( 309 ) 



Fourth case: P' precede.-^ P, and Q' follows Q. 



A). Q' is separated by r from the infinite. We construct the polygon 

 '^' with its skeleton arcs in tlie same way as in the third case. We 

 want to find the total angular variation <i\ of the inverse transfor- 

 mation vector for a positive circuit of ^P', and we represent by x^ 

 the total angle described by the inverse 

 transformation vector along the segment P' Q' 

 of ^' ; by Xs ^he total angular variation of 

 a nowhere vanishing vector of which the 

 origin runs from P' to Q' along '^' , and 

 the endpoint as a continuous function of the 

 origin from P to Q along path arcs nowhere 

 passing outside ^' ; by tpj the total angle 

 described by the inverse transformation vector 

 from Q' to P' along r' ; by il^j the total 

 angular variation of a nowhere vanishing 

 vector of which the origin runs from Q' to Fig. 4a. 



P' along r' , and the endpoint as a continuous function of the origin 

 from Q to P along a curve obtained by replacing in r each part 

 lying outside ^' by the segment of ^\ joining the same endpoints. 



From the equations 



Xi = X, + 2n7r [n ^ 0) 



<^i = Xi + ^Pi 

 X, + t^, ^ 2.-T 



then ensues coj = 2n:r {n ^ 1), so that inside ^' there would have to 



lie an invariant point. 



B). Q' is not separated hy r from the 

 infinite. We construct the polygon ^' with 

 its skeleton arcs in the same way as in the 

 second case under B). We want to find the 

 total angular variation m^ of the inverse 

 transformation vector for a positive circuit 

 of ^' , and we understand by (o, the total 

 angular variation of a nowhere vanishing 

 vector of which the origin describes ^', and 

 y^\\ U the endpoint as a continuous function of the 



origin runs first from P to Q along path 

 arcs nowhere passing outside ^', and finally 

 Fig. 46. describes r. 



