338 BELL SYSTEM TECHNICAL JOURNAL 



for 7} even. When n is negative, the coefficient may be found by using 

 the relation 



Amn - (- l)"^mn; (26) 



for all other values of the subscripts excluded the coefficients are zero. 

 A recurrence formula for computing products of higher order is 



A = ^ 



m + n 



2( ^— ^ + (w - 1)k )^„_i. „_i 



- (m + n - 4)Am-2, „-2 (27) 



with ni — 2 positive. Comparison of equation (24) with equation (25) 

 reveals that if in the former k is replaced by ki, 



A, = jAu„ (28) 



so the equations (21) and the recurrence formula (27) suffice for com- 

 puting all the coefficients in the series for the induction. 



Calculation of the Induction — Case 2 



For case 2 the two branches of a minor loop and an adjoined portion 

 of the major loop are combined into a Fourier's series whose coefficients 

 are functions of position in the major loop. By developing these 

 coefficients into Fourier's series, a double series in time is obtained. 

 For this case, as for the other just considered, the induction thus is 

 developed in the form of equation (4) and the coefficients are deter- 

 mined through the third order. 



When minor loops are formed throughout the lower frequency cycle, 

 an expression for each minor loop and the portion of the major loop 

 joining it to the next one is found relative to an origin at the junction 

 of the major and minor loops. A succession of such loops is then 

 referred to the origin of the major loop by transforming the coor- 

 dinates of a typical minor loop, the transformation being a function 

 of the position of the minor loop. 



Attention first will be devoted to a single high-frequency cycle 

 occurring while the lower-frequency component of the magnetizing 

 force is decreasing. Let the time of occurrence of the maximum in 

 the higher frequency component of the magnetizing force during this 

 cycle be designated by r. By restricting application to characteristics 

 with sizeable minor loops, i.e. k <^ I when p > q, consistent with the 

 stipulation that the minor loops do not vanish anywhere, this maximum 



