278 HENRY A. HOWL AND 



2. This effect is not due to hysteresis but to the variation in the mag- 

 netic permeability. 



3. The harmonics increase with the increase in magnetization of the 

 iron and nearly vanish as the magnetization decreases, although it is 

 doubtful if they ever quite vanish. Hence, an increase of resistance 

 will decrease the harmonics. 



4. In the method of introducing the hysteresis into the equations, it 

 enters as an addition to the resistance in the term Ra { -f- Anb, where 

 R is the resistance, a^ the maximum current, A the coefficient of hyster- 

 esis, which is dependent upon the amount of magnetization of the iron, 



n the number of turns of wire, and b= is 2- divided by the time of 



a complete period. 



The introduction of the hysteresis into the ordinary equations, there- 

 fore, presents little or no difficulty. 



Many observers have noted that the current curve in a transformer 

 was not a sine curve and Prof. Ayrton has shown the presence of the 

 odd harmonics but gives no explanation. Mr. Fleming has attributed 

 them to hysteresis, but I believe the present paper gives the first true 

 explanation. 



Problem 3. To find the work of hysteresis. Let the .resistance, R, 

 be zero. The work done will then be the integral of the current times 

 the electromotive force, or 



(1 P fit 

 dt a 



the integral to be taken for one period of the current. 



27T 



f*** I d*u dii 1 



w= I- bA sin (bt + e,} y + Bny / + (7n s 3 y 2 - 7 f- + \dt 



I II \ ' if a J fjf -J fit 



/ V |_ Ui J 



w = A ~a\. 

 o 



All the other terms are zero. 



In a unit of time the energy absorbed is 



Steinmetz has found by experiment that this varies as the 1-6 power 

 of the magnetic induction. Of course the present theory gives nothing 

 of this but only suggests a way of introducing the hysteresis into cal- 

 culations of this nature. For this purpose replace A by A 1 ^- 6 and the 



