22 Proceedings of the Royal Irish Academy. 



The collection of all parts of the contour integral and the equation of 

 their sum to zero yields the equality 



2t|log P(Z,)- log A"; ■ 2ip log P| ; + Jar- logF{£, £, d log <?(£) = 0. 



which may also l>e written (15) 



2t!log : -log.fir} + 2»p log *•(>,&) 



- | JT-' lOfi / ■.; : . / 1 g r i l\rr~' log P 5 £ |<» = 0. (16) 

 £ = -« j = -co 



9. Let 2 the complex conjugate to sq, and let a function £(£, £ ') be 

 defined by t lie equation 



- - : •-*>*> ,, 



this being the fonn appropriate when Z is real and £ Mini let the 



function be defined i"i the half-plane on the posili the real axis of 



that the continuation from t lie above form shall be by paths which 

 never cross th>- axis and never quite pass through tin- points i a. Then the 

 form appropriate to Z real and C ! > »' is 



«.' + a' + «*-o 1 )* (£."-**)* 

 This function G(t,Z») has important relations to the function F(£, £,) in 

 the pari -•• when . When Z is real ami h : < "'. the complex 



... When . and £• > « : , the complex 



conjug 



. - i has no mines or branchings in the relevant half-plane 



the irrelevant side of the real axis) and so log £(£,£,') has no 

 bran.. ,ni half-plane. Therefore the integral 



- - - (19; 



must have tl when taken round a contour differing from that 



shown in only by the omission of the nowunneei ity and cut. 



•Inst a» in the case i ered in the preceding article, the contribution 



die integral made by the gn lcircle has zen> limit, and the 



infinitesimal Bemicircle round _ - . . Thus the result of 



the contour integration, in its limit fern 



- ' 



. . . • . . - - . . 0. (20. 



. -« 



