Browne — On an Integral Equation proposed by Abel. 61 



As £ tends towards infinity, the left-hand side of this equation approaches 

 the limit 



/(«) 



K{t)I{tx)dt. 



Suppose B to be the line jj = §, where - 1 < S < y ; then the limit of the right- 

 hand side is equal to 



Lt J_ r+I 



|=oo lirl) -| 



ij 



s+ie -i-s-ie , , s 7 



x y ip{y)dy 



J 



idO, 



which, when we put y = xe~ r , becomes 

 Lt ± dO 



I = oo ZlT J -| J log 



Vt 





This is a case of Fourier's integral. Integrating first with regard to 9, we 

 find 



Lt 



J = co 2tT_ 



= Lt. 1 



| = oo ir 



• + oo gi T { _ g-i T { 



log- *T 



" + °° sin r£ 

 log- r 



e rS \p (joe-*) dr 



tS ^ (cee-r) dr. 



For < x < a, log - is negative ; also the value of e rS \p (xe- T ) for t = 



is i// (a;) . The function | e r5 ^ (#e- T ) | is absolutely integrable even when 

 the upper limit is + oo, since we have, for very great values of r, 



\erZ\p(xe-T)\ < Met*(xe-r)y = Mxi e T ^~y), 

 and S - 7 is a negative quantity. Hence we have 



/(»)- 



K(t)I(tx)dl = ^(x), 



which shows that (a;) = I(x) is a solution. 



3. We must now consider if the expression I(x) has a meaning or not. 

 We have 



1 



t«K\t)dt *=° u 



t*K(t)dt 



t*K{t)dt 



and we will prove that the integral 



i. 



1 



' ~ 2iri 

 is equal to the expression 



t«K(t)dt 



t*K(t)-dt 



x« I y l -«\[/(y)dy da 



& (as) - j* .' . . [ i"(*i) 2T(*») . . . -STfo) i£(*i 4 . . . t p x) dt, dt. 



dt p . 



[9*] 



