308 



a double point in A and this has been shown on page 114 to be 

 impossible. 



It only remains to consider the case that the angle between the 

 semiplanes aE^ and aD^ is equal to 180°. Above it was shown 

 that in every semip\a,ne through a in which A is not isolated, two 

 branches must meet at A hence in the plane formed by aE^ and 

 aD^ there are four possibilities : 



1. A is double point. 



2. A is ordinary point with d for tangent. 



3. A is cusp. , 



4. A is isolated. 



To complete the demonstration of the existence of a plane in 

 which A is isolated, we shall show successively that J, 2 and 3 

 lead to contradictions. 



1. Let y be the plane of a E^ and a D^. From the double point 

 A four branches depart in this plane, successively: AP, AQ, All 

 and ^*S. The line a lies again between AS and AP, hence also 

 separates AQ from AR. The semiplanes a and ^, in which A is 

 cusp are again supposed to lie below y. In the complementary semi- 

 planes A is isolated hence above y, AP is connected with AQ and 

 AR with AS, below y, AS with AP and AQ with AR. These last 

 two connections are via the branches meeting at the cusps in « and 

 i3. Let (I be a line in y through A, separated from a by the branches 

 RAP and QAS, and let ö be an arbitrary plane through d. 

 In plane d two branches meet in A from above /. Both these 

 branches have d for tangent, because A is isolated in every semi- 

 plane through a above y. Hence in 6, A is ordinary point with d 

 for tangent. In the vicinity of A the curve in d lies above y, hence 

 below y the point A is isolated in d. This however holds for any 

 position of <i (through d), but then it is impossible that the branches 

 meeting at A in « (or |i) are connected inside every neighbourhood 

 of A with the branches meeting at A in y. 



2. Let y again be the plane of a E^ and a D^. In this plane A 

 is ordinary point with a for tangent. The semiplanes with cusp in 

 A are again supposed to lie underneath y. Let d be a line in / 

 through A having three different points in common with E"* and 

 let Ö be a plane through d{=\=y). In every semiplane through a 

 above y, A is isolated and in every semiplane through a below y, 

 A is not isolated, hence if a semiplane turning round a converges 

 from below towards the semiplane of y in which A is isolated then 

 in these semiplanes ovals, passing through A and having a for 



