Hysteresis Loops and Lissajous 9 Figures, 427 



analysed into an harmonic series of Lissajous figures of the kind 

 considered in the §§4 and 5. 



A number of examples of hysteresis loops were chosen, 

 and subjected to harmonic analysis, to ascertain what con- 

 stituents were present. The loops chosen relate to various 

 kinds of iron and steel, hard and soft, solid and laminated, 

 taken by various methods ; a wide selection being made in 

 order to ascertain the physical significance of the several 

 constituent terms. 



In carrying out the analysis the author used the simple 

 approximate method described by him to the Physical Society, 

 Dec. 9, 1904, vol. xix. Proc. Phys. Soc. p. 443, based on an 

 arithmetical process originated by Archibald Smith and 

 generalized by Runge in the Zeitschrift fur Mathematik und 

 Phi/sik, vol. xlviii. p. 443, 1903. It was found that for the 

 present purpose it sufficed to ascertain the harmonic sine and 

 cosine terms up to the eleventh, and therefore to employ 

 twelve equidistant ordinates in the half-period. The work 

 proceeded on the lines of the simple schedule given by 

 the author on p. 448 of his former paper, with a slight 

 modification to enable the origin of abscissa? to be taken 

 not at the point where the ordinate has zero value, but 

 at that point where the ordinate has its negative maximum, 

 At first the values of the twelve ordinates required for the 

 analysis were taken from the current curve plotted, as ex- 

 plained above in § 2, from the hysteresis loop. But it was 

 seen that it was unnecessary to draw the current curve, and 

 that the values of the ordinates might be taken direct from 

 the hysteresis curve, by taking them not equidistant, but at 

 places corresponding to equidistant points in the axis of 

 abscissae of the wave-curve, which points, when the curve is 

 wrapped round a cylinder, will no longer appear equidistant. 



§ 7. The following are the results : — 



Example I. fig. 14, PI. VI., Ewing's hysteresis loop for 

 pianoforte steel wire, in state of normal temper, being fig. 11, 

 pi. lviii. of Philosophical Transactions, 1885. 



The analj-sis of the values of ^ gives the following co- 

 efficients of th^ harmonics up to the eleventh order :■ — 



Sine Terms. 



Cosine Terms. 



A l = 32-2 



B x = -45-4 



A 3 = 7-1 



B z =-20-6 



A- = 0-7 



B h =-10-8 



A 7 =- 0-7 



B 7 =- 5-7 



^1 9 = - 0-05 



J5 9 -~ 4-1 



A n =- 0-06 



B n =- 3-4 



