It is shown that the H -functions for isotropic scattering in a semi-infinite plane-parallel atmosphere may be readily and accurately obtained by direct quadrature of an integral form. Methods are described which enable the accuracy of the calculations to be maintained in the neighbourhood of singularities in the integrand and its first derivative. The final form has been programmed for the IBM 704 computer, and a total of 160 H -functions have been obtained for values of the albedo ϖ = 0(0.01) 0.90 (0.005) 0.95 (0.001) 0.99 (0.0005) 1.0 Good agreement has been found in checks involving the zero-order moment and the first and second moments. The H -function obtained for the conservative case ϖ = 1 agrees to within one unit in the sixth decimal place with that computed by Placzek from a modified form of the Wiener-Hopf integral. Tables of the H -function are presented for μ = 0(0.05) 1.0 and selected values of ϖ , the tabular values being rounded to the sixth decimal place. To enable H -functions to be obtained for any required value of ϖ , the complete set of computed functions has been approximated for μ = 0 (0.05) 1.0 by polynomials in ϖ using Chebychev polynomials and the method of least squares, several ranges being used to cover the entire range in ϖ from 0 to 1. The H -functions computed from these polynomials are correct to within one unit in the fourth decimal place. The first-order moment has also been obtained in polynomial form.