238] INDUCTION OF CURRENTS. 479 



circuit. A quantity varying in this manner is said to perform a 

 harmonic oscillation, with the amplitude E . 



The electromotive-force takes on all values between E and 

 E Q and returns to its original value in the time that the vector 

 takes to make a complete revolution, T= 27r/&>. The time T 

 is called the period, and its reciprocal, the number of periods in 

 unit time, n = co/Sir, is called the frequency. 



In like manner the quantity Ae i<at is represented by a vector 

 revolving with the same period, of length equal to the modulus of 

 the complex quantity A. Since the argument of a quotient is 

 equal to the difference of the arguments, the vector representing 

 Ae i<at lags behind that representing E Q e i<at by the constant angle 



But from the equation (3) we find that this ratio is the complex 

 quantity, R + iLco, whose argument is tan" 1 Lco/R. The current, 

 being represented by the projection of the second vector on the 

 real axis, is said to differ in phase from the electromotive-force by 

 the amount a, the difference in this case being a lag. The ampli- 

 tude of the current, being the modulus of A, is the quotient of the 

 moduli 



Expressing these results analytically we obtain equation (5). 



The quantity J, by which it is necessary to divide the ampli- 

 tude of the electromotive-force in order to obtain the amplitude of 

 the current, is called, as proposed by Heaviside, the impedance. 

 If the circuit has no self-inductance, or if the current is steady 

 (o> = 0), it becomes the resistance. 



It has been proposed by Hospitalier* to call the coefficient of 

 i in the ratio E /A, the reactance. 



The mean value of a quantity varying harmonically taken 

 over any exact number of periods is zero, while in virtue of the 

 formulae 



I rT 1 rT 1 1 f r . 



~ I cos 2 wtdt = m I sin 2 cotdt = ~ , ^ sin cot cos cotdt 0, 

 -L Jo J- Jo 2 JL JQ 



* Hospitalier, L'Industrie Electrique, May 10, 1893. See also, Steinmetz and 

 Bedell, Trans. Am. Inst. EL Eng. 1894, p. 640. 



