68 



Proceedings of the Royal Irish Academy. 



The expression 



,(x) = I(x) = — 



xp. \° ir Y - a ^{y)dy 



da 



is therefore a solution, in the interval < x < a. of the equation 

 i(x) -*(c) + llK(t)4,(tx)dt. 



We remark that to the hypotheses already made about i// (x) and K(t) 

 we must add that these functions are absolutely integrable and satisfy any 

 of the ordinary conditions for development in Fourier series in the intervals 

 (0, a) and (0. 1) respectively. 



In order to have the general solution, we must add to I (x) an expression 



of the form 1 X" V (a) da 



27i J c T^]lt«K(t)dt' 

 where C is a contour enclosing all the roots of the equation 



1 -l\t«K(t)dt = 0, 

 and lying entirely to the right of the line q = - 1, while v a) is a function 

 which is holomorphous, but otherwise arbitrary. 



5. The expression H u) + I'x) includes all the solutions of the integral 

 equation which satisfy the same conditions as ip (x). To prove this, we have 

 only to show that for any function <j> (x) satisfying those conditions, the 



expression _ "<• r fj l 



<p(y)- ^K{t)<f>{ty)dt^dy 



^(*)^ 



i#-i. 



da 



fr K{t)dt 



differs from <p (x) only by an expression of the form H(x). 



We suppose that </> (x) is of the form xfr\ [x], where »? (x) remains less than 

 a fixed number N for < x < a, and p - $ is positive. We can approximate 

 to jj (x) by a polynomial b u + b t x + . . . b n x", which we call »j„ (x). Putting 

 xpij„ <Xj = <p n (x), the difference | <p (x) - <p n (x) | can be made less than tx?, 

 where t is arbitrarily small. We have 



_1_ i" 



'2-n-i J j) 



R (x) = 



t*K(t)dt 



2rri 



1 



^>y 1 - a <p(y)dylda 



c* | ^ y 1 — [ * (y V t*K[t)d£- I"' K(t)<p(ty)dt 

 f ' 1 m i i ° r 



t*K(t)dt- 



dy \ da 

 l K(t)4,(ty)dt 



dy 



•j: 



- da. 



t°-K(t)dt 



