Browne — On an Integral Equation proposed by Abel. 71 



second. The direction of integration round the contour is clockwise, and 

 we obtain 



B n (x) = xf (b * b, x + . . . + b n x") + S„ (x) 



= 0„ [x) + S n {x), 

 where S n (x) is an expression of the form 



2 r x«* (A nr + B nr log a; + . . . + N nr (log*)*), 

 a r being a root (of multiplicity s + 1) of the equation 



t«K(t)dt = 0, 



and the 2 extending to all the roots. Hence we can find an expression S„ (x), 

 containing a finite number of terms such that 



| B (x) - $ (x) - S„ {x) | < b*», 



s being arbitrarily small. But this is evidently impossible unless B(x) - <j>(x) 

 is itself of the form 



2r *"*■ {A r + B r log *+... + N r (log x)'), 



which is the same as saying that B(x) - <p{x) is of the form H{x). We have 

 proved therefore that the expression 



-If - 

 2iri J c 



X°- v(a) , 1 



(fa + 



l \y' 1 ' a ^(y)dy 



da 



r t«K(t)dt 2iri > D \-\ x t*K[t)dt 



includes all the solutions of the equation 



4>{x) = xp(x) + I ' K(t) <j>(tx) dt. 



J 



This expression contains linearly as many arbitrary constants as there are 

 roots (each counted to its degree of multiplicity) of the equation 



1 - I 1 t*K(t)dt = 0. 



6. The equation of Abel 



[ l G{t)f{tx)dt = g{x) 



is the same as 



. . xg(x) [ l G'(t) .. . 1± 



* {x) - G wm)* {tx)dt ' 



