432 PROCEEDINGS OF THE AMERICAN ACADEMY. 



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

 putting t = in (c), we obtain 



o + o i m 2 + 1 , m % — 1 . m 2 — 1 



//» 2 +l, m 2 -l , m 2 -l\ 



.'. T = — — r — log m log — . 



\ 2 & 2 * —'2 m J 



When Z = a, z — o + di. 



■ / m 2_i i_ m 2 1 - m 2 \ 



— C log — — m log — : — 



\ 2 • ° — 2 s 2m J 



To find />, we notice that where t is very small we may write 



dz \/t(t — a), ,- dt 



— = - — dt as V 1 — a - + 



C t-\ t—1 



= m [log (<—!) + coust.]. 



• • • 



z 



Const. = — i ir, 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 — 1) — iir]. 

 When t = 1 + c, z = — cc — i A ; 



„*. — go — ih = (7/« [log lei — z tt], 



or h = C m-rr, = . 



m7r 



Hence 7 = ^— - = K, say m > 1. 



h 2 m 



