The quartz piezo-electric resonator is examined experimentally and theoretically with special regard to an equivalent electrical system which can represent it.

It is shown that, as theoretically predicted by Butterworth, such a resonator can be represented by an inductance, a resistance and a capacity all in series. These are pictured as in parallel with another small condenser and the whole is in series with a third condenser, the additional condensers representing air-gaps. The equations for the current in an oscillatory circuit, to which the resonator is attached, are developed and it is found that almost perfect agreement exists between the forms of current curve obtained theoretically and experimentally. This agreement is found to hold for longitudinal resonators of as low a frequency as 44,000 and for transverse resonators of as high a frequency as 15,000,000 periods per second.

It is next shown how the logarithmic decrement of the resonator may be obtained from a rectified line plotted from observation on the current in the oscillatory circuit as a function of frequency width across the crevasse which pierces the summit of the resonance curve.

The methods of analysis of the equivalent mesh into its components are next developed and it is shown that this analysis can be effected by carrying out a series of observations of the current at resonance when the air-gaps are varied by known amounts, or when the effective resistance of the oscillatory electrical circuit is given different known values.

The theoretical and experimental results are found to be consistent. The value of the shunting condenser is somewhat smaller than that measured at a neighbouring frequency outside the region of resonance. A possible explanation is offered.

The effects on frequency of response of variation of air-gap are studied and the difference between prediction and observation is discussed. This difference is only a few parts in a hundred thousand. In the case of transverse resonators remarkable effects occur when the air-gap is varied through the regions where its length is an integral member of half-wavelengths of the supersonic air waves produced by the vibration of the quartz.

The temperature coefficient of frequency of a considerable variety of resonators is examined over a range of temperatures up to 40°C. It is found that very diverse results are obtained and probable explanations are offered.

The effects of displacement of the resonator from the position of centrality are examined. These are shown to be small, but not quite negligible.

The current taken by the quartz mesh is then examined in some detail theoretically, and one or two experimental curves are given, together with a graphical method of deducing the curve of current from the constants of the quartz. A selection of theoretical curves for the conditions under which the experiments of Cady were made have been calculated. The curves are in all respects of the shape and form experimentally found by him.

A number of conclusions and suggestions terminate the Paper.