307 



Now let the semiplaiie aE revolve round a towards aF and aD 

 towards 176^ as indicated by tlie arrows. For nE there is either a 

 last position in which A is, or a first in which A ü not isolated. 

 Let this be aE^. In the same way aD^ for (^/T). If the angle between 

 aE^ and aD^, in which aF and a 6' are situated, is < 180°, then 

 at once planes can be found in which A is isolated. Thus it remains 

 to consider the cases in whicii the angle is ^180°. 



In every ó'é/mplane through a, in which A is not isolated, two 

 branches must meet at A, because if there was only one, the pro- 

 longation of this branch would be situated in the complimentary 

 semiplane, and these two branches would be connected inside any 

 vicinity of A, on both sides of this plane, hence through a there 

 would be no semipianes at all in which A is isolated. 



For this reason, if the angle between aE^ and aD^^ were ^180°, 

 then there would be a finite angle inside which every plane through 

 a has a double point in A. Let y be a plane inside this angle. The 

 semipianes through a in which A is cusp are supposed to lie under- 

 neath y. If y be turned round a in either direction, A at first 

 remains double point. In y four branches depart from A, successi- 

 vely AP, AQ, AR and AS. Let a lie between AP and AS and by 

 consequence also between AQ and AR. Let 6 be a line through A in y 

 between AP and AQ, hence also between AR and AS. Lastly let 

 /J be an arbitrary plane through b. In ^ two branches arrive at A 

 from above y, because above y, AP is connected with AQ and AR 

 with AS. The alternative that above y, AQ is connected with AR 

 and AS with AP, is excluded, because in the planes through a in 

 which A is cusp, the branches meet in A from below y. 



Now the two branches in /? meeting at A from above '/ cannot 

 form a cusp in A, because in that case, A could at the utmost be 

 isolated in only one semiplane through a. On the other hand the 

 branches in ,3 meeting at A cannot form an ordinary point at A with 

 b for tangent, because then we could turn y round a to a position 

 y' in such a way that the line of intersection of y' and ^? would 

 have three different points in common with the curve in ^. But in 

 y' the point A would remain double point (provided the rotation is 

 small enough) hence this line of intersection of y' and ^ would 

 have at least four points in common with the curve in y'; a con- 

 tradiction. 



In (-i two branches arrive in A from above y, but we found that 

 in /? the point A can neither be cusp nor ordinary point with b for 

 tangent, hence A must be double point in /?. /? however, was an 

 arbitrary plane through b hence everi/ plane through b would have 



