The design of machines for medium-frequency power circuits, increasingly used in industry and in aircraft, demands a knowledge of flux-penetration phenomena as affecting both the iron losses and the utilization of the available iron section. The paper shows how hysteresis influences all important quantities, especially the eddy-current loss, the apparent flux, and the utilization factor. Theoretical expressions for these are, on certain assumptions, rather simply derived and presented in a form convenient for use in laboratories and design offices. Curves are given for this purpose, and the ranges of frequency and thickness indicated over which the influence of hysteresis on the other quantities is pronounced. Systematic collection of test results showing their departure from the theoretical values would assist in elucidating the relation of the discrepancies to the grain structure and the metallurgical history of the sheet.