430 Dr. Silvanus Thompson on 



current and potential at that instant. If the voltage is re- 

 presented by the expression 



V = V sin pt, 



where p stands for 2tt/ , j and if the current, being some 

 periodic function of the time, is represented as 



then the element of work imparted to the circuit during time 

 dt being CVdt, the work given to the iron (if the copper 

 resistance is negligible) daring one cycle will be 



V 1 sin pt.TJr(pt) . dt. 



But ^r(pt) consists (see § 3) of a series of harmonic sine 

 and cosine terms. The quantities which will be formed by 

 multiplying the members or that series by sin pt, and inte- 

 grating each product over a whole period, will fall under 

 three kinds, the values of which are known, viz.: — 



(i.) I sinpt . A n sin npt . dt=Q, (except when n = l); 



(»•) 



(Hi.) 



I sin pt . B n cos npt . dt =0, (in all cases) ; 



I A 1 . siii 2 pt . dt=A x . ~~ . 



Jo 



That is, the only ivork done in the cycle is that done by that 

 constituent of the current which is in phase with the voltage, 

 namely, its fundamental sine-term. All other constituents 

 are wattless. And since the area of the loop represents the 

 work done, it follows that the area of the hysteresis loop is 

 equal to the area of the orthogonal ellipse which is its funda- 

 mental constituent of the sine series. The true and funda- 

 mental form of every hysteresis loop is therefore an orthogonally 

 placed ellipse. All departures from that form are wattless — 

 are mere distortions which involve no expenditure of energy. 

 The area of the hysteresis loop is proportional to the maximum 

 value of 23 and to the amplitude of the first sine term into 

 which the values of f^ (corresponding to the values of the 

 current) can be analysed. If the value of the amplitude of 

 that sine-term be denoted by H&. 1} then the area of the loop is 



accurately given by the expression j- x 2S max . X?h; being the 



