( 81 m ) 



so that we arrive at the absurd result that inside ^V must lie an 

 invariant point. 



B). s' is not separated by s from the infinite. Then the two end- 

 points of .9' as well as the two endpoints of s lie on o^. Defining 

 f'^i» /n X21 ^/'ij '/a' ^1' V'a in the same way as just now, we arrive here 

 at tlie following equations : 



;/j z=z -/^ -f 2»rr {n > 1, because between P' and s' lies the Schnitt S^) 



'li—U, — 2rr 



A\ = 4% 



^i = Xi + ^/i +if'i 



Xï + 'f\ + V'. = -'T- 

 Thus again Wi = 2?zjr (?z ^1), so that inside ^^' there would have 

 to lie an invariant point. 



Second case: F' follows P, and Q' precedes Q. 

 A). Q! is separated hy r from the infinite. We construct the 

 polygonal lines ^\ and ^\, and tiie polygon ^' with its skeleton 

 arcs in the same way as in the first case. Then the counterimage 

 of ''P' is a simple closed curve ^P bearing skeleton arcs wiiich, like 

 those of ^', cut neither ;* nor r' . We want to find the total angular 

 variation />■, of the transformation vector for a positive circuit of '^-V 



We represent by E' the endpoint of ^'j 

 on t; by E the counterimage of E'; by Xi 

 the total angle described by the transforma- 

 tion vector along the segment PE of ^P; by 

 X2 the total angular variation of a nowhere 

 vanishing vector of which the origin runs 

 from P to E along '^. and the endpoint as 

 a continuous function of the origin from P' 

 to E' along path arcs nowhere passing out- 

 side ^ ; by i|.'j the total angle described by 

 the transformation vector along the segment 

 Fig. 2a. EP of ^ ; by V?^ the total angular variation 



of a nowhere vanishing vector of which the origin runs from E to 

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

 along a curve obtained by replacing in the segment £"7^' of P' each 

 part lying outside ^ by the segment of r joining the same endpoints. 

 Fiom the equations 



X, = X. + 2n7r {n ^ 0) 

 ^, — ip, 

 ^1 = Xi + ^. 

 Xï + ^2 = 2^ 



