311 



or with C or with the third point of 7^^ on a, sitnated beyond C. 

 Tlien however, for every q such that AC^q^i) there must exist 

 a point P of F^ in the limiting plane, such tiiat AP:=q and this 

 point P must be situated either on b or a or in the lower angle 

 between b and a. Thus, once more a contradiction is obtained. 



Third case. This case is treated in entirely the same way as the 

 second. 



It has been shown that a plane through A exists in which that 

 point is not a point of inflexion. Let f( be this plane. If in a the 

 point A is not isolated, doable point or cusp, the only remaining 

 possibility is that A is ordinary point in a. Let a be the tangent 

 at A in a . a has, besides A, another point C in common with F\ 

 When a revolves 'round a, the point A continues to count double 

 and C to count single on a. Assuming that in no plane through a 

 the point A is isolated, double point or cusp, it follows that in 

 every plane through a, A is ordinary point with a for tangent. 

 Now when a revolves round a and if we consider both semiplanes 

 in which a divides «, it is obvious that at least once a semiplane 

 in which A is isolated must be limiting element of a sequence of 

 semiplanes in which cojivex arches are situated, passing through A 

 and having a for tangent. Let a be this semiplane and «,',«,', «g' .. 

 a converging sequence. In order that A be isolated in the semip\a,ne 

 a', it is unavoidable that the curves passing through A in ((\, «', a\. . . 

 are ovals contracting towards A (it must be understood that the 

 converging sequence is started fai* enough). Let 6 be a line in «' 

 through A having two other points B and 6' in common with F'. 

 Let ,i be the plane through b ± a' , and let b^,b^,bt ... be the lines 

 of intersection of /i and (t\, a\, u\ . . . respectively. Every line 6,, b, . . . 

 intersects the oval in the corresponding plane at a second point, 

 different from A. When the ovals contract, these points converge 

 towards A. Hence in plane /? the line b would be tangent at A, 

 but this is impossible, because b has two other points B and C in 

 common with F'^. This completes the required demonstration. 



Second part. Let A be ordinary point in a plane a and a oi-dinary 

 line of intersection through A in a. 



llieorem. 1 : If a sequence of lines in R, converges towards 

 line a, then points of F' on these lines converge toioards A. ') 



1) This theorem and its demonstration hold also when A is sitnated on a line 

 of i^'^, provided this line does not lie in «. 



