314 



with the line at infinity and in the limiting plane « the point A 

 could not be isolated, because a sequence of connected sets of points 

 each having breadth > some tinite value ;; cannot converge 

 towards a single point. 



Theorem 5: A hi/perbolical point of F* can only be limiting point 

 of hyperbolical points. 



A^ A^ . . . . converge towards A. Corresponding tangent planes 

 «1 ((^ . . . a. A is supposed to be hyperbolical. We shall show that 

 if .4,, is assumed to be elliptical or parabolical for every n, contra- 

 dictory results are obtained. 



The points of space inside a sufficiently small but finite vicinity 

 of A which are riot situated on F^ belong to either of two regions 

 6^1 and G, which are not connected inside that vicinity of A. 



The results obtained when proving theorem 1 show that if we 

 move round a hyperbolical point in the tangent plane we alternalely 

 pass through G^ and G^, to be more exact : twice we pass through 

 G^ and twice through G^. Moving round a parabolical point in the 

 tangent plane we pass once through 6', and once through G,. In 

 the tangent plane of an elliptical point however, a finite surrounding 

 of tiiat point belongs entirely to only one of the two regions, for 

 instance to G^. Hence an arbitrary line through an elliptical point in 

 the tangent plane departs on both sides in the same region. This 

 also holds for a parabolical point, provided we exclude the cuspidal 

 tangent. 



From the converging planes «j «^ . . . . we choose a component 

 sequence «„j «„^ . . . such tiiat in each of these every line through 

 Am A„^ . . . departs on both sides in the same I'egion, for instance 

 (?! (again cuspidal tangents excluded). In the limiting plane a we 

 choose two points B and C of G^ diametrically situated with regard 

 to A. Let b^ and 6, be spheres round B and C, all internal points 

 of which belong to G^. Let a be the line through C, A and 5 and 

 «m (1,12 • • • a linesequence respectively passing An^ A„^ . . . and situated 

 in «„J «„^ . . . , converging towards a and containing no cuspidal 

 tangents (again a,., a„, . . . can be fixed by a simple condition). In 

 the end the lines a^ a,,^ . . . will pass through the spheres b^ and 6, 

 but this means that a line through a point which counts double, 

 departs on both sides of that point in G^ and further on on both 

 sides carries points of G^. Hence on both sides F^ must be crossed 

 again and lines would have been constructed cai:rying four points 

 of F\ 



