Browne — On an Integral Equation proposed by Abel. 69 



The first of these expressions is equal to p (x). The second vanishes on 

 account of the already proved relations 



_1_ 



2-rri 



t« K(t) dt 



y- l -«<p{y)dy\ da 



1 (-1 



K^) K(t 2 ) . . . K(t r ) <p {t,t 2 ... t^dt,.. . dt r , 



2iri 



t« K(t) dt 



[Xa. 



y-l-a 



K(t) t(ty) dt 



dy)da. 



J 



For the last term, we know that we can take m such that the integral 



. D 



U K{t) dt 



da 



is less than a finite quantity Q. Then, if we call B n (x) the expression got 

 by substituting <p n (x) for <p (x) in R (x), we have 



R (as) - R n (x) = <p(x)- «/>„ (x) + J n (x), 



where 



J„(x) = - 



A- 



■m 



•• 



r 



i 



t*K(t)dt 

 



1 -J 



-°\L(y)-<l>»(y))[t«K(t)dt 

 L\ /Jo 1 



K{t)(^{ty) - ^ n (ty)\dt\ly 



D 





1 - 



t«K{t)dt 







da. 



We have 



If t«K(t) 



I J 



eft 



<p(y) - <p»(y) \ + 



^K(t)L(ty)-<l> n (ty)\dt\<e'xP, 



where e' can be made as small as we please. Hence, if we have on the line D, 



1 



then 



-fWio 



j 



dt 



l-WI <*^-xA a y-^S + rdy 

 ZQt'x^af - 5 



2tt (p - S) 



[10*] 



