FERROMAGNETIC DISTORTION OF A WAVE 327 



and sufficient evidence of behavior for extending them in such circum- 

 stances is not available at present. Some experiments by Lehde ^ 

 and others indicate that subsidiary loops do not quite close at their 

 junctions with the major loop and often show departures both ways 

 on different parts of the same complex loop. This sort of behavior is 

 not explainable by inequality between an and 2ao2 for the subsidiary 

 loops, as it would cause the departure of one specimen to be always the 

 same way depending upon which quantity was the larger. In all 

 cases, even near saturation, Lehde's results show this departure to be 

 small and the connecting branches between successive subsidiary 

 loops to form approximately a simple loop. On this basis Madelung's 

 results can be considered confirmed to a sufficient degree of approx- 

 imation. 



The ratio aii/ao2, which has been taken as a measure of the validity 

 of Rayleigh's relation in single-frequency theory, becomes a criterion 

 of the usefulness of Madelung's propositions concerning loop form in 

 multi-frequency theory. Those substances which most closely accord 

 with Rayleigh's analysis can also be expected to be in best agreement 

 with Madelung's results. The relation between coefficients required 

 on the basis of Madelung's and Rayleigh's experiments will be used 

 hereafter to simplify the analysis. The simplification will be evidenced 

 by the customary nomenclature, in which 



A^O — fllO» V = ao2 = 2'^ll' 



Any attempt to distinguish here between the two latter constants 

 would be meaningless because beyond the scope of Madelung's em- 

 pirical rules. Fortunately the constants of most commercial materials 

 conform closely to the above equality. 



Types of Two- Frequency Loops 



The aspect of a hysteresis loop formed by a two-frequency wave 

 changes greatly with the frequencies and their amplitudes. Different 

 pairs of frequencies having equal ratios give rise to families of loops 

 which are identical except as affected by eddy currents. These, for 

 the purpose of this study, are supposed to be so small that the flux is 

 substantially uniform over a cross-section of the magnetic circuit. If 

 the two frequencies have a common source or are synchronized, the 

 hysteretic phenomena are singly periodic and subject to simpler 

 treatment than developed here for independent sources. 



For detailed analysis of the effects of hysteresis with two applied 

 frequencies from independent sources, the phase angles of both may 



"^ Rev. of Sci. Instr., Vol. 2, pp. 16-43, Jan. 1931. 



