Part 1 of the paper is concerned with the theory of a.c. bridges in which the ratio arms consist of tightly-coupled inductors, and refers particularly to bridges in which two sets of such ratio arms are used together in order to obtain very large ratios between the unknown and standard impedances. Reference is made to the important property of inductively-coupled ratio arms which enables the direct impedance between two points to be measured independently, within limits, of any impedances connected between a third point and the two terminals of the impedance being measured. The logical development of bridges with two such ratio arms follows. The material in Part 1 of the paper is the original work of the late A.D. Blumlein, B.Sc.(Eng.), Associate Member, and the present authors have used verbatim a considerable portion of a hitherto unpublished memorandum written by him in January, 1941.Part 2 of the paper describes Blumlein's application of the principles described to a general-purpose mutual-admittance bridge for the measurement of capacitance and resistance. This has a wide range of impedance measurement, and the facility of being able, in many circumstances, to measure componentsin situwhen it is difficult or impossible to isolate them from other components of a circuit.