434 PROCEEDINGS OF THE AMERICAN ACADEMY. 



z 1 + m^ ^ {\ ^ mY , 2 m 



— = OT H — log m los 



C '2 « 2 ° t-\ 



1 + m^ m^ — 1 m^ — 1 



2 ® 2 ^ — 2m 



1 + »^^ (w+ 1)2 , 4m2 , , 



= ^ + — ^ — log ^ r - "* log 1 2 + "^ log (< - 1) ; 



2 ° 7?H — 1 1 — m'' 



log (« - 1) r= — 1 Z log --^- + log ; 



* ^ ' Cm 2 m ^ m — \ ^ \ — m^ 



— X \ -{- nr m -\- \ 4 m 



■2 



\og(l-t) = -—-\- -— — log + log 



C?n 2 m ""* m — 1 ' ° m'^ — 1 



Substituting this value of log (1 — f) in (c) we obtain 



I' r h f 1 + 77i~ m + I Am'- \ 



^ = - 471 L^ + ;^ (,' + -^^ ^°s ^^m - ^^g .^Tzrr j. 



which is the required expression. 



When a =: m = 1 



which gives the case of a semi-infinite plate near an infinite plate 



(^1) 



00- 



CO cc 



this case (d) reduces to 



which is a correct result. 



Where a = — co m = + ■\/cc 



this should give the expression for the electricity on a square corner 



near an infinite plate 



CO ' 



CO 00 



(d) becomes : 



— r r h / V^ \ 



