745 



considering that only in the tangent plane a a second branch can 

 pass throngh A. This completes the demonstration of theorem 2. 



Let an oval in a cross tlie line a at A. This oval and the linea 

 have a second point of intersection B and the points A and B have 

 the tangent plane « in common. If A moves continuously along a 

 then, according to theorem 2, the tangent plane « also changes in 

 continuous fashion and the point B also moves continuously. ^) From 

 this follows that a point A at which an oval crosses a, can only 

 be limiting point of points of a possessing the same character. 

 Besides it is easy to prove that the tangent to the oval at A also 

 changes continuously. This result however will not be needed, but 

 we do want the following: 



Let A^, A^ . . . on ti converge towards A. Tangent planes «j, «, . . . «. 

 If the oval in a crosses a at A it follows from the above that for 

 n larger than some finite value the plane «» also shows an oval 

 which crosses a at An. 



Now suppose all these ovals in «„ turn at An their concave sides 

 to the left. The oval in « is the limiting set of the ovals in «„ and 

 considering a sequence of finite concave branches cannot converge 

 towards a finite convex branch, it follows that the branch in tt 

 through A also turns its concave side to the left. 



Taking these results together we obtain : 



Theorem 3 : .4 point of line a in the tangent plane of tohich an 

 oval crosses a, can only be limiting point on a of points having the 

 same kind of tangent plane also tuith regard to the side to vjhich 

 the ovals through those points are concave or convex. 



Theorem 4: F^ cannot exist if the restcurve does not degenerate 

 in any plane through a. 



We consider the case in which the curves of the second order 

 in the planes a^, «^ • • • • (passing through a and converging towards it), 

 contract towards part of a. We call internal region of these ovals 

 that region which contracts to nothing but a. We found that the 

 points of a belonging to this limiting part must be situated in the 

 internal region of the oval in «„ for n larger than some finite 

 number. From this follows that the part of a belonging to the 



^) This theorem and some others which shall be formulated presently concerning 

 the directions in which A and B move, have already been given by Juel, Math. Ann. 76, 

 p. 552. The existence and continuous changing of tangent planes is simply 

 postulated by that author. 



