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A theory is presented which predicts the behavior of any self-limiting oscillator in the presence of an injected sinusoidal voltage or current of small but constant magnitude. The internal mechanism responsible for synchronization is not needed, and the theory is thus applicable to any source of alternating current. Experimental verification of the theory is presented for the case of a low-power Hartley oscillator operating at 11.5 Mc. The theory is extended to include the mutual synchronization of two oscillators of arbitrary properties, and applied to a number of examples to indicate briefly the properties of a synchronized oscillator when used as (a) a linear voltmeter for small voltages, (b) a field-intensity meter, (c) a linear a.m. demodulator for small signals, (d) an f.m. demodulator, and (e) a synchronous amplifier-limiter. The use of a synchronized oscillator is of particular interest because microwave generators can be used in addition to the more conventional triode oscillators.