1921-22.] Linear Differential Systems and Integral Equations. 43 
V. — On the Connexion between Linear Differential Systems 
and Integral Equations. By E. L. Ince, D.Sc., Lecturer in 
Mathematics in the University of Liverpool. Communicated by 
Professor E. T. Whittaker, F.KS. 
(MS. received October 29, 1921. Read January 23, 1922.) 
§ 1. Introduction. 
This paper summarises the results of an attempt to extend the theory upon 
which the relationship between linear differential equations and integral 
equations is based.* The case in which the nucleus K(;r, s) of the integral 
equation arises as a Green’s function is well known ; the nucleus is there 
characterised by its having discontinuous derivates when x = s. The 
method here dealt with is virtually an extension of Laplace’s and 
analogous methods for solving linear differential equations by definite 
integrals, and leads to nuclei which are continuous and have continuous 
derivates for x = s. 
§ 2. The Relationship between Linear Differential Systems of 
THE Second Order and Integral Equations. 
Let us, in the first place, consider the homogeneous and self-adjoint 
linear differential equation of the second order 
+ Ctu = 
dx 
du 
dx 
+ [(X = 0 
( 1 ) 
in which k{x) and l{x) are defined for and are such that k(8) = k(a) 
and l(/3) = l(a); let us adjoin to (1) the periodic boundary conditions 
u(/3) = u{a)^ 
u\/3) = n(a)J 
The system (1, 1') is in general incompatible, j* i.e. admits of no solution 
not identically zero. There may, however, exist a sequence, finite or 
infinite, of characteristic numbers a^, a^, . . . such that the system becomes 
* An abstract of the theory as it stood in 1910 is given by Bateman, “ Report on the 
Theory of Integral Equations,” British Association, 1910, § 21. References to later work 
are given in footnotes in the present paper. 
t For the general theory of the compatibility of linear differential systems, see Bocher, 
Legons sur les methodes de Sturm (1917), chap, ii. 
