418 PROCEEDINGS OF THE AMERICAN ACADEMY. 



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

 we shall have then for t = — 1 



+ oi = _ i + ^±^- 2 - ft log (- i) + r. 



Therefore the constant of integration is : 



r = i_£±!>! + i iog(-i). 



When t is very small the principal term is : 



~ = — ft log i! + • • • J > 1. 



Now as long as £ 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 z = — C (— ft log t — ft tt i) 



which fulfils the above conditions, 



and C b it — h C > and real. 



OTT 



Also when * = b z = g + hi, 



then 



9+ ki = - cfb-b\ogb+ 1 _(^+ !)"_ 57r A, 



again, and which gives : 



g = - C(b (1 - log ft) + 1 - ^^Y 

 In the w-plane the diagram is : 



V — V 



CO 



t = 



t = QO 



v = 



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

 angle at t — 0, 



dw dt r ., 



-=r = — . .'. w = JJlogt = </> + z \p. 



