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This paper concerns the exact solution of the nonlinear differential equations which describe the time dependent behavior of multistage cascade separating processes for separation of two very similar molecular species. The Rayleigh separation law is postulated for each stage. The main questions of interest in the nonstationary operation of cascades involve (a) their behavior while the concentration of the required component is being built up to the desired value; (b) the behavior of the cascade during the transition from one stationary mode of operation to another; and (c) the manner in which local concentration fluctuations are propagated through the cascade. We have derived analytical expressions that are suitable for the discussion of all of these points in square cascades. The cases of finite, half infinite, and infinite cascades are considered with and without product withdrawal. A brief discussion of the linearization of a class of nonlinear second‐order partial differential equations is given in Appendix I.