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Part III is a circuit analysis of cryotron ring oscillators. Ring oscillators have been constructed so that the dynamic behavior of film cryotrons in circuits could be studied. The analysis is concerned with the frequency-L/R time constant and circuit resistance-gate resistance relationships so that the results of measurements on oscillators can be properly interpreted. Two analyses, based on different ideal characteristics, are made. The first treats each stage as a linear amplifier, and the second treats each stage as a switching circuit. Although the two analyses start with rather different assumed ideal characteristics, the results agree in many respects.