Eggers 



where ± stands for 17 positive or negative. By partial integration regarding f 

 and i , observing the Laplace equation for G as stated in A and making use of 

 the fact that F = at the integration limits if C < , we then have 



ir I ^^ndS' --^ \\ F(^,o[0x(G^G-) + 0z(Gj + G-^) 



d^d^ 



^j F(^,0^(GYY + G;Y)d^d^ +^ |J0(G;-G;)d^dC 



-^ f 0F(^,O) (gJ + G") d^. 

 We can now transform the line integral, obtained previously, as 



-^ J (^xG-G^0)d7^ = - ^ r Lj^(G* + G-) -0(GJ + G-)] Fx(^,0) df 



(6) 



4t77o 



k /^(^•°' 



V^Xx(G' + G-) - 0(G^^+G^^) 



d^ 



and then combine all components, observing B for G, as 



= - ^ II F(^,0(G^ + G^)d^d^ + 4^ 11 S(^,77)Gd^ dv 



4777^ J_ ^ 



if. 



■f 



F(^,O)0xx(G*+G-) d^ - ^ I" F(^,0 



>Ax(G^ + G^) + 02(G^+G^) 



(7) 



d^dTj 



^x(^,V,0)G -4^(^,V,0)G^ 





+ >Axx(^-^'OG- 0x(^'-^.OG^ 



da d77 



f=x. 



f = x. 



^^[^[f. 



eF(^,0- ^ 



(G;-eFG;^) - (G;+eFG;^)] d^d^ . 



(8) 



If we here neglect the contributions of S^ and S^, the remaining expression, 

 proper behavior of &(X,Y) assumed, really makes this omission legitimate due 

 to the properties stated under F for the function G. 



654 



