1921-22.] Linear Differential Systems and Integral Equations. 45 
It follows that I(;r) satisfies the system (1, 1') for the characteristic 
number <x,.. Let us assume for the moment that l{x) is not identically 
zero, that is to say, that Ui{s) is not orthogonal to K(a?, s) in (a, /3). Let us 
confine our attention to the more general case in which the index of 
compatibility is 1. The system (1, 1') has then no solutions other than 
multiples of for the characteristic number a^, and therefore l{x) is a 
multiple of tCf(x). In other words, u^{x) satisfies the integral equation 
u{x) = J 1^(^5 s)ic(s)ds ..... ( 4 ) 
for a particular value A,, of A. This integral equation is therefore satisfied 
by all solutions of system (1, 1') which are not orthogonal to K(ir, s). In 
particular, if the nucleus K(rr, s) is closed, the integral equation is satisfied 
by all solutions of the system. We may go further and define a con- 
ditionally closed nucleus as one not orthogonal to functions of a specified 
type. If the nucleus satisfies such a condition, the integral equation will 
be satisfied by all the solutions of (1, 1') which are of that type, e.g. by all 
the even solutions of the system. 
3. Conditions that a Solution of the Integral Equation 
SHALL SATISFY THE DIFFERENTIAL SYSTEM. 
The theorem now to be proved is the converse of the theorem of § 1. 
The proof depends upon the Hilbert development of the nucleus K(^c, s) 
in terms of the fundamental functions. 
It may easily be verified that the iterated nuclei Ki(cc, s), K2(a3, s), . . . 
satisfy the same partial differential equation (2) and the same boundary 
conditions (2', 2") as the original nucleus K(x, s). Let us assume that 
K.{x, s) is symmetrical ; the treatment of a skew-symmetrical nucleus would 
follow on similar lines. 
Let the development of K(a3, s) be 
K(x, s) = 
4>r(x)cfir(s) 
+ 
then the expansion of the iterated nucleus of rank p — 1 will be 
K^_i(;r, g)= Ap + 
p = 2, 3, 4, . . . 
(5) 
(5a) 
Let us assume that the series for ~K(x, s) and its second derived 
series are uniformly convergent throughout the square a^x^ 
the same will be true of the iterated nuclei. [If this condition be not 
