126 



Proceedings of the Royal Irish Academy. 



Deferring consideration of that part of (47) which depends on the value 

 of its first term at the lower limit, t' = 0, the other terms are. therefore, 

 after an ob\ious rearrangement, eq^ual to 



1) ~ A Jg 



*..(i')-^>..(A)^^(Q, ^-..(g)-yA )^^(,) 



B-\ 



D -\ 



- r !'^n(A)/i(<') - ^.-„{\)Mt')]eKt-ndt'. (53) 



Jo 



But, by virtue of equations of type (20), the terms after the first two are 

 identical with 



$ii(^) f ' eM' - ^■)Ut')dt' + $,1 (D) f eW - 0/,(^) dt'] , (54) 



■'o 'Jo ' - 



which will be sufficiently indicated for the general case by the form for two 

 unknowns, viz. : — 



Jo 



(55) 



On now bringing in the value of the first term of (47) at the lower limit, 

 and placing it on the right-hand side of the equation below, it foUows 

 that (46) may be written in the form 



B-X 



e\t 



Jo 



\'e-M'f,(t'}dt' 4>JB) 



= e^i 





B-\ 



-^,i(\)U+ \<p.^{B)- 



- 022(^)12/ ^ /y-) 



^56) 





D-\ 



, 92:i(,Aj 



< = o. 



Now multiply both sides by d\ / A (A) and integrate round the contour at 

 infinity. The first term on the left gives IrAxt ; the second gives zero. 



