64 



Proceedings of the Royal Irish Academy. 



where d(r) is a function which is finite and continuous for all values of - 

 As A diminishes, this integral can be made less than 



where e' is arbitrarily smalL 



Taking all these results together, we see that the integral 



C t 



-\ ...f di 1 ...dt p K(t l )...K{f p ) 



""Jo Jo 



log 



sin ri 



^=-? *rS $ fat, . 



tit, . . . t„x 



tpxe-^) dr 



B m (x) 



da 



tends towards zero with ^ ■ 



5 



Hence we have I p = ^ p («) * 



-4. It remains to prove that m can be taken such that the remainder 



2nri J* l-^fK [t)dt 



has a meaning. As we have said, the integral 



r ! 



t a K(t)dt tends towards zero with — ,; 

 and since the line D does not pass through any root of the equation 



1 - \ l t*K (t) dt = 0, 



1 



l-l\t*K{t)dt 



is of limited magnitude all along B. Also the expression 



|z°j\r 1 "°'/'(y) < fy| 

 is less than 



the expression 



JLfrs 



| y-l-S*jdy = 



Mz* (17-° 



y- S 



Hence it suffices that m be taken such that the integral 



I'M 



c-jJJWco* 



* When the equation can be solved by iteration, the solution is evidently 



♦ W =♦(*)+*!(*) + ... + *> (*) + -. • 



