where N-N₁ are the numbers of spaces, and _r_-_r_₁ are the radii of single spaces in the respective soils, and T-T₁ the times required for a unit volume of water to flow through the soils under the same head or pressure. The space within the soil is rarely filled with water in agricultural lands, and the most favorable amount of water for the soil to hold, as Hellriegel and others have shown, is from thirty to fifty per cent of the total amount of water the soil can hold if all the space within it were filled. If the space within the soil be only partly filled with water, as in most arable lands, the water will move in a thin film surrounding the soil grains and according to the fourth power of the thickness of the film. The mean thickness of the film surrounding the soil grains may be theoretically determined by the following formula, which is based on the conception that the film is cylindrical and of uniform size throughout: _t_ = _r_(1 − √(_s_)/(_s_ + _p_)) where _s_ is the per cent by weight of water which the soil will hold when the empty space is filled with water, _p_ the per cent of water actually contained in the soil, _r_ the radius of a single space, and _t_ the mean thickness of the film surrounding the soil grains. The relative rate of flow of water through the soils will then be according to the following formula: T₁ = (N(_t_)⁴T)/(N₁(_t_)₁⁴)) It must be remembered that these formulæ give only approximate and comparative values for comparing one soil with another. The structure of the soil is altogether too intricate to expect ever to obtain absolute values. If the observed rate of flow varies widely from the relative rate calculated from the mechanical analysis, it will indicate a difference in the arrangement of the soil grains, or in the amount or condition of the organic matter in the soils. In the older agricultural regions of the United States, south of the influence of the glacial action, the great soil areas appear to have sensibly similar arrangements of the soil grains, and sensibly uniform conditions of organic matter, save where these have been modified by local conditions.