114 



breadth, lienee each has an infinite limiting branch. In the limiting 

 plane (f the branches departing from A at tirst face h with their 

 convex side {b is tangent at an ordinary point). But a sequence of 

 finite concave branches cannot have a convex limiting branch 

 hence a contradiction is obtained. The possibility might be ])ut 

 forward that on the converging branches points of inflexion may 

 converge towards A, but a curve of the third order with double 

 point has only one point of inflexion ^), hence it may be assumed 

 that only on either the left or the right hand branch points of inflexion 

 converge towards A and tlie contradiction remains with regard to 

 the other branch. 



We now proceed to show that not all planes through b can 

 have a double points in A. Again AE and AD are supf)osed to be 

 joined by II below a and AC and AF by IV below a (fig. 3). 

 AR and AS are situated on II. We found that if a be tnrned round 

 b in such a way that the right hand side moves downwards, then 

 at tirst A remains double point and the branches arriving in ^ from 

 below remain situated on IV. In the same way as AC and AF 

 are connected by IV below «, the branches J 7? and ^.S are connected 

 by a component region of II on the right hand side of /?. Taking 

 in consideration this analogy it is obvious that if ^ be turned round 

 b in such a way that the lower half moves to the right, then at 

 first A not only remains double point, but the branches meeting at 

 A from below a are still situated on II. This may be expressed as 

 follows : There cannot be a last plane in which the branches are situated 

 on 11, and the same can be said of IV. 



Let us now consider the set of semiplanes through b and situated 

 below a. If every plane through b has a double point in A, then 

 in each of these semiplanes two branches would arrive in A from 

 below u. It was found that if these branches are both situated 

 on II, then the same holds for the branches in all semiplanes situated 

 more to the left. In the same way if both branches lie on IV this 

 is also the case in all semiplanes more to the right. Besides the set 

 of semiplanes with branches on 11 cannot have a last element on 

 the right side and those with branches on IV cannot have a last 

 element on the left side. But all semiplanes have two branches 

 below a, hence the two kinds of semiplanes with branches on II 

 and IV respectively must be separated by a seraiplane with one 

 branch on II and one on IV, and this is impossible according to 

 page 111. Thus the assumption that a// planes through è have double 

 points in A leads to a contradiction. 



1) JuEL loc. cit. Acad, of Denmark § 5. 



