common with the trace d.^. 



If 6', and d., had three ]>oints in common at /-* then d.^ and the 

 ))rojection p^ of C^ on V out of any arbitrary point A would liaxe 

 three successive points in common. This projection p^ is a cubic 

 cnrve of class four. The three inflexicmal tangents of />., are the traces 

 of the three osculating planes through A, thus they are also tangents 

 to d.,. As the contact of the second order in P must count for three 

 common tangents and each of the three inflectional tangents of p^ 

 for two common tangents, two curves respectively of class two and 

 class four would lia\e nine common tangents, which is im[)0ssible. 

 So the curves ('., and d have at P no three points in common. 



