V.—On the Connexion between Linear Differential Systems and Integral Equations

Abstract

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(x, 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.