Expansion Problems Associated with a System of Integral Equations* By

may be reduced to a single linear integral equation whose kernel is defined on a^x^a+n(b — a), a^s^a+n(b — a). Greggi§ considered a system of the form (1) and by use of the transformation introduced by Fredholm showed the form of the resolvent matrix for the system; for the symmetric system where Ku(x; s) =Kn(s; x) (i,j = \,2, ■ ■ ■ , n) he also stated theorems analogous to those proved by Schmidt|| for a single integral equation with symmetric kernel. System (1) also comes under the class of systems treated by Plâtrier.^ Weatherburn** has treated the system (1) without using the transformation introduced by Fredholm, but by vector method throughout. He states all results for the case « = 3, but his method of procedure is equally applicable to the general case. In the present paper a special system of the form (1), to which is applied the term "definitely self-adjoint," is considered and the existence of a countable infinity of real characteristic numbers is established, together with expansion theorems in terms of the characteristic solutions of the system of integral equations. A definitely self-adjoint system of integral equations includes as a special case the symmetric system with closed matrix kernel. It also includes the system of integral equations to which a boundary value problem for a system of ordinary linear differential equations of the first order which