Low Frequency and their Resonance. 425 



hysteresis and Foucanlt current-losses approach the order of 

 magnitude of the losses due to ohmic resistance. The natural 

 period of such circuits, especially when tuned up to a fre- 

 quency of over 200 periods per second will be given very 

 accurately by the formula 



T=27ryTC 

 To such circuits only the following discussion refers. 



III. On the Electrical Flow in a Resonant Circuit. 



Let a simple harmonic e. m. f. of period T act upon a cir- 

 cuit having localized self-induction and capacity, coil and con- 

 denser being connected in series. By the generalized form of 

 Ohm's law we have in the usual notation 



L-^ + Kcc + P=E sin pt (6) 



The integral obtained by well-known rules is 



x= , x - sm(pt-q)) (1) 



A/(l-j» 2 CL) 2 +yC 8 R 2 V ; 



' . 1-p-CL 



where tan qj= — ^-= — 



pKC 



which can also be written 



E • , 



x = — , — sin (pt+q),) 



v> 2 l;+ R2 



pjj, 



tan <p.—-&- 



The integral written in this last form shows, as Oliver 

 Heaviside first pointed out, that a condenser of capacity C in 

 series with a coil changes the impedance of the circuit in such 

 a way as if the condenser had a negative coefficient of self- 

 induction equal to —p.* It produces also a shifting of phase. 



The impedance is reduced to ohmic resistance when 1^=0 or 

 j9 2 LC=l, that is when the period of the impressed e. m. f. is 

 equal to the natural period of the circuit, or in other words, 

 when the two are in resonance. 



The current and therefore the amplitude of the charge of the 

 condenser reach then their maximum value. 



* It is well to observe here that later on in the analysis of more complicated 

 circuits possessing localized self-induction and capacity, I simplify my calculations 



very much by substituting Lj = — L + — — for the coeffic. of self-induction and 



treating the circuit then as if it had no capacity. 



