420 



PROCEEDINGS OF THE AMERICAN ACADEMY. 



— 1 v 



-r— (<t>t= P - </><=-i) = 7— 2 lo g (~ tp)' 



4 7T "± 7T 



V f — X 



1 - 2b\ 

 : 4*\~Cb + ^2b~ ) 



_±_ r r + h ( 2b -' [ Y\ 



~47rh[_ X 7r\ 2b J J 



If b is infinite this becomes 



+ v ( . h \ 



If 5 is unity the equation becomes 



VI h 



q — ; I X + 



(- + ^> 



4 7T /< V ~ 2 71 



which agree with the known special cases already worked out. 



Problkm II. 



Consider the problem of a square edge facing a semi-infinite plane, 

 which may extend beyond the edge as per diagram. 



The polygon consists of the sides 

 ' = " °° DABCE. 



E 



v — v 



t = 



B 



t = 



« = + SO 



t = a 



D A 



The transformation for this case is 



i dz = V 7=b{t-b) = T - l f x (£-o)\ 



Gdt t — c \ t - c J 



which integrates into 



^=1(t-b)l+2(c-a)Vr^b+(c-a)V'^~(>log 



U yt — b+yc — b 



Let us put a = a, b = — 1, c = ; then 



y/t — b—v'c — b 



Z = § (t + l) 1 - 2 a \/t + 1 + alog ^ + T. 



C 



0+1-1 



