FERROMAGNETIC DISTORTION OF A WAVE 343 



therefore immaterial, so /?X can be replaced by pt in equations (40) 

 and (46). Combination of equations (45) and (46) with equation (40) 

 results in an expression of the induction on the upper half of the 

 complex loop in terms of time. 



For the lower half of the loop a mean position must be found. 

 Having started with incommensurable frequencies at zero phase 

 angles, the reversals at the lower tip of the major loop will occur at 



pt — 2mw + R, 



where < i? < 27r and all values of R within these limits are equally 

 probable. The lower frequency component at the instant of reversal 

 will have a time angle 



qt = (2» + l)7r + S, 



where —{q/p)Tr < S < {q/p)Tr and all values of 5 within the limits 

 are equally probable. The expected medians of the time angles are 

 therefore (2w + l)7r and (2« + l)7r for the higher and lower frequency 

 components respectively. These values and the point symmetry of 

 the characteristic specify that the induction during the ascendency of 

 the lower-frequency component of the magnetizing force will be equal 

 in magnitude and opposite in sign to the induction for the descent 

 with the phase of each component increased by its expected median. 

 The induction for an increasing lower-frequency component is therefore 



given by 



B£pt., qf] = - Bilpt + TT, g/ + tt], (47) 



where the right-hand member is evaluated for a decreasing lower- 

 frequency component. 



The coefficients of equation (40) may be altered accordingly to 

 furnish a set for use when the lower-frequency component is increasing 

 by replacing qt by {qt -f tt) and pt by {pt + x). In series form, then, 

 the induction on the lower half of the loop is 



B^ = Ibo" + bi" cos pt -{- h2" cos 2pt -f h^" cos Zpt 



+ ai" sin pt + a^" sin 2pt -\- a^" sin Zpt. (48) 



One pair of coefficients is necessary to specify completely the 

 amplitude of each component of the induction when it is split into 

 in-phase and quadrature terms harmonic in pt. Coefficients of cor- 

 responding terms in equations (40) and (48) are all functions of qt, 

 each series applying over one half of a lower-frequency cycle. Each 

 pair of coefficients can therefore be developed into a Fourier's series 

 in qt. so that the single expression 



