76 Proceedings of the Royal Irish Academy. 



for Oiri log - • In this interval M is the maximum of | p (x, t) \ , and 



v\s-y 



- j is the maximum of e T i s -y). Hence this part of the integral is equal to 



f \p (x) + cxy ( 



where c is a finite constant independent of cc. Hence we have 



2^ 



X a f y-i-«i(y)dy 



Uo 



1 . 



da = \p (x) + e'j-5, 



where t approaches zero with ■=, independently of x. And in general 



=+» i 



2° A y<ji r x> \°r^(y)dy 



da 



= i£(ar) + A^ (ju.r) + . . . + A r i/- (pTx) + . . . 



+ t'x& [i + ,v + . . . + (\psy + ...], 



which, as D n becomes infinite, tends towards the convergent series 



-,// (x) + X\fj (fix) + . . . + X r \p (/j. r x) + . . . . 

 If, on the other hand, we have | \fi7 | > 1, then will | A^u" | be > 1, and 

 the expression 



1 f x'\° !/- l - a $(!/)dy 



%iri J Dn 1 - A/*« 



may be written as follows : — 



da 



Ti 



,=-1 2in 



A r (ji r xy 



I 



y-l-« ^ (y) dy 



da. 



"When x lies between a and /«*, all these terms vanish in the limit, since 

 every n r x is greater than a. And in general, when n s a > x > /i'* } a, we have 



Lt i f x a ] a r l - a 4>(y) d y 



£>„= x 2iriJ Dn 1 - \n a 



da 



= A"' $ (ji-'x) + A" 2 1/» (w" 2 *) + . . . + A~» i£ ( M "V). 



This function is discontinuous at the points fia, fi 2 a, etc. 

 We remark incidentally that the expression 



i c x" I yi-*if,(y)dy 



gives a solution of the equation for <f> (x), 



<p (x) = i/, (a;) + Atf. (,ua;). 



