Inductance of a Wire to an Oscillatory Discharge. 435 

 so that 



#{X) 



= xh (10) 



Equivalent Resistance and Inductance for Oscillatory Dis- 

 charges. — To effect the object of this paper we must now 

 apply equations (5), (6), and (7) to the case of logarithmically- 

 damped alternating currents where all the functions are 

 proportional to e^~ k)pi . 



The value of E so obtained must then be separated into 

 real and imaginary parts as in (8), and then, together with 

 the imaginary quantities, must be collected a proportionate 

 part of the real ones so as to exhibit the result in the form 



E = B"C+(e-%L // C (11) 



The quantities denoted by B/' and \J' in this equation will 



then represent what may be called the equivalent resistance 



and inductance of length / of the wire to the damped periodic 



currents under discussion. For, the operand being now 



e d-k)pt^ jfa Q time differentiator produces (i — k)p, and not ip 



simply as in equation (8) for the sustained harmonic currents. 



dT 

 Thus (5), (6), and (7), on elimination of 8, Z, and -jj, give 



E _/;_ k \ n „\ | <Kip*p-kp*fi) (m 



Now we have 



thus 



^)- 1+ 2-l2 + 48-180 + *-" * {l6) 



cj)(ipufJL — /cpa/jb) 

 <fi (ipufjL — kjjafA) 



+i-[ip^+ |^v- 1 -^p z «V - m ]^py w } . (u) 



Hence, substituting (14) in (12) and collecting the terms 

 as in (ll), we find that 



R" 1 + i 2 s A(l + A«) 3 , 3 l-2F-3/t 4 4 4 



TT =1 + T2-AV+ — jj-W fgo AV • ••-, (15) 



