The transfer of solutes into and out of living cells takes place under varying conditions, defined by the relative movement of outer medium and suspended cells. In a tissue the cells are bound to a fixed position and the outer fluid is in steady motion. A specially complicated case is represented by the walls of biological tubes, such as the intestine, the ducts of glands and the renal tubular system, where an epithelial layer separates two fluid compartments, namely the tubular contents on one side and blood and extracellular fluid on the other. The theory of diffusion through the wall of a cylinder has been treated by Jost (1952) for two cases, where the contents inside are either at rest or are flowing in an axial direction at a constant rate. The first condition is approximately realized in the intestinal tract, in view of the slow movement of its contents. For most excretory organs no clear-cut information is as yet available, whether the flow is constant or variable (Bro-Rasmussen, Killmann & Thaysen, 1956). For the kidney, however, it is well known that both the composition of the solution at the end of the tube and the axial flow vary considerably, a case not included in Jost's treatment. For this special case, Dole (1943) first derived a mathematical expression relating the renal clearance of solutes to the rate of urine flow.In the present paper we shall develop a theory for the movement of solutes in cylindrical tubes, through the wall of which exchange may take place in either direction. The general equations which will be derived will then be applied to a special problem, renal excretion. Application of the theory to experimental observations will be presented in a later paper.In order to follow the change in concentration of a given solute with time, let us make the following not unreasonable assumptions: (a) The movement of particles in free solution is much faster than their transfer through the membrane.

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