The method of hodographs, recently developed by Hamel for the analysis of ground‐water flow systems containing free surfaces, is applied numerically to six cases of the seepage of water through dams in which the ratio of the inflow fluid head to the width of the dam at the base varied from 2.04 to 1.08. In one case the velocity distribution was computed both along the inflow and outflow faces, as well as along the base of the dam. The pressure and velocity distributions along the bases and the velocity distribution along the inflow faces were found for 5 other cases, in 4 of which the outflow fluid head was zero. The heights of the surfaces of seepage and the total fluxes were calculated for all six cases. These calculations show that the Dupuit‐Forchheimer theory of the free surface and velocity distributions in gravity flow systems is definitely wrong, and that the extension of the empirical results of Wyckoff, Botset and Muskat, obtained with radial gravity flow systems, to linear systems is unjustified. The fluxes, however, given by this extension or the Dupuit‐Forchheimer theory turn out to be in close agreement with those calculated by the exact theory. The evaluation of the fundamental integral over the argument of the modular elliptic function involved in the calculations is extended beyond the range given by Hamel and Gunther so as to include all real values of the modular elliptic function.