( 307 ) 



then ensues t)-^ = ^njt {n> 1), so that inside ^ there would have to 

 lie an invariant point. 



B). Q' is not separated by r from the infinite. We construct between 

 to' and m' a polygonal line approximating 7 -[-»'+ ^'' + *'" at a distance 

 8, cutting off from tu' resp. m' the segment h' resp. ft', and forming 

 with f', r', and ft' a polygon ^13'. The deter- 

 mination of f, and the construction of the 

 skeleton arcs of P' take place in the same 

 way as in the first case. We want to find o^^/ 

 the total angular variation i>i of the trans- 

 formation vector for a positive circuit of the 

 counterimage "P of ^V^ and we understand 

 bj ^5 the total angular vaiiation of a no- 

 where vanishing vector of which the origin 

 describes ^V, and the endpoint as a continuous 

 function of the origin runs first from P' to 

 Q' along path arcs nowhere passing outside 

 ^P, and finally describes r'. 



Then we have : 



^j =: ^, + 2ujt {n ^ 0) 



^, = 2jt 



Hence O-^ = 2?2.t (/z ^ Ij, so that inside '^ there would have to lie 

 an invariant point. 



Fisr- 2^. 



Third case: P' foUoius P, and Q' follows Q. 



In this case r contains a skeleton arc d separating Q' from the 

 infinite, and accessible from the infinite without a crossing of 7 -\-r-\-r' . 

 We determine c, t, and f, and we construct ^1^'s, 'P'4, *^', 'P, and the 

 skeleton arcs of these polygons in the same way as in the second 

 case under A). 



The last point of intersection with '13 of the skeleton arc s' of 

 ^'3 separating Q from the infinite, we represent by L' ; the counter- 

 image of L' we repi'esent by L, the counterimage of s' by s, the 

 endpoint of ^'3 on t by E' , the counterimage of E' by E. 



A), s is separated by s' from the infinite. Our aim is to find the 

 total angular variation v>j of the transformation vector for a positive 

 circuit of p, and we represent by /i t'^e total angle described by 

 the transformation vector from P to L along ^; by Xj tl^G total 

 angular variation of a nowhere vanishing vector of which the origin 

 runs from P to L along ^, and the endpoint as a continuous function 



