We present a new quasi-analytical technique for solving the flow equation. It has affinities with Parlange's method, but offers the following advantages: freedom to choose an initial assumed flux-concentration relation,F1, greatly improves the possible accuracy of the first approximation, and the higher approximations preserve integral continuity and therefore behave more stably. The first of these advantages is of practical importance, but the second is more basic. This paper treats only solutions subject to concentration conditions; the related technique for solutions subject to flux conditions will be developed in a later paper.The technique is studied analytically and numerically forone-dimensional sorptionsubject to constant concentration conditions. It is found to be convergent for a wide range of shapes of the diffusivity function. For the unfavorable case of the ‘linear’ soil, the mean error is 3 percent after two iterations and 1 percent after three. For absorption in Yolo light clay the corresponding figures are 0.57 percent and 0.07 percent.The general iterative scheme forone-dimensional infiltrationsubject to constant con—centration conditions is presented. Three choices ofF1should yield useful first approximations: (A)F1A= limt→oF(t is time) =Fabs, theFfor the analogous absorption process; (B)F1B= limt→∞F= &thetas;; and (C)F1c, an interpolation function which is exact in the limits ast→ 0 andt→ ∞.F1Ashould lead to a goodlower boundfor the infiltration rate functionq(t),F1Banupper bound, andF1Caclose upper boundfor all except very larget, and the quality of the estimates of moisture profiles should be comparable. Detailed calculations for Yolo light clay bear out these expectations; the three estimates are wholly consistent with the power series solution of Philip (1957b). The error of the approximation based onF1Cincreases from 0 percent att= 0 to about 1 percent att= 106sec. This first approximation is accurate enough to render iteration unnecessary for most purposes. Parallel calculations confirm the nonconvergence of Parlange's method when applied to infiltration.General iterative schemes are given also fortwo- and three-dimensional sorptionsubject to constant concentration conditions.