420 



PROCEEDINGS OP THE AMERICAN ACADEMY. 



9 

 ? 



— (<^,=P- «/>,=_,) = 4^2 log (- ip)' 



— V 



X \ —2b 



+ 



4 7rnCA ' -lb 



~ 4 TT // L"^ TT V 2i ylj" 



If & is infinite this becomes 



+ V 



(^ + ;)• 



^ 47rA 

 If h is unity the equation becomes 



V ( . h 



^ 4 TT // \ 2 TT 



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 



B 



t = 



< = 



i = + 00 



t = a 



D A 



The transformation for this case is 





(-'^> ' 



which integrates into 



^=l(t-by+2{c-a)Vt-b+(c-a)Vc-b\og^-^=^^^^^+T. 

 ^ yt — b+yc — b 



Let us put rt = a, 6= — 1, c = 0; then 



^^='i(t+ \y-2aVt+ I +alog ^^J"J '^ ^ +r. 

 t^ V ^ + 1 — 1 



