Expressiny Solutions of Laplace's Equation. 203 



singularities tire the roots of 



f2-'2tfM-\-l = 0; 



i. e. t = H'± n/z^' — I? 



which are branch-points. 



The function t-/jL+ ^/t''-'^fJit + l is not zero anywhere, 



unless /JL= +1. 1 7 7 u 1 



Since the con tour is supposed carri ed round b ot/i branch- 

 points t=^±y/JiF^^, the function ^t'--2fit-\-l resumes its 

 initial value after describing the contour. 



Hence 



^ j_r e d 



i^i] ^t-,+^r'-2>.t+vr^^ It it-,^Vt'-:^t,+m. (31) 



Integrate by parts : then because the contour goes round 

 both branch-points the integrated terms vanish, and the 

 integral 



= A_-r p___./f. . . . (32) 



Let us differentiate this with regard to yu- in this new form: 

 n\ d , - "^ 





^^^'- (33) 



But from (31) and (32) 



■•■ (^ri((^-^)-^"«'^))= ^^g^(^--r=?p:-(M); 



.-. (1-^2)-— p:+>(^)= ^-1(1-^^')-^' p;«(^) (34) 



