737 



We call internal region that one which contracts towards (/ or part 

 of a only. Now if A continued to belong to the external regions, 

 the curve in plane (i would show a cusp in A with both branches 

 arriving from the same side of the tangent, and this is excluded. 

 The possibility might be put forward that for every n the oval in 

 «„ has the line of intersection h,, with ^ for tangent in An., in other 

 words that the two points of Z'^', which 6„ carries besides ^, coincide. 



Fig. 11. 



As follows can be shown that this possibility is excluded. In the 

 second communication {Second paii, theorem 1) we proved : 



If a line a in a plane « intersects the curve in that plane at an 

 ordinary point A, then lines which converge towards a end up by 

 carrying points of F^ converging towards A. The demonstration 

 we used there, also holds if A li^s on one or more lines of F^, 

 provided A is not situated on such a line in plane «. We proceed 

 to apply this theorem to the case of fig. 11. In plane ,-? the line è„ 

 intersects the curve (which is no straight line) at the ordinary point 

 An- tn plane «„ however it would be possible to find a sequence 

 of lines converging towaids h,, but carrying no points of F^ which 

 converge towards An'- a contradiction. 



Hence A will end up by belonging to the internal regions of the 

 ovals ^) and considering this region together with its boundary con- 

 tracts towards a or part of a it follows that every plane through A 

 not containing line a has a point of_ inflexion at A nnth tangent in a. 



^) We exclude the possibility that A continues to lie on the ovals themselves. 

 The cases in which A belongs to an oval in a plane through a will be dealt with 

 sub 2 and H. 



