so that 



in tJie Theory of Integral Equations 181 



8. A further extension is obtained when the kernel k includes 

 more than one parameter t; thus let 



/i (x) K [6 {x, ti , Q} dx = -f, (^1 , t,), 

 /„ (x) K [d {x, t„ t^] dx = yfr. (t, , t.^, 



J a.2 



\ fi (!/) « [^ {!/> f^ (^> ii> Q> V {x, ti , t)}] di/ 



J a, 



= -v/tj \/j, {x, ti, t^, V {x, ti, t.^\ 



and 



f Vi (//) « [^ y^ f^ (•'•' ^i> Q, V {x, t, , f,)}] dy 



= \/ro [^ {x, t,, ti), V {X, t,, Q). 



Now consider 



/i (•^-'O -^/^a [/A (*', ^1, t;), V {x, ti, t^)} dx 



= f V"i (^) ( I ' /3 (i/)/^ [0 [y, ti (,*•, ^x, t^}, V {x, t, , t,)}] dy) dx. 



■J tti ^ ■ fl2 ' 



This double integral is equal to 



if ^ {y/, /x (.r, t„ t,), V {x, t„ t.^} = e [x, iM (y, t„ t.;), V (y, t„ L)}. 

 Now suppose 



fl (X, t„ «,) = <^,-' [f{x) + (/>! {t,) + (/>! (^2)}. 



^ {x, t„ L) = </)-! \2f(x) -f- (/>! (^0 + </), (t,)} ; 

 then 6^ {y, /j.{x, ti, t,), v{x, t^, t,)} 



= </>-! {2/(2/) + 2/ (*■) + (/), (t,) + 01 (t,) + (/), (^0 + 0. (g). 

 This is symmetrical in x and y, so that we may write 

 /^(^■, ^1, ^2) = 0rM/3(*')+ 03(^1, 4)j, 

 /. C^-, t„ t,) = 0,-^ 1/4 (^0 + 04 (^1, ^2)}, 

 (a^, t„ t^ = g [f, (x) +f, (x) + 01 {t,) + 0, (QK 

 leading to 



5'{/3(^)+/4(i/)+/;(*')+/4(^O+03(^l, ^ + 04(^1, t.^\, 



