Experimental evidence is cited that, for very small hysteresis loops, the remanent magnetic flux density Bris not proportional to the square of the amplitude of the magnetic field excursion H1(as given by the Rayleigh relation), but rather to the first power. It is shown that (subject to one very general assumption), were this dependence exact, it would lead to negative hysteresis loss. It is also shown that (subject to the same assumption) any power-series dependence of on will always give . An approximate first-power dependence is found to follow naturally from the spring model. A new Rayleigh relation is proposed which contains a logarithmic term which may not be expanded in a power series about zero argument.