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hand side of AN, because the component region of II which forms 

 the direct connection between AR and AS, is situated on the right 

 hand side of /?. This bianch on II cannot have AL for tangent 

 because in that case the branch on IV would also have AL for 

 tangent and cusps where both branches arrive from the same side 

 of the tangent, are excluded. Hence the branch iii y on II foruis 

 at A finite angles with both A2f and AL. 



The line c has, besides A, another point in commoji with F", and 

 for this reason can never be tangent at a double* point. Hence the 

 branch in y situated on IV cannot have AL as tangent, so it must 

 arrive in A under a finite angle with AL, and it follows that if the 

 plane (< be turned round line b in such a way that the right hand 

 side moves downwards (fig. 3), the point A will at first remain 

 double point. The above results may be taken together as follows: 

 a cannot be limiting plane of planes through b in which A is not 

 double point. But by reversing a and ^ in our reasonings, the same 

 can be said of plane /?. Hence: If a be turned round b in either 

 direction, A at first remains double point. In neither direction can 

 there be a last plane in which A is double point, so either there 

 is a first in which A is not double point, or all planes through b 

 show a double point in A. 



In a first plane in which A is not double point, there still arrive 

 two branches in A from above « (one on I and the other on III) 

 hence in such a plane A would be either ordinary point with b 

 as tangent or cusp. The case that A is cusp shall be dealt with 

 sub 2. So at present only two assumptions need be made, namely 

 that there is a first plane in which A is not double point, but 

 ordinary point with b for tangent, or that all planes through b show 

 a double point in A. We shall successively show that both these 

 assumptions lead to contradictions. 



Let rJ be first plane in which A is ordinary point with b for 

 tangent and 6^, 6^, 6^ ... . a sequence of converging planes (all pas- 

 sing through b) in which A is double point. In rf a finite neighbour- 

 hood of A exists containing no points of i^' on ojie side of the 

 tangent b, in this case below b. Considering F^ is a closed set, this 

 is only possible when in the converging planes the loop of the curve 

 (that is the part of the second order) ends up by being situated in 

 the semiplane of on which converges towards the lower semiplane 

 of 6. Besides these loops must contract towards A and nothing but 

 A. Hence for n^ some finite number the branches in ff„ belonging 

 to the ])art of the third order depart fiom A above b. At first the 

 concave side of these branches faces b. Both branches have infinite 



8 



Proceedings Royal Acad Amsterdam. Vol. XX 



