Mathematics. — "On Elementary Surfaces of the third order'. 

 (Second eommnnication). By B. P. Haalmeyer. (Communicated 

 by Prof. Brouwer). 



(Communicated in the meeting of June 30, 1917). 



\ 4. If A is cusp in one and not more than one plane «, then a 

 is tangent plane. 



Let a be cuspidal tangent and A", and K, the branches meeting 

 at A. The Jordan theorem for three dimensional space tells us that 

 within any vicinity of A the branches K^ and K^ are connected 

 by a set of points I and by another set II, both sets belonging to 

 F\ Again I and II have no points in common and are the con- 

 tinuous (1,1) representations of plane regions I^ and llj which have 

 the character of Jordan regions in the vicinity of the point A^ 

 corresponding to A. Inside a finite neighbourhood of A all points of 

 F' belong to I -\- U -\- K, -f- K^. Let EF be a linesegment in a 

 crossing both branches K, and K,. Suppose I and II were situated 

 on- the same side of a. If a parallel linesegment converges from 

 that side towards EF, it would end np by having at least two 

 points in common with I and also two with II : a contradiction. Hence 

 I and II lie on different sides of «, for instance I above and II below «. 



Let /? be a plane through .4 not containing the cuspidal tangent 

 a. A cannot be cusp in ^ (iH=")' ^"^ the results of § § 1 and 3 

 show that A cannot be isolated or double point in ^. Hence A is 

 in /? ordinary point or point of inflexion. On the line of intersection 

 of « and /?, A counts double in « hence according to the tiieorem 

 of p. 117—118, A also counts double on that line in i? and this excludes 

 the possibility that A is point of inflexion in [3. Thus only remains 

 the possibility that A is ordinary point in [j and the line of inter- 

 section with « is tangent because .1 counts double on this line. 



Thus has been proved that in any plane not containing the 

 cuspidal tangent in « the point A is ordinary point with tangent in a. 



Remains to consider a section of F^ in a plane ^(=h«) through 

 the cuspidal tangent a. Let 6 be a line through Am^{=\=a). We 



consider a sequence of planes [i„^„ through h and converging 



towards it The lines of intersection of « and ^^,^^ /? are 



respectively denoted by «,,«, a (all passing through A). In 



