The theory of equilibrium staged parametric pumps with discrete fluid transfers has been presented in previous papers for total reflux. Here, it is extended to open, fractionating systems, operating in cyclic steady state. The cases of and enriching section, a stripping section, and a complete fractionating apparatus are successively considered. Providing adequate variables are defined, the result is a diagram which plays the same role as the McCabe-Thiele diagram in distillation. The optimal feed stage location, the minimum reflux, and the separation factor for given operating conditions are determined analytically and graphically in the case of linear equilibrium isotherms. The method is then generalized to non-linear equilibria, particularly of constant relative volatility type.