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This is a comprehensive treatise on the properties of the ordinary and generalized functions of Legendre, with a final chapter on functions of Lamé. The work is all that one would expect from a writer who has himself made many original contributions to the subject, and whose style as an expositor is well known through his text book on real variables. As an up-to-date handbook of spherical harmonics, the new book will undoubtedly be welcomed by both pure and applied mathematicians, and become the standard reference in its field. PHILIP FRANKLINdoi:10.1090/s0002-9904-1932-05449-0 fatcat:422y4hdytzac5lmro7ykdi6mue