AND THEIK CONNECTION WITH THE THEORY OF INTEGRAL EQUATIONS. 421 



may be looked upon as K(S, t) 6 ltri (s ; Si) 0, ir ,(t ; s l )(dsdt) taken over Q, or, as it is 

 usually written,* 



J *(*;*) 0.^(8 ; s,) e.^(t ; ,) (ds dt) 



and, from what has been said in the preceding paragraph, that portion of the latter 

 which arises from the part of Q exterior to d n is zero, while that arising from </ u is 

 simply 



[ K (s, t) (ds dt). 

 Jn 

 We have, therefore, 



f f K(s,t)0., 1l (s;s 1 )0. t ,(t;s l )d8dt 



J a J a 



= f K(*,t)(dsd*)+l K(M)^;*iK,(;*i)('fr< / 0- . . (2) 



Jg Jil 



Again the total area of d n is 4(2Tj + e), and so, if M is the maximum value of 

 | K (s, t) \ in Q, we have 



If K(*,t)0. t ,(s;* l )0, i ,(t; 



I J <ln 



also the remaining integral on the right-hand side of (2) can be replaced by 



rs, + >i r^+Tj 



K: (.s' ; t) ds dt, 

 "' '' 



wliich is evidently equal to 



f 7 * I" 1 



K (^i + u, s l + v) du dv. 



J -T) J -n 



Thus it follows from (2) that 

 [ f K(s,t)0 tt .(s;s 1 )0 t M;s 1 )dsdt-F f K (s l + u,s l + v)dudv <4(27 ? + e)M. . (3) 



|JoJa J_,,J_, 



Now let us suppose it possible for K (s lt s^ to have a negative value, say a ; then, 

 because K (s, t) is continuous, we can choose a value of 77 so small that 



for all values of u and v whose moduli are not greater than 17. We shall therefore 

 have 



n n 



K(S I + U, St + v) du dv > 2r) 2 a. 



J - 



Recalling our hypothesis that K (s, t) is of positive type, it follows from this and 

 (3) that 



Of. HOBSON, 'The Theory of Functions of a Keal Variable' (1907), p. 416. 



