The paper deals with the accepted methods of representing ferromagnetic properties and with the vector diagrams and equivalent electric circuits commonly employed to indicate the behaviour of iron-cored coils under a.c. excitation. The. point of view adopted is that such vector and circuit representations must not be interpreted literally when, for example, fundamental measurements of the properties of iron are undertaken by a.c. methods. For instance, the wattless component of no-load current of a transformer—frequently termed the magnetizing current—is shown to be in no sense a measure of the amplitude of the magnetizing force experienced by the core, even though, in the r.m.s. system of measurement, it must be used in assessing the so-called reactance of the winding. An inductance coefficient deduced from this reactance is an artificial quantity because it infers a permeability value which cannot be related with the true reversals curve of the core material at any point. It is demonstrated by experiment that the total no-load current (r.m.s.) is a closer measure of true peak magnetizing force, though this current needs to be corrected for eddy-current effects and, even then, has still a very limited application in fundamental investigations.Attention is directed towards precise methods of measurement of magnetic properties using alternating current, and a means of expressing results is suggested which avoids the worst anomalies of the r.m.s. equivalent circuit and vector diagrams. The mathematical treatment of the excitation ellipse associated with the latter is extended and its full implications are discussed. The relations between the ellipse and the true hysteresis figure are shown by an analysis of the harmonics of excitation. The true cycle of permeability is derived, thus enabling an expression to be formed which is general for all conditions in the core, so covering the cases of remanent magnetism and of excitation by reversed direct current such as is customary in ballistic measurements.The aim throughout the paper is to re-establish a clear outlook upon the elements of cyclic ferromagnetism, in the belief that the half-truths of the common a.c. vector treatments have an undue influence upon the engineer's mode of thought.