Browne — On an Integral Equation proposed by Abel. 

 We have also 



P t«K(t) dt "a" {' y-l-« | ►, {y, t) 9 (ty) dtldy 



79 



Zirt 



Vt'KU) 



1 



da 



dt 



:£!>* u: 



>i(y,t)6(ty) 



dt\dy 



-I 



t°-K{t)dt 



t«K(t)dt 



Ida 



which can also be made less than ^Ax s , by choosing x sufficiently small. 



Hence | SO {.r) | can be made less than qAx s , and the convergence of 

 the series assured. 



The more general equation 



<p{x) - h (z) <p {fix) =\p(x) + K(x, t) <j> (tx) i 



J M 



is the same as the following 



\dt 



<f> (x) - \<p {fix) 



\p{x) + w (x) (j) {fix) + r){x,t)<p {tx) dt 



J u 



f K{t)${tx)dt, 



where A = h (o), w(x) = h {x) - h {o) ; this equation can be solved by a 

 series in a similar manner. 



9. I gave in my thesis some methods for showing the existence of 

 solutions of all the types of equations treated in this paper. These methods 

 largely depended on the approximation to the kernel K{t) by a polynomial, 

 from which a differential equation was deduced and solutions found by 

 successive approximation. The procedure was long and troublesome, and did 

 not apply to similar functional equations involving multiple integrals. 

 The new formulae can be extended to those cases. Let us take, for example, 

 the equation 



VV G(t, r)f {tx, ry) dt dr=g (x, y). 



Putting x( a; »2/) = f( x >y)dz, multiplying by x, and integrating by 

 parts with regard to t, we obtain 



£ (1, r) x («, ry) - 1 1 JL G {t, r) x {tx, ry) dt dr = xg {x, y). 



