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The primary relationship underlying this investigation is the exponential law relating plate current to plate voltage in a diode when operated in the so-called "retarding-field" region. By extending this classical mode of operation to multigrid tubes, the tube yielded further operating conditions in which plate current is accurately proportional to the product of a linear function of plate voltage and an exponential function of grid voltage over wide ranges. An analysis is given of various multiplying circuits using a single multigrid tube under this mode of operation. Two methods are described for compensating accurately the inherent plate rectification due to the exponential grid curvature. The circuits are most suited for electronic wattmeters having exceptionally good features and predictable performance over the frequency range from 20 cycles up to the neighborhood of 50 Mc.