This study concerns a boundary-value problem approach to the action of a lens system upon a coherent light beam. For the sake of simplicity, we assume the lenses to be cylindrical and to be composed of homogeneous perfect dielectric substances. In addition, we assume that the cross sections of the lenses are those of two intersecting circles. The solution to Maxwell's equations is sought by means of Green's identity, which permits us to express the electromagnetic field within and outside of each lens in terms of the field values on the lens surfaces. The application of the continuity relations across these same surfaces then yields a system of coupled Fredholm-type integral equations of the second kind. These are reduced, by means of the numerical quadraturing technique, to a finite-order matrix equation, the unknowns of which represent the sampled field values along the lens surfaces. For a large number of lenses and for a large number of sampling points, the inversion of the matrix equation can be accomplished, when computer memory size becomes a limiting factor, by suitable partitioning. No theoretical approximations intervene at any level of the analysis; thus the accuracy of the result is only limited by the number of sampling points and by the word-length capacity of the computer. This approach permits a quantitative measure of the behaviour of the type of diffraction-limited systems which occur in all fields of optics, especially in microscopy, and provides a tool for investigating new types of optical systems which cannot be studied by conventional optical theory.