in the Theory of Integral Equations 183 



5. Let US next consider the case of three integral equations 



!'\f\{x)K{dU;t)}dx = f,{t), 

 J «, 



/; (a:) K [e {x, t)] dx = ^/r, (t), 



J a,, 



r\f,(x)K{d(x,t)]dx=f,(t). 



We have 



'yA''^)f.{OOr,t)\ir,{d{x,t)}dx\ 



= r Mx)^}r,{e{x, t)}f,{e(x, t)] dx\ (11), 



= fV;cr) ti {^(^'> 01 ir,{0(x, t)} dx ] 



if certain conditions are satisfied. For 



'"' f\{x)y^,[e{x,t)]ylr,[e{x^t)]dx 



= I '' ./; i-'^) f ' /. (z/) '^ [^ (>/> t)\ dy f V; {z) K\e{z, t)\ dzdx, 



and the equalities (11) will hold good if, for example, k (x) = x'^ and 



d[y,d{x,t)\.e[z,e{x,t)] 



is unaltered by the circular substitution {xyz). 



Now suppose that ^ 



e{x,t)^f{x)t-^ (12;; 



then [y, {x, t)] . 6 {z, d (x, t)\ =f(y)f(z) {x, t) 



Hence if k(x) — x'^ the relation (12) satisfies the conditions. The 

 generalisation to the equality of n integrals is apparent, and in 

 that case 



0(x, t)=f{x)t''-'^ 

 is a solution. 

 We have also 



/i (.'•) f; 1^ (*•, 0} ^3 {^ (.'<-■, t)] dx 



fb, 

 = f, (.'/;) yjfs {\ (x, t)} ^fr, {X (x, t)] dx 



J rta 



