418 PROCEEDINGS OF THE AMERICAN ACADEMY. 



If we make the point < = — 1 correspond to the origin in the z-plane 

 we shall have then for t ^= — \ 



o + oi = -i + ^^^- - ^Jog (- 1) + r. 



Tlierefore the constant of integration is : 



r = i-(^%5.og(-i). 



When t is very small the principal term is : 



^= — Jlogif + • ' • i^ I. 



Now as long as t is negative and its absolute value very small, the 

 imaginary part must be equal to zero. But when t passes through the 

 value zero from negative to positive, z increases by hi. 



Hence s = — (7 ( — ^log t — h tt i) 



which fulfils the above conditions, 



and C b TT =^ h C > and real. 



oir 



Also when t — b z = g + hi, 



then 



g^hi = - cfb-b\ogb+ 1 - (^+ 1)" _^^ A. 



h= Cbir 

 again, and which gives : 



g=-c[b{i- log b)+\- ^^^^y 



In the w-plane the diagram is : 



t = 



t = oo 



i' = 



and the corresponding transformation is, since the polygon has one zero 

 angle at f = 0, 



dw dt _,, 



— r=y. .'. W = D\0gt = (l>-\- lip. 



