313 



«„, or is composed of this limiting set together with an isolated point. 



Because F^ is a closed set, the section in a contains tlie entire 

 limiting set of the sections in «„. 



From theorem 1 follows that an ordinary point in « must be 

 limiting point of the sections in a,,. For a point of injle.vion in « 

 this is proved in a strictly analogous way. That a double point or 

 cusp in « is limiting point of the sections in «„ can easily be shown, 

 if we remember the connection existing between the branches 

 arriving at such a point. This connection has been treated in the 

 first part. 



Remains only a possible isolated point in «. That such a point 

 need not be limiting point of the sections in «„ may be seeji from 

 examples of cubic surfaces. 



Theorem 3 : llie tangent plane changes continuously icitJi the 

 corresponding point of F^. 



Let the points A^ A^ . . . . of F^ converge towards A. \i n^ «, ... u 

 are the corresponding tangent planes we have to show that «i «, . . . . 

 converge towards a, and nothing but «. Suppose «^ «,.... had a 

 limiting plane ,i, different from a, but of course passing through A. 

 Foregoing results show that ^ is in /? either ordinary point or point 

 of inflexion, hence anyway a line a through .4 in /i can be chosen 

 having three different points A, B, and C in common with F". Let 

 a^ a, . . . be a linesequence respectively passing through A^ A^ . . . . 

 and situated a^ «^ • . • • ^'^^ converging towards the line a in the 

 limiting plane ^ (these lines a^, a^ . . . . can be fixed in different 

 ways by a simple condition). From theorem 1 follows that for n ^ 

 some finite number the lines a„ carry points Bn and Cn of 7^' 

 converging towards B and C respectively. The point A,, however 

 counts double on any' line in the tangent plane rr„, hence lines would 

 be constructed having four points in common with 7^^ : a contra- 

 diction. 



Theorem 4: An elliptical^) point of F^ can only be limiting point 

 of elliptical points. 



Let the points A^ A^ . . . . of F^ converge towards A. Corresponding 

 tangent planes «j «^ • . • • «• Suppose An were for every n double 

 point or cusp in «„. Then in every «„ a branch would connect A„ 



') Points of F^ which are in the tangent planes isolated, double points or cusps 

 we call respectively elliptical liyperbolical and parabolical points. Except these 

 F" can contain one exceptional point, the character of which has been dealt with 

 in the first part. 



21 



Proceedings Royal Acad. Amsterdam. Vol. XX. 



