The general theory of piezo-electricity is summarized, and the equations applicable to quartz are stated. The "concentrated mass theory" of longitudinal vibrations in rods is then developed, and the properties of the "resonance circle" discussed. On this theoretical basis the equations for the reaction of a piezo-electric resonator upon a circuit are derived, and solved by a graphical method. From the vector diagram, important parameters, including the capacity and resistance of the resonator at any frequency, may readily be found. The construction and operation of quartz and of quartz-steel resonators is described, and an account is given of their application as wave length standards and as frequency-stabilizing devices. Other practical applications are also suggested.