105 



and C of F^ are cliosen, both internal to h' and on different sides 

 of «. The corresponding points B.^ and C'l aie situated inside c^ and 

 can be joined by an open .Jordan curve not passing througii ylj and 

 entirely internal to t\. The set of points K corresponding to this 

 curve is closed and connected (both these properties are invariants 

 for (1,1) continuous transformations). A' is situated entirely inside/;, 

 contains points on both sides of u but no points of « itself (/I is 

 the only internal point of c belonging to 7^'). Hence /iT is composed 

 of two closed sets of points, one on each side of «, but this is im- 

 possible, because K is connected. 



The above results may be taken together as follows : 

 Through the line a passes a plane «, in which A is isolated, and 

 a plane ^, in which A is not isolated. Besides, inside a sufficiently 

 small neighbourhood of A the surface F^ lies entirely on one side of 

 a, let us say below «. Hence inside that neighbourhood of .1 the inter- 

 section of i? and F* lies entirely below a (always excepting the 

 point ^itself, which is situated on a). Considering the possible forms 

 of elementary curves of the third order, there remain two possibilities: 



1. A is ordinary point in /? with a as tangent. 



2. A is cusp in /I 



Let A be cusp in /? with ó as cuspidal tangent. In no plane through 

 h can A be isolated, because the two branches meeting at the cusp 

 in i? furnish points of F^ on both sides of each of these planes 

 inside every vicinity of A. But above « there is a finite neigh- 

 bourhood of A containing no points of F^, hence in every plane 

 through h, A is either cusp or ordinary point with the tangent in 

 «. We proceed to show that if A is cusp in /i it cannot be ordinary 

 point in two other planes through h. 



Let rr,, «2, «3 .... be a sequence of parallel planes each of which 

 lies above all preceding ones and which have n for limiting plane. 

 Let the points of intersection of b and «i, r(,, <f, . . . . be respectively 

 B^, B,, B^ . . . . If the sequence is started high enough every plane 

 «i,«2... has a point in common with each of the branches meeting 

 at the cusp in (3. Let these points be B^' and B^", B^ and B^', 



BJ and B," Nons of these points B,', B," can be 



isolated in the planes a,, «,.... considering the branches meeting 

 at the cusp in /i furnish points on both sides of each of these 

 planes in every vicinity of B^', Bi". . . . 



A sequence of connected sets of points, each having a breadth 

 > p, has for limit a connected set of points with breadth ^p. 

 From this follows that when n increases the points BJ and BJ' 



