§ 1. Introductory—In a former paper (Proc. Roy. Soc. Edin., vol. liv, 1934, pp. 135–144) the author discussed the evaluation of a number of integrals of Associated Legendre Functions, regarded as functions of their degrees. The methods employed depended mainly on contour integration, and most of the integrals were evaluated in terms of infinite series of Associated Legendre Functions. In the present paper the methods employed are of a more elementary character, depending mainly on the use of Dirichlet Integrals; the results obtained are more general; and the integrals and the corresponding series are evaluated in simpler forms. The same notation is employed as in the previous paper. The Mehler-Dirichlet Integral

where o < θ < π, μ > – ½, is used throughout.