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The plate current at constant screen and plate voltages in a tube having two well-shielded control grids acting in tandem on the same electron stream is shown to be equal to the product of two functions, each a function of the potential on one grid only. When the second control grid has a uniform structure, the plate current is practically zero for second-control-grid voltages more negative than a value obtained by dividing the screen voltage by a suitable μ factor. For the pentagrid mixer hexode as modulator and heterodyne detector, simplified equivalent circuits are shown to hold, whereby the desired-frequency output currents and voltages may be obtained by calculations similar to those for class A amplifiers. These calculations may be quickly and accurately made. The timesaving is great since a multiplicity of series terms need not be calculated. Where the outer control grid has a uniform structure, the outer pentode has a definite cutoff amplification factor. In terms of the grid-voltage functions and measurements of plate current and plate resistance at known control-grid voltages, the performance of the tube may be calculated when the grid voltages are known. For the pentagrid mixer hexode as modulator and detector, simplified equivalent circuits similar to those of an amplifier are shown to hold. A method is given whereby the desired-frequency output currents may be obtained by calculations involving simple trigonometric functions. These calculations may be quickly made and yield accurate results.