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certain direction round a. The points .4 and J9 then also move along nr 

 continuously'). Two assumptions are possible: A and B can move 

 in the same or in opposite directions. T.el the direction be the 

 same. In the time that B has described the original segment i^ .4, the 

 point A has gone further, hence ail this time we keep tangent planes 

 with ovals having two different points in common with a. 



When B arrives at the original place of A the plane « must 

 have turned an angle of 180°, but if the bi'anch through B has 

 originally been concave to the left, it must now be concave to the 

 righthand side, and this is not possible as on the way the concave 

 side in B cannot jump round and no' change from concave to convex 

 can have taken place via a degeneration of the oval in two straight 

 lines (according to our assumptions). 



The second possibility was that ^1 and B move in opposite direc- 

 tions. Let the tangent plane successively turn round a in opposite 

 directions, then we obtain two different points in which A and B 

 meet. Such a meeting takes place either wiien the two points of 

 intersection of a and the oval converge to one point oi' when the 

 entire oval contracts to nothing but a single point on a. In both 

 cases the concave sides of the branches through A and B face each 

 other. A priori it seems possible that before the meeting the convex 

 side of the branches through A and B face each other, but then 

 these branches would be connected on both sides via the line at 

 infinity and in the limiting plane the oval would degenerate in two 

 straight lines ^) through the point where A and B meet and this 

 contradicts our assumptions. 



Now we start from the original position of A and B and we 

 observe A only. Let the branch through A turn its concave side to 

 the left. If we turn the tangent plane in such a way that A moves 

 to the right, then the concave side goes on being turned to the left. 

 But before the meeting with B takes place the concave side must 

 be turned to the right (that is in the direction in which ^4 moves) 

 and this means a contradiction because the curvature cannot change 

 its sign discontinuously, neither can it change via a degeneration of 

 the oval in two straight lines (according to our assumption). This 

 completes our demonstration. 



Remark. Above we spoke about the possibility that the oral 



^) If a goes on turning in the same direction A and B obviously cannot change 

 the direction in which they move for then points of a v^^ouid exist with two 

 different tangent planes. 



2) The oval does not converge towards a. 



