The integrals of §§ 5, 6, and 7 of the following paper were first established by C. de la Vallée Poussin in a memoir Sur quelques applications de l'intégrale de Poisson (Ann. de la Soc. sc. de Bruxelles, vol. 17, 1892–3). An analogous integral to that of § 5 was discovered by A. Hurwitz, who seems not to have been aware of de la Vallée Poussin's memoir, and will be found under the title Sur quelques applications des series de Fourier in the Annales de l'École normale, vol. 19, 1902. In view of the value of these integrals for the theory of the Fourier series, the discussion now given, which follows different lines from those of previous proofs, may be of some interest. The discussion turns chiefly on the Second Theorem of Mean Value which is quite as applicable to Poisson's as to Dirichlet's Integral.