Browne — On an Integral Equation proposed by Abel. 75 



This gives us 



2-jri D„ 



r=o 2iri 



1 - A/u a 



da 



J On 



T 



^(y)dy 



da. 



Let us take the first integral of this series, namely, 



2^1 



x a 



/>„ 



y-l-a ^ (y) dy 



8«. 



We have already shown, supposing 2£ to be the length of D n , that it is 

 equal to 



1 

 2^ 



+1 



IB 



dr 



log- 



c t(s+»9) ^ (%e-r) 



'+<*> sin £r 



e^S if, (xe-r) dt 



log- 



a 



x y (•+» s in g r 



*" Jl X T 



log- 



6 t(S-7) p (x-, t) rfr, 



where p (x, r) is limited for all values 



^ x < a, log - :£ r ;< + oo , 



The quantity 8 - 7 is negative, hence 

 x y f+ co sin £r 



«t(8- 7 ) p (^ r ) di = J* Y [p (.r, 0) + e] 



= H( x ) + \ ex > 



where e approaches zero with -^ , independently of x. 



s 



The other part of the integral, 



"° sin £t 



xy 



77 



e r{S-y) p [x, t) dr 



log' 



presents difficulties as * approaches zero (for then log - ->- - 00 ). We sup- 

 pose, however, that ip (x) satisfies Dirichlet's conditions for < x < a ; then 

 \p(xe~ T ) and p[x,r) will always have a limited number of maxima and minima 



B.I. A. PBOC, VOL. XXXII., SECT. A. [11] 



