78 Proceedings of the Royal Irish Academy. 



It can be shown that, when x is taken sufficiently small, the solution of this 

 equation is given by the series 



<£ [z) = <p (z) + fr (x) + . . . + $ n (x) + . . ., 

 where 



*,,(*) = £$_,(*), 0=1,2,3...). 



We can choose the line D and the domain of z such that 



I *,(*)| < MxS, 



J/ being a constant. For the convergence of the series 



it suffices to show that if | 6 (.r) | < Az s > A being a constant, the domain of 

 x may be chosen such that 



I S6 (z) I < qAzS, 

 where < q < 1. 

 We have 



"W-53 



J 1 + I t«K(t)dt + .. . + 



I j. 



t"K(t)dt 



t"K(t)dt\ 



D { *■ J" y- 1 - [£ u (y, t) 8 (ty) dtjdy]da 



Jo V 



+ aw*]/, 



£ hx-(*j *]"•»£ r 1 - [£ 1 (y. 9 e to) #}& 



<?a. 



1 - 1 t*K'4)dt 

 Jo 



where we have 



-H. (■'-) = f l n far, 0(te)<ft, 



H r (x) = J 1 JT(<) H T j, (to) <&, (r = 1, 2, . . . m - 1), 

 and m is taken great enough for the convergence of the integral 



[ \[ X UK(f)dt\ m \da\. 



We have 



*=o«M> = 0; 



hence we can take z small enough to have 



H,(x) + H x {x) + ... + H n _ l {x) 



<lAzK 



