80 



Proceedings of the Royal Irish Academy. 



Now, putting 



'y C x fy 



<t> (», y) = x (*. y) ty = f( x >y) ^ %. 



Jo J J 



multiplying by y, and integrating by parts with regard to t, we get 



G{1, l)<p(x,y) - £ | £(l,r) <p{x,ry)dT -£ | 0(f, 1) ${tx,y)dt 



(•i n 32 

 + J o J o 9^" ^' ^ *^' T ^^ ^ = ^ ^ ^ 

 Supposing 67 (1, 1) =)= 0, this equation reduces to one of the form 



<p («, y) = ^ (•*•> y)+\ kit) $(tx, y) dt + \ a'O) 0(*> r v) dt 



Jo Jo 



+ (' [ K(t,T)^(tx,T-y)dtdT 



J J 



By analogy we can at once write down a solution as follows : — 



sp-yfi «-i-a <o-i-p ^j {u, v) du dv 



*(«.y)--s? 



4l- 



rfa f//3. 



-[ t*k{t)dt-\ l T*K{T)d T -\ l [* t^TPKy^dtdT 



Jo Jo Jojo 



To explain this formula, we suppose that 



\^{x,y)\ < Mxfy, 

 M being a constant, p > - 1, <i>-1, < x < a, < y < I. Putting 

 a = ^ + iij l5 /3 = £, + iij 2 , 



we suppose that .D is the infinite straight line £, = 8, in the plane of a, and A 

 the line £ 2 = S 2 in the plane of j3 ; also - 1 < S, < p, - 1 < S 2 < n. The 

 integrals on D and A are taken in the same manner as the integral on D has 

 been hitherto. We choose the line D so as not to pass through any of the 

 roots of the equation 



k{a) ^V t-k(i)dt = 1. 



As before, with very general assumptions about k it), it may be proved that 

 these roots are finite in number. The function | 1 - k (a) | has then on 

 the line D a minimum which is not zero ; and when a is taken anywhere on 

 this line, it may be proved in the same way that the roots of the equation to 

 determine /3 



l-j(a)-l(j3)-f(«^) = 0, 

 where 



j(/3) = r^(r)rfr, K(a,(i) 



t*rPK(t,T)dtdT, 



are finite in number. As a varies on D, these roots will describe a finite 

 number of curves. We must draw the line A so as not to touch or pass 



