434 PROCEEDINGS OF THE AMERICAN ACADEMY. 



z 1 + m 2 , (1 + m) 2 . 2 m 



— = m -\ — log m log 



C 2 ° 2 °t— 1 



1 + m 2 , m 2 — 1 , m 2 — 1 



log 1- m log — - 



2 ° 2 ° — 2 m 



1 + ffl 2 , (m + 1 ) 2 . 4 m 2 



-^ — 1°£ — o ; m 1°S ^ 



2 e w 2 — 1 D 1 — m 2 



= m + -o- lo S "3s T _ m l0 S 1 ^2 + m lo g (< - ! ) 5 



z 1 + m 2 m + 1 , 4 m 2 



log (t — 1) = — 1 log - — + log 5 ; 



to v ' Cm 2 m & w — 1 & 1 — m 2 



— x 1 + m 2 , m + 1 . 4 m 2 



log (i - o = t> — i — ? — log — ^r + lo § -> — r- 



& v (7m 2 m ff m — 1 ° m 2 — 1 



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



v h ( 1 + m 2 . m + 1 . 4 m' 2 \ 



which is the required expression. 



When a = ?n = 1 



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



(d) 



co ■ 



co ■ — - — ■ Qo 



this case (d) reduces to 



— v 

 which is a correct result. 



4 IT h \ TV j 



Where a = — oo m = + v"^ 



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



near an infinite plate 



CO 



co ■ 



co oo 



(d) becomes : 



-[. + i( 1 + ^„,_ ¥l 4)]. 



