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A network theory is derived for circuits constructed of linear rheostats driven from a common input shaft. A correspondence between these networks and RL networks is used to demonstrate that driving point impedances and transfer functions are real rational fractions in a real variable, the input shaft rotation. Requirements on the pole-zero characteristics of these functions are established. Methods for developing functions approximating a given input-output relation in physically realizable form are discussed. Modern methods of transfer-function realization can be applied to these fractions to synthesize the physical networks. Networks for sine and log transfer functions obtained by the technique are presented.