The mathematical equations for the transport of nonelectrolytes across cell membranes are critically examined and cast in forms suitable for solution which involve fewer approximations than has heretofore been commonly done. For the use of red cells, the equations are developed to include the effect of the variation in apparent nonosmotic water owing to the variation in hemoglobin concentration as the cell swells or shrinks. Two methods of solution of the equations are developed and studied and sample calculations are provided. It is shown that the solutions to the linearized equations commonly found in the literature are insufficiently accurate for some purposes and this inaccuracy is avoided by the methods given here. The importance of retaining the effects of variations in apparent nonosmotic water and in solute volume in the cell is demonstrated.