106 



cannot continue to be situated on odd curves in «„, for an odd 

 curve is never entirely internal to a finite region (in other words: 

 always has infinite breadth), so the limiting set would be a curve 

 in « passing through A. But if for n larger than some finite value 

 the points ö„' and Bn" can neither be isolated nor situated on odd 

 curves, they must lie on even curves, which in this case must be 

 ovals. Obviously these ovals contract when n increases and A is the 

 sole' limiting point. Let y and (f be the planes through b in which 

 A is supposed to be ordinary point (with the tangents in a). 



Let (1,1, c„ and d,, be the lines of intersection of «„ and /?, y, ff res- 

 pectively. Obviously a,, intersects 

 the oval in plane «„ at Bn' and B„". 

 Bn is a point of the cuspidal 

 tangent in ,? and Bn and Bn are 

 points of the branches meeting at 

 the cusp from different sides of 

 the tangent, so on line a,,, Bn is 

 situated between Bn and Bn', 

 hence Bn is internal point of the 

 oval in «„.from this follows that 

 the lines c„ and dn passing through 

 B„ have each two points in common 

 with the oval, one on either side of Bn- Let these points be CV, d" 

 and Dn,D,;'. 



In plane j? A is cusp with h as tangent, but in y and d A is 

 supposed to be ordinary point with the tangents in «. From this 



follows' that by taking n large enough the ratios ^777 ^"^ n ' 



Bn Cn Bn J^?i 



may be made as small as desired, even of the second order with 

 respect to the distance of the planes « and «„. Besides the angles 

 of fin, Cn and dn are the same for every n, hence for n large 

 enough, the linesegments Cn 0„' and /?„ BJ will have no point in 

 common and this result contradicts one of the fundamental proper- 

 ties of ovals. 



The following question arises: Is it possible that A is ordinary 

 point in y and d and cusp in ;/, but with a cuspidal tangent not 

 coinciding with b? We shall show that the answer must be negative. 

 The notation of points of intersection etc. is kept the same as above. 

 In /? the branches meeting at the cusp would arri/e from the same 

 side of b, but in 7 and f) the branches meeting at .1 arrive from 

 different sides of b. Hence for n large enough the oval in «„ would 

 be sucii that on the lines c„ and d„ the point Bn is situated between 



Fig. 1. 



