214 Dr. A. C. Crehore on the Theory of the 



The solution of this equation might be obtained under 

 certain conditions, but it is not the present purpose to 

 examine the effects o£ inductance in the external circuit. 

 Assuming, therefore, L to be small and negligible (21) is 

 reduced to the following equation of the second order, 



*•+(*.+ S^ + n » 2 *'=| e =l/(«)=^' • • (22) 



Upon comparing this with (19) it appears that the only 

 difference is in the coefficient of <j)s. Denoting the new 

 coefficient by k s we have 



fa=h+~, (23) 



which means that the effect of closing the external circuit is 

 to increase the coefficient of air damping k by an amount 

 h 



-~, which latter may be called the coefficient of electro- 

 magnetic damping. 



The Impressed Electromotive Force. 



We will now consider that the impressed electromotive 

 force in the external circuit is a simple harmonic function of 

 the time, since any form of periodic force may be resolved 

 into a series of such simple waves. Let 



e=Ecos^=/(^) (24) 



With this value substituted in (22), the solution for the 

 periodic portion, or particular integral not containing 

 arbitrary constants, is 



AJsine s , * 



$ s = — t cos(g>£ — € s ) . . . (25) 



where 



sine = [(n/ _j;r + , sW]J • • • • (26) 



cose=- 1. . . . (27) 



[W-w 2 ) 2 +Wp 



tane=-^-2 (28) 



The values of n, and h s are given in (13) and (18) above, 

 and 1= =5- , the maximum value of the harmonic current. 



