AND THEIK CONNECTION WITH THE THEOKY OF INTEGKAL EQUATIONS. 441 



Hence, as the latter converges uniformly for a ^ s < b, a t ^ b by what has just 

 been said, the result follows. 



26. Let us denote the sum of the first m terms of the series on the right of (19) 

 by S m (X ; s, t), and the remainder after these terms by R m (X ; ., t). We have 



. 



n = i A n A 



and hence, keeping m fixed, 



^^(X ;.,*)- -3. 



A ^- no n .= 1 A n 



Thus, since (19) can be written 



K A (s, t) - R m (X ; s, t) = K (s, t) + S m (X : *, t), 

 we obtain the equations 

 L[K A (*,0-IM*;M)]= L pK A (,,)_E te (X;*,e)]K(,0-2 !fcLi ^ fc ^. (20) 



A-*- ic \ ----* ,1 = 1 A B 



Tliis relation holds for any continuous function K (s, t), but we now add the further 

 limitation that the function shall be of positive type. Then, since 



777 TT , 

 A n ^A n Aj 



we shall have 



R m (X ;*,*)< 0, .......... (21) 



for each negative value of X. 



Let us, in the next place, investigate the values of K A (*, s) for negative values of X, 

 it being supposed, as above, that K (s, t} is of positive type. If 6 (.s) is any continuous 

 function defined in the interval (a, b), it follows from (19) and the theorem proved at 

 the end of the preceding paragraph that 



f" f K X (*. t) e (s) e (t) dsdt = f f K (s, t) e (*) e (t) d s dt + s - - \f ,/, (*) e (*) d*T. 



JJ JoJo 11 = 1 A,, (X n A) |_Ja J 



Recalling HILBERT'S theorem, it will be seen without difficulty that this reduces to 

 f f K A (s, f) 6 (.s-) (t) d* dt=t^- ["[V (*) B (s) dJ. 



J(lJa n = l A n A |_J J 



VOL. CCIX. A. 3 L 



