,/ X r' 'm(x')\nr dx' ^ [/ -.2 21^'^ 



<t>(x,y)=\ where r = [(a; - x )'' + y^J 



[3] 



, . d<i> If' m(x)[x — x]dx [a] 



U(X,y) = ^- = -^T— \ J-/ ^T2~, 2T~" "-^ 



dx 2n .Lc [(x- X r + y ] 

 6ij 2-n}-c \^%-xY +y^] 



It may easily be shown that 



M(a;,0)=-- [6] 



27r j-c (z -X ) 



and, using the substitution tan p 



, „. m(x) 

 i;(x,0)=-Y- [7] 



For the integral in Equation [6] Cauchy's principal value must be taken. This is true 

 for all improper integrals appearing in the present analysis despite the omission of specific 

 symbolism to that effect. 



The mixed boundary conditions on the a;-axis will be satisfied if 



m{x)=2U, ^^{x) -c<x<0 [8] 



dx 



dvo / .\ , ' 

 — + t;— —, 77" = —^r- 0<x< I [9 J 



and 



27rJ_c {x — x') 2;rJo (x—x') 2 



where Equation [8] has already been incorporated in Equation [9], which is an integral equation 

 for the remaining unknown part of the source distribution. The problem may then be reformu- 

 lated: To find m(x) such that 



dVo 



fl mlx')dx' r° TJ^^^^dx' 



and 



