107 



th^. points of intersection with tJje oval, but on (lie line a„ both 

 points of intersection lie on the same side of H„. This means that 

 Bn is at the same time internal and external to the oval, and this 

 is impossible. 



The above results may be taken together as follows: yl is supposed 

 to be isolated in plane «, and 6 is a line through A not situated 

 in a. Nojo, if A is ordinary point in tino dijferent planes through 

 b, it cannot be cusp in any other plane throu,yh b. But if /I is ordinary 

 point in a plane through b, the branches meeting at A in this plane 

 furnish points of F'^ on both sides of every plane through /; inside 

 every vicinity of A. Hence in no plane through b can A t»e 

 isolated. Besides above « there is a finite vicinity of A contain- 

 ing no points of F^ so in no plane can A be double point, point 

 of inflexion or ordinary point with tangent not situated in ((. Hence 

 when A is supposed to be isolated in «, and b is a line through A 

 not situated in cc, the final result may be formulated as follows: 

 // through b 2)ass two different planes in tuhich A is ordinary point, 

 then in every plane through b, A is ordinary point and all the tangents 

 are situated in a. 



Above we found that in ^ the point A is either: 



1. Ordinary point with a as tangent. 



2. Cusp. 



Let the first possibility be assumed. We turn the tangent a in the 

 plane ii round the point .1 in both directions to the positions a' and 

 a". Provided these rotations be small enough the lines a' and a" 

 have each three different points in common with i^' '). Hence in no 

 plane through a' or a" can A be isolated point, double point or 

 cusp. Points of inflexion are also excluded, because one of the branches 

 meeting at such a point would arrive from above «, but above a 

 there is a finite neighbourhood of A containing no points of 7^''. The 

 only remaining possibility is that in every plane through a' or a", 

 A is ordinary point and the tangents must all be situated in « because 

 above a there is a finite neighbourhood of ^ containing no points oï F\ 



Let c be an arbitrary line through .4, not situated in ^ or /•?. The 



1) JuEL, loc. cit. Acad, of Denmark. When points of intersection are counted 

 as explained, an elementary curve of the third order and an arbitrary line in its 

 plane have in common either three points or one point. Hence a tangent at an 

 ordinary point A carries one point more of the curve. Now if this tangent be 

 turned round A over a sufficiently small angle, A is replaced by two points of 

 intersection A and B each counting single. But there must be still another poini 

 of intersection, as there are to be three altogether, so the line in its new position 

 has three different points in common with the curve. 



