746 



internal region of tiie oval in «„ must diminish for increasing n, 

 because if the oval in «„ crosses a at An and B^ then «„ is tangent 

 plane at A„ and in case An ended up by being situated inside the 

 ovals, u also would be tangent plane at J„: a contradiction. 



Hence if the ovals have the entire line a for limiting set, none 

 can have points in common with a. An idea of this case may be 

 got by imagining a sequence of hyperbolas of which the angle of 

 the asymptotes (inside which the hyperbola is situated) tends towards 

 180° and such that the centre is situated on a and both asymptotes 

 converge towards a. 



In this case everything is in favour of counting a as a triple hne 

 in a. In no plane through a a brancii would depart from any point 

 of a, and except a, F^ would contain no straight tine. 



A second possibility we wish to consider separately is that in the 

 tangent plane of every point of a the oval has a for tangent. Again 

 let A be a point of a with tangent plane n. The line a divides « into 

 two semiplanes, in the one A is isolated and in the other an oval has 

 a for tangent at A. 



Now let A move along a. The plane (( turns rounds. If ^ moves 

 on in the same direction the plane a goes on turning in the same 

 direction, for otherwise two points of a might be found with the 

 same tangent plane and this cannot be as in either point an oval 

 in the tangent plane must have a for tangent. 



If A goes round the entire line a the tangent plane meanwhile 

 turns 180° round a. The ovals in the tangent planes merge conti- 

 nuously into each other, hence after turning 180° the branch having 

 a for tangent is situated in the wrong semiplane. This means that 

 on the way the branch must change from the one semiplane into 

 the other and this is only possible either via a tangent plane in 

 which the restcurve consists of two lines through .4, or via a tangent 

 plane in which the oval has contracted to nothing but point ^. The 

 first possibility is excluded according to our assumption and the last 

 would mean that a sequence of ovals in the con\erging planes 

 contract towards a point of d not belonging to the internal regions 

 of the converging ovals. This was found to be impossible hence the 

 assumption that every point of a has a tangent plane with oval 

 having a for tangent, leads to contradictory results. 



Leaving apart both cases treated above, there certainly exists a 

 plane through a in which an oval has two different points A and 

 B in common with a. Let this plane a revolve continuously in a 



