( 308 ) 



Fig. 3a. 



of the origin from P' to L' along path 

 arcs nowhere passing outside ^ ; bj 

 cp^ the total angle described by the 

 transformation vector from L to E 

 along ^; by (f.^ the total angular 

 variation of a nowhere vanishing 

 vector of which the origin runs from 

 L to E along '^, and the endpoint 

 as a continuous function of the origin 

 inside ^ from L' to E' along an 

 arc of simple curve p; by if^i the 

 total angle described by the trans- 

 formation vector from E to P 

 along ^■, by \\\ the total angular variation of a nowhere vanishing 

 vector of which the origin runs from E to P along ^|3, and the 

 endpoint as a continuous function of the origin along a curve obtained 

 by replacing in the segment E'P' of ^^ each part lying outside ^ 

 by the segment of r joining the same endpoints. 

 Then the following equations hold: 



Xi = X, + 2njr (n ^ 0) 



^1 = Xi + (Pr + V'l 



Xï + y, + ^\ — 2jt. 



Hence i^j ^= 2?zjr (n ^ 2), so that inside ^^3 there would have to lie 

 an invariant point. 



B). s is 7iot separated hy s' from the 

 infinite. Then the two endpoints of s 

 as well as the two endpoints of s' lie 

 on 0,. Defining ^p /i, Xa' ^i' V^»' ^'i' V'j 

 in the same way as just now, we arrive 

 here at the following equations : 

 /^ = -/j -f 2nTi (?2 ^ 1, because between 

 P and s lies the Schnitt aSJ 



^Px = ^ï 



^i^Xi+yi + V'i 

 X, + 'P, -f ^\ = 2^. 



Thus again ^, = 2wjr (n ^ 2), so that inside ^ there would have 

 to lie an invariant point. 



Fig. 3&. 



