We apply the method of image charges from electrostatics to the study of the Neumannfunction for the Laplace operator, equivalent to the Green's function with a Neumannboundary condition imposed. Such an analysis has previously been given for the moregeneral ellipsoidal case; however, the azimuthal symmetry of the prolate and oblate spheroids simplify the analysis and the results may enjoy an undiminished application, as many naturalphenomena conform to these geometries. Our results are twofold: we first use classical methods to derive a series form of the Neumann function given a point source located in both the interior and exterior of the twospheroids; we then use each series form to develop a corresponding system of images that replicates the boundary conditions and yields an equivalent integral solution.