420 Dr. Silvanus Thompson on 



hysteresis loop, projecting it by first turning its abscissa 

 through a quadrant about the centre 0, then tracing along 

 horizontally to the point Q on the flux-density curve, where 

 a vertical line QM is dropped. Then a horizontal line pro- 

 jected from P cuts QM in R, giving thus the corresponding 

 point on the current curve, the peak of which, corresponding 

 to the cusp of the loop, occurs at the time when the flux- 

 density curve is a maximum, and when the voltage curve is 

 at its zero. 



It will be seen that the wave-form of the current curve 

 reflects, in a certain way, the form of the hysteresis loop. If 

 the loop is sharply cusped, the wave-curve will have corre- 

 sponding sharp peaks. In fact, the loop consists of the two 

 halves of the wave-curve, folded back one upon another, but 

 with the ordinates differently spaced, exactly as if the wave- 

 curve had been wrapped around a cylinder* and projected 

 upon a plane cutting the cylinder diametrically through the 

 two peaks of the curve. 



Now this current curve can be subjected to harmonic 

 analysis, and its harmonic constituents discovered. Eacli 

 constituent will be a pure sine-curve or cosine curve. If 

 each such constituent be drawn, and then be projected back 

 by reversal of the process by which the wave-curve was 

 obtained, the several constituents will reappear as separate 

 closed curves ; and by the summation of these the original 

 hysteresis loop can be reconstituted. It thus appears that 

 any hysteresis loop can be analysed into an liarmonic series of 

 closed curves corresponding to the various terms in the analysis 

 of the current wave. An examination of these constituents 

 of the hysteresis loop is the principal object of this com- 

 munication. 



§ 3. In this graphic process, which is equivalent to 

 wrapping the periodic curve around a cylinder, the area of 

 the projected curve is equal to the integral, over the whole 

 period, of products ohtained by multiplying each ordinate by 

 the sine of the angle at which it stands in the wave-curve ; 

 abscissas in that curve being reckoned as values of angles. 

 (The origin of the cycle is taken where the curve has its 

 negative peak.) In symbols this is equivalent to 



J 



'"sin 6 . f(0) . d6. 



* As in the graphic method of harmonic analysis of Clifford described 

 by Perry in Proc. Phys. Soc. vol. xiii. 



