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The method developed by FredholmI for the solution of a non-homogeneous linear integral equation is comparatively simple in case the corresponding homogeneous equation admits no solutions not identically zero; in terms of the determinant and the first minor for the kernel of the given equation a resolvent function is defined, by means of which the solution is readily found. If, on the other hand, the homogeneous equation admits non-zero solutions, the development of necessary and sufficient conditions for the existence of solutions of the non-homogeneous equation by Fredholm's method, while similar in character to the simpler case, involves the introduction of minors of higher order, thereby necessitating somewhat tedious and complicated algebraic manipulation. Other methods of approach have been devised in this case—for example, Schmidt's detour through the easier theories of equations with kernels which are symmetric, which remain small in absolute value, or which consist of the sum of a finite number of terms of the form <p (x) $ (y), to that of the equation with general unsymmetric kernel.\ ,Such methods, although they present many points of interest in themselves, involve a considerable amount of work before arriving at the few essential facts regarding the solution of the integral equation. It is the purpose of the present paper to furnish a scheme for obtaining these essential facts rapidly and directly, without the introduction of any new concepts beyond those involved in the simpler case; by an easy artifice, the solution of a given integral equation is reduced to the solution of another, for whose kernel a resolvent function exists;

On the pseudo-resolvent to the kernel of an integral equation