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Voigt's theory of the diffraction of polarized light by a black halfplane. The method of many-valued wave-functions and Riemann surfaces is employed. The standard of knowledge expected of the reader of this work is that of a graduate student who has completed the usual courses in analysis and electromagnetic field theory. Twenty-three exercises are provided in the first three chapters. The text is replete with footnote references to papers that have appeared in the literature up to 1939. By rigor of logical treatment and careful attention to detail the authors have produced a critical treatise which will undoubtedly become a standard reference work.