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The problem discussed in this paper is that of the scattering of electromagnetic waves by a spherical obstacle. The classical theory is well known. Stratton's ``Electromagnetic Theory,'' for instance, contains a very good summary of that discussion. The computation of the scattering cross section, according to this theory, leads to some difficulties. In the case of large spheres, where geometrical optics should apply, the rigorous theory yields a scattering cross section equal to twice the actual cross section of the sphere! The discussion presented in this paper explains this strange result and shows the role played by the shadow and by the diffraction fringes surrounding the shadow. A reasonable system of approximations yields the well known ``Babinet's principle.'' The physical interpretation is of such general character that it must certainly apply to a variety of similar problems in acoustics or wave mechanics. The spherical shape of the obstacle is essential in the present discussion, but similar results would certainly be found for other shapes of the obstacle. The case of a circular cylinder, investigated by E. B. Moullin and L. G. Reynolds, also leads to a total cross section twice as large as the actual cross section, for large cylinders. The explanation is the same as for the spheres. Experiments very carefully designed by Dr. Sinclair and Professor V. K. LaMer gave a very precise check of the theoretical predictions.