275 



Further K {c,}/) be a s^'ininetrical kernel foi- a < .i">b and a < y < h 

 and this kernel be hvo (inies continuouslv differentiable with respect 

 to ./; (and of conrse also with respect to y). We assume the identity 



LyK{x,y) = L,,K{x,y) . .....( 1 0) 



Then, i(j(.r) being a conlinuous function in {<t,b), it appears liiat 



b b 



a r rbK{x,y) 



— I A {x,y)^(y)dy = I ^ My)d?f 



d.v J J Ox 



a a 



and 



b b 



d' r rd'K(x,y) 



~ j K{x,y)x\,{y)dy =. J - y ~- ^ivYhh 



II a 



as may be sho\^n bv means of integration of these equations with 

 respect to .r, as this integration, on the assumptions made, may i)e 

 effected under the sign of integration. In the same way it appears, 

 that every characteristic function <{(-«) of A' (.r,_v) may be different iated 

 twice. 

 We put 



bK{iuy) _ dK(x,ii) 



the sign of substitution relating, as always in the sequel, to >;. 



We now have the following 



Theorem I. If (f {.v) be a chartfcteristic functloii of K (.»■, //), 

 i.e. ivhen 



L{x,y ) = 



/>(>/) Kix^ri) 



K{r,,y)\ 



(11) 



cf'{x)=iX \K{x,;i)(^{y)<hi, 



(12) 



?/y huve 



AM-^) = ?.lK{x,y)L„fp{yyiy >.'lA{'V.>/)r/iy)dy . . (13) 



fi a 



Proof. We have 



b b 



h,.qix) = ?.i A,-h'{x,y)rfiy)dy =).i\A:,K{x,y)\ rfiy)dy 



ri a 



from (10). Now foi* two arbitrary, twice continuously differentiable, 

 functions the so called formula of Grken is valid ; in the case 

 of functions A' (.'",//) and 'f{y) it takes the form 



ƒ 



\K{x,y)L,/p{y)-fi {y)L,iK{x,y)\dii r= 



P(\) jA ('N^)'/ (>/)- ^ '/(»i) 



