760 Sir Oliver Lodge on 



the insertion of capacity in series ; so that the general equation 

 for a circuit, with u as current, 



u 

 Jjil 4- R« + o- = E cos pt 



becomes simplified, when the current is properly attuned to 

 the driven frequency, i. e. tuned in such a way that p 2 LS= 1. 

 It then, after a brief moment of growth, assumes the 

 simple form 



• E 



because the first and third terms neutralise each other under 

 tuned conditions ; since then U+p 2 u = 0. 



(7) If a current is started and stopped in such a circuit, its 

 rise and fall are governed by the customary laws ; for it must 

 be remembered that there are free oscillation terms in the 

 complete solution which control the initial and final stages 

 of such a maintained current, and that these are discarded 

 from the permanent theory as being too rapidly evanescent to 

 deserve much notice. The initial (and final) free oscillations 

 are rapidly killed out of the complete solution of the above 

 equation by an exponential time-coefficient involving R/2L, 

 and forced oscillations alone survive. But the free are 

 not wholly to be ignored, and in some cases may be found 

 to give trouble. The current starting from zero, when it is 

 attuned to the maintained oscillations and not over-damped, 

 rises to its final stage according to the equation 



E, * 

 u = — ( 1 — e 2L ) cos pt, 



as clearly explained in Lord Rayleiglr's ' Sound,' vol. i. 

 §§ 47, 48. Hence, to attain rapid response, it may even be 

 desirable to introduce a little extra resistance into some of 

 the circuits, so as to diminish the time-constant 2L/R by 

 which these momentary perturbations are regulated. It may 

 be noted, as a practical consequence of the completer theoiy, 

 that the frequency factors should be chosen with capacity 

 large and inductance small ; for it is the latter factor alone* 

 that retards the attainment of the permanent state. 



Of course a current takes time to develop in a circuit, 

 even without any self-induction, — the velocity of light must 

 be involved and a travelling wave-front ; but in ordinary 

 practice these can safely be ignored, 



