82 



Proceedings of the Royal Irish Academy. 



11. We have still to establish the existence of ~LtU„(x, y). We suppose 

 for simplicity, though it is not necessary, that the functions k(t), k{t), K(t,r) 

 are finite in the domain < £ < 1, ^ t •*= 1. Then, as we have already 

 proved, the integrals 



]"J*(«) ' \da\. } a Z(j3) "l^l 



have a meaning ; and the same may be proved in a similar manner for the 

 integral 



We have 



\ j) ^\K(a,{S)\*\da\ | dp | 



1 - h(a) - Kifi) - K (a, /3) [1 - k (a)] [1 - K (/3)] - R (a, /3) 



where 



putting 

 Hence 



J2(a,/3) = *(a)*(0) + 7T(a,/3) = V H *« i» R{f,r)dtdr, 



J J 



5(*,t) =&(0 + JT(t)+ 7T(*,t). 

 1 1 5(a,/3) 



l-/=(a)-ff(/3)-^(«,/3) [l-A-(a)][l-X(/3)] [l-A-(a)]=[l-^(/3)J- 

 [1 - k (a)] 2 [1 - K (0)]' [ I - A (a) - X :/3) - K(a, 0)] 



Now, putting 



if, (a, |3) = x" y ? I m-1-» tri-3 1/, (m, f) d?t dv, 



J o J o 



we obtain, by former methods, 



Ja 1 - a-(/3) ^ 

 0(m, ;/) being a finite function. Hence 



«-!-« 6 (it, y)du, 



«-!-« (m, ;y)d« 



„ _*l a 'PJ d„J(l = J o da, 



which integral exists. 



Again, since the integral 



Li \^(a,Pi\'\da\ | d0 | 



has a meaning, there is no difficulty about the integral 



[R(a,/3)?j,(a,fi)dad& 



u. 



* [1 - ft a;]= [1 - I-(0)P [1 - k (a) - K{P)-K(a, ji)] 



