743 



branches arrive at ^1 in Sn, forming tiiiito angles. These l)ranches 

 are connected alternately oji different sides of 8,1. Through .4 in 6„ 

 Ave can at once find a line through which pass two planes having 

 a cusp in A and such that in both the cuspidal branches arrive 

 from the same side of «„. Then in the same way as before we can 

 once more obtain a contradiction. 



It now remains to consider the pos.sibility that for every ji the 

 oval in f„ has a for tangent at A. For increasing n these ovals 

 contract either towards A only or towards a connected part of a 

 containing A. 



If A is the only limiting point, then the contracting ovals would 

 give to A the character of a point of a twodimensional continuum 

 and a sequence of points on a «having A for limiting point, could 

 not be fitted in anymore. 



If on the other hand the limiting set is an interval on line a then 

 the internal points of this interval would, in planes not containing 

 a, be cusps with both branches arriving from the same side of the 

 tangent and this is also excluded. 



We thus have proved that every plane through A, containing 

 neither line a, nor the tangent at A to the oval in a, has an 

 ordinary point m A iviih tangent in a. The planes through the tangent 

 at A to the oval in a have point of injieMon in A ivith tangent in a. 



There remain to be considered the curves in planes through a. 

 These shall be dealt with presently. 



3. We now come to the third possibility mentioned on page 736. 

 The restcurve in a consists of an oval having a for tangent in A. 

 In a there depart from A two branches AB and AC on a and A E 

 and AD on the oval. In almost the same way as before it appears 

 that here AC is connected with AD, AD with AE, AE with AB 

 and lastly AB with AC. The connecting sets of points are again 

 situated alternately above and below a. This being established the 

 further reasoning used for case 2 holds here without any alteration 

 (again we remind the reader of the assumption that no second line 

 of F^ intersects a). Results: In every plane ivhich does not contain 

 line a, the point A is ordinary point roith tangent in « (in all these 

 planes the branches depart from A to the same side of «). 



The curves in planes through a must be considered still. 



In each of the three above cases, «. was found to possess the 



