432 PROCEEDINGS OF THE AMERICAN ACADEMY. 



Let the origin in the z-plane be chosen at the point B, < = ; now 

 putting t = in (c), we obtain 



o + o i m^ + 1 , )tr — 1 frr — 1 



-^ - ^— log-^- - '^log ^^ + r. 



//«2 -1-1 m^— 1 , m^— 1\ 



•'• r = — T — log r m log 7 — . 



\ 2 ° 2 ^ — 2m J 



When i = a, z = o + di. 



+ d? = CM r — log — m 1 



1 - m^\ 



OS, ; I 



* — 2m y 



^, //«■ — 1 , I — m^ . 1 — 771- 



— ^ ;^ lo^ ; '" los — i 



\ 2 • ° — 2 * 2 »« 



^ f m^ -{- 1 . 2 m 77?'^ Otti ^ ^^2 



2 ^ ' 



To find h^ we notice that where t is very small we may write 



• • • 



dz \/t{t — a) ,- dt 



-c= t-i ^^ ''^^ Vi-«r^ + 



.-. 7^ = ?« [log (< — ]) + const.]. 



Const. = — i TT, and is so chosen that the imaginary part of z is zero as 

 long as t is less than unity. 



.•. z = m C [log (t — l) — i tt]. 



When t =z I -{- e, z = — cc — ih; 



/. — 00 — ih = Cm [log Id — ^ tt], 



or h =■ C m TT, C = . 



mir 



TT d (m — I)- „ = ^ 



Hence 7 = ^ — = A, say m > 1. 



h 2 m 



