( 305 ) 



to ^ 2 of the third communication ') ; by (f^\ the total angle described 

 by the inverse transformation vector along the segment L' E' of 'P' ; 

 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 from L tot E along a curve p 

 lying inside ^"'); by \]\ the total angle described by the inverse 

 transformation vector along the segment E' P' of r' ; by i|-., tlie 

 total angular variation of a nowhere vanishing vector of which ihc 

 origin runs from E' to P' along r' , and the endpoint as a con- 

 tinuous function of the origin from E to P along a curve obtained 

 by replacing in the segment EP of r each part lying outside '^' by 

 the segment of "P'^ joining the same endpoints. 

 Then tiie following equations hold : 



X, = X. + 2n.T (/. ^ 0) 



(fi = Ts 

 tf\ = rp^ 



^1 ='Ai + ^/i -f il'i- 

 Now Xs + '/a + V'a represents tlie total angular variation of a 

 nowhere vanishing vector of which the origin describes the polygon 

 ^P' in a positive sense, and the endpoint as a continuous function 

 of the origin a closed curve nowhere passing outside ^]^', so tiiat 

 we have: 



Ai + ff, + V'2 = 2-T. 



Hence 



o>i r= 2n.T (w > 1), 



Fig. lb. 



1) See these Proceedings Vol. XUl, p. 770. 



2) If L' lies not on "'P', but on one of the paths connecting ^' and (f, we must 

 take care that p does not meet this path. 



