Browne — On an Integral Equation proposed by Abel. 83 



Finally, the integral 



Z>JA[1 -*(«)]* [l-x(P)f 



is equal to 



VV E{t,r)F{tx,ty)dtdT, 



J J 



where we have 



, r {tx)°- (ryY \ | ' u-i-<> v-l-Pip (u, y) du dv 



F{tx,ry)=\ ' hh data, 



J"J A [1 -*(«)]* [l-*G3tf 



which is proved by the same method as before to be a finite function of tx 

 and ry. Hence the existence of Lt U n (x, y) is established. 

 When the equation 



<t> ( ;r . y) = ^ 0> y) + 1 h (0 <p ( tx > y) dt + \ k 0) (*, ^) d T 



Jo Jo 



+ [' f ' K(t, t) <6 [tr, T y) dtdr 



Jo J 



is derived from the equation 



il G(t,r)f(tx,r !/ )dtdr = ff(x y y), 



If, JO 



then from the solution given for <j> (x, y) we obtain by the same method 

 as before 



-. & r r x^^yP+1 j u- 1 - a v-' i --^g('u,,v) dudv 

 f r r y \ = t— 2 J o J " rf a rfft 



47r 2 S^%J a jA (a + lj(i3+ l ) jN'%a 7 se(i J r)rf^r 



It is clear that to the solution for ^ («, y) we may add expressions of the 

 form 



P Q8) /Afr* 



a;<* 



yfl 



Jc, 1 -ft(o)-x-O) -K{*,P) 



q(a)x a da 

 C,l-k( a )-K((3)-~K(a,f3)' 



(a, fi) X" y& da dfi 



JcJdl -hla) 



■(al-j(/3) -JC( ,3) 



ft, ft, ft, ft being any closed curves in the planes of a and j3. an d p (/3), a (a), 

 and r (a, /3) arbitrary holomorphous functions ; a is arbitrary in the first 

 expression, and j3 in the second. I have not been able to discover whether 

 the addition of these expressions gives the complete solution. They are 

 evidently solutions of the homogeneous equation 



f(x,y) = \ Jc (t) (tr, y) di +\ K(r)<i.{x > ry)d T +\ K\t,r) (p[ts,ry, dtdr. 



Jo Jo Jojo 



R.I.A. PROC, VOL. XXXII., SECT. A. [12] 



