AND THEIR CONNECTION WITH THE THEORY OF INTEGRAL EQUATIONS. 445 



30. From this theorem several interesting results may be deduced. For example, 

 replacing K(S, t) by the series (27) in (19), we obtain 



(28) 



where the series on the right is Uniformly convergent. Again, if we write s = t in 

 (28), and integrate with respect to s between the limits a and b, we obtain 



*(,)cfo = S - l - ......... (29) 



n = 1 A n A. 



Provided that X is not positive, the terms of the series on the right are all positive 

 and less than those of the series 



which, by writing X. = in (29), is seen to converge. It thus follows that, lor X < 0, 

 the former series is uniformly convergent. Integrating. (29) between the limits and X, 

 where the latter is negative, and recollecting FKEDHOLM'S formula 



it is easily seen that 



D(x) = n(i-A) (xso), 

 n=i \ t 



since D (0) = 1. It now follows that, ;is the right-hand member of this equation is 

 an integral function of X, we may drop the restriction X:S 0. We have thus expressed 

 D (X) as an infinite product. 



Finally, we may remark that if | X | is less than the least of the numbers X 1( X 2 , . . . , X n , . . . 

 the right-hand side of (29) may be expressed as a power series in which the coefficient 

 of X m is 



ij,. 



= 1 A n 



Also, by employing NEUMANN'S expansion for K A (s, t), it is easily seen that the 

 coefficient of X on the left is 



( b 

 K m+l (s, s) ds, 



J a 



where in the usual notation 



rb rb rb 



K m+i (.*') t) I I K \ y > *'i) f(#ij *'a) K (*' t) OWjCWj... rt*' m (Wi i), 



J a J a a 



and 



KI (s, t) = K (s, t). 



