Abstract : The Schwinger variation method is removed from its customary setting in Green's function theory, and is approached from the integral equations viewpoint. The traditional Schwinger theory is improved upon in two respects. Error bounds are provided for the approximations to A(f), a function determining waveguide transmission and reflection coefficients due to diaphragms containing windows. A method, involving quadratures only, is provided and will improve the accuracy of this approximation as much as desired.