182 Major MacMahon and Mr Darling, Reciprocal Relations 

 which is symmetrical in x and v/ ; and hence it follows that 

 I J[(x)ylrn {/m(x, ti, L), v{x. t^, ig)} dx 



■ ' a, 



= I fo,{x)'\^i {/"-(*■> t\, ^2)) v{^> ^1) ^2)) dx. 

 As a particular case we may write 



^L {x, t, , L) = cf^r' (./X^O + ^1 (A) + ^1 (^2)}, 



V (X, t, , t.;) = (/),-' {/(*■) + 02 (^1) + C^2 (4)}, 



(x, t„ Q = g {2/(*0 + 4>, (t,) + 4>, (01, 



and again 



^l{x, t„ Q = ct>-' {^,f{x) + (i>(Q + cj)(t,)], 



{x, t„ Q - ct>-' {/(x) + (^0 + (/) (t,)}, 



the case where /x i^ y and each resembles as much as possible. 

 It is evident that the case in which the kernel includes any number 

 of parameters may be treated in the same manner and presents 

 little difficulty. 



4. The method may also be extended to double integrals. 

 Thus let 



/i {^> y) K^ [^ (*'. y> ii> 4)1 dxdy = f^ (t,, to), 



J »! J a,' 



/■2 (^-^ y) « [^ (^, 2/. ii, 4)1 dxdy = -f . (^i, 4) ; 



[b, rb,' 



then / /i {cc, y) i/r^ {/x (^■, y, t^, 4), ^ (a-', y, 4, 4)} c^^'f^^/ 



J «, >/ a,' 



6/ 



/2(^, 2/)'fi l/^(«> y. 4, 4)> ^(^S ^» 4, 4)1 dxdy 



if ^ {^r, w, yu, (a;, y, ^j, 4), v {x, y, t^, t^)] 



= [x, y, IX {z, w, ti, t.^, V (z, IV, ti, 4)1- 

 If A, B, G, D, E be functional symbols, one solution is 

 ix{x, y, t„ O = A-' [B{x, y) + C {t„ t,)] 

 v{x, y, t„ t,) = D-^[B{x, y) + E(t„ t,)} 

 (x, y, t„ t,) = B (x, y) + kA (t,) + ^D (t,). 



