Previous mathematical developments of so-called two-term infiltration equations have been approximate instead of exact and thus provide little insight to account for any experimental shortcomings of such equations. In contrast, it is here shown that one form of two-term infiltration equations is obtainable by integration from an exact solution of the one-dimensional downward Richards equation subject to the customary initial and boundary conditions. The mathematical trial solution is a variables-separable combination of product and additive forms. Intrinsic to the solution are two further stipulations: (1) the unsaturated hydraulic conductivity function must be linear with the water content, and (2) the ponded-water head on the soil surface must increase as the square root of time from initial water application. The absence of both these stipulations in customary ponded-infiltration measurements thus accounts for why the two-term equation frequently fails experimentally. Nonetheless, the mathematical simplicity of the new solution makes it useful pedagogically.¿ 1997 American Geophysical Union