109 



1. A and B are both isolated. 



2. A and B are both cusps. 



3. A is isolated and B is cusp. 



4. >4 is cusp and B is isolated. 



But no elementary curve of the third order can have two isolated 

 points, two cusps or one of each, hence the required contradiction 

 is obtained. 



§ 3. If A is double point in a plane a and cusp in not more 

 than one plane, then a is tangent plane. 



The points of F' situated in the plane « form an elementary 

 curve of the third order K, which has a double point in A. A is 



the point of intersection of two convex 

 arches K^ and K,. Let c be a circle 

 /7 round A in u, such that all points of 

 K which are internal to c belong to 

 /ij + /v, and besides c must be such 

 that it has only two points-. C and E 

 in common with K^ and only two 

 points: B and F with K^. All these 

 conditions can be fulfilled by taking 

 c small enough. 



Now the branches AC, AD, A E 

 and AF are connected by four sets of points I, II, III and IV, 

 having no points in common, all belonging to F^ and each of 

 which is entirely situated on one side of «. Respecting these four 

 sets of points, the Jordan theorem for threedimensional space ^) 

 leaves only two possibilities. 



The first possibility comes to the following: AC and AD are 

 connected by I, and AD and AE by II, AEJ and AE' by III, and 

 lastly AF and AC by IV. If the concave side of EC faces F, let 

 us assume for a minute that III and IV are both situated above «. 

 Now if a pai'allel linesegment converges from above towards E' C' 

 it would end up by having at least two points in common with 

 both III and IV, and this is impossible. Hence III and IV cannot 

 lie on the same side of ii. If the concave side of DF faces E then 

 II and III must also be situated on ditferent sides of k. Hence II 

 and IV lie on the same side of «, but then I is certainly situated 

 on the other side, for suppose all three were on the same side then 

 a parallel linesegment converging from that side towards PQvfOu\é 



1) Brouwer, Math. Ami. 71, p. 314. 



