FERROMAGNETIC DISTORTION OF A WAVE 341 



For combining these three expressions a Fourier's series may be 

 developed applicable over the entire interval — tt < p\ < tt, and 

 the induction expressed by this series can be referred to the central 

 axes, B, h, of the major loop by including in its constant term the 

 value of the induction at the junction of the major and minor loops. 

 The series is 



Bi = ^bo + bi cos ^X + b2 cos 2p\ + b^ cos 3p\ 



+ a/ sin ^X + ^2 sin 2p\ + a^' sin 3p\. (40) 



The coefficients are determined by the integrals 



aj =- r B sin np\d{pX), b r! =- C B cos np\ d{p\) , (41) 



'I —TV fJ —tr 



where 



B = B' + Br. 



Br is the induction at the junction of the major and minor loops, 

 found by inserting equation (37) in equation (38). The resulting 

 quantity together with expressions previously found for B' over various 

 parts of the cycle are substituted in the integrands of equation (41), 

 and the integrations are performed, using h' given by equation {33). 

 All terms of order higher than the third and those containing the 

 square of the frequency ratio as a factor are rejected as they occur. 

 The resulting coefficients are functions of qr both explicitly and also 

 through A and pk. 



To determine p\ as a function of gr, the vanishing of h' at the tip 

 of the minor loop on the major loop gives an equation for use along 

 the descending branch of the major loop. By equation {33) 



P(l -cosM) = 4a('i -f^), 

 (1 - cos M) = (27r - pK)Ksm qr. (42) 



An approximation to the general solution can be got by transforming 

 equation (42) into a quadratic algebraic equation and solving. This 

 is done by means of the first two terms of the power series expansion 

 for cos ph.. The approximation will be best for small values of the 

 angle, but very good over all its admissible range. This reduction 



gives 



{pK)'^ + 2ic sin gr {ph) - 47rK sin gr = 0, (43) 



the roots of which are 



pK = — K sin gr ± Vk^ sin^ gr + ^ttk sin gr. 



