52 Proceedings of the Royal Irish Academy, 



and (43 ), the vanishing of t^f is expressed by the equality 



/= 2i S [P log sin {wX"' (w. - v\)\ - ttX'' Q cot UX"' (v.'o - (f,)) + etc.] 

 + i{f{a)^f(a + it) -2/(10,)', 



^1 



tT' log sin { ttX-' {ir, - w) }fX«i) dtc, (44) 



a+iC 



which, when ir is substituted for ir^ after integration, constitutes an expres- 

 sion for i/ as a function of ir. The limit of the right-hand side for ^ -> + co 

 may be a comparatively simple form. 



With a view to investigating this limit, it can be verified that, for xp great 

 and positive, 



- log sm - (!/-^ - 2c) =- log (-|i)+^ («;„-«>)-- ^ ^ ^^P ( ~x~ ("" ~ ■^^») } • 



(45) 

 Let it be assumed that /'(ic) is such a function that it can be represented, 

 for great positive values of i^, by the series 



/'(«•) =2 7. exp ! - 27rtX-' (n-s)w\, (46) 



1-0 



where n is an integer ; and let it be assumed tliat this series is integrable, so 

 that, for \P great and positive, 



n - s 



i\ - . ^ y. 



-3=0 »=ntlJ 



/(jt-) = 7„w + K- 2 +2 -i^ exp I- 2!riA-'(n - s)w]. (47) 



As jf'{w)dw, taken round the rectangular contour, equals 'Z2niP, to which 

 integral the contributions of the sides (f> constant, and s/- zero, are respectively 

 zero and 2jr, it is necessary that 2ir - -ynX = '22-iTiP, so that 



> = 2,rX-Ml-2iP). (48) 



With the above assumptions, for i/. great, 



TT-' log sin S jrX-' (Wg - w) j f'[w) = E ^ C + ^(w,) + iX"' (Wg - w^f'iw), 



where E consists of exponential terms, C is a constant, and 



F[w,^ = - 7r-» 2 -^ exp (- 2Tri\-''{n-s)w,\. (49) 



H - S 



