This paper describes a form of alternating-current resonance bridge suitable for measuring circuit resistances up to 30 ohms at frequencies up to 0.5 megacycle per sec. and up to 10 ohms at frequencies up to 10 megacycles per sec. The complete theory of the bridge and analysis of the errors is given, together with an account of a series of special check measurements designed to reveal the accuracy actually obtainable. The limit of error appears to be in the neighbourhood of 0.01 ohm for resistances up to 10 ohms at frequencies up to 10 megacycles' per sec.