410 PROCEEDINGS OF THE AMERICAN ACADEMY. 



treatment of composite simple series circuits with component condenser 

 discharges : — Form the equivalent single-element series circuit (Fig- 

 ure 1). Impress the same initial emf. on the single condenser as would 

 be impressed on the component condenser. The discharging oscillations 

 of the single-element system will then be identical with those that would 

 occur in the composite system. After the oscillations have subsided, 

 there will be in the composite system a residual electric energy to take 

 into account, which does not appear in the equivalent single-element 

 system. 



Thus in the case of Figure 16, let ,^i = 5 X 10^ Sa = 14 X 10*, 

 .53 = 6 X 10* darafs, h = 0.05, k = 0.02, 4 = 0.03 henry ; / = 150 

 ohms, /' = 50 ohms, and let an initial charge of 0.02 coulomb be given 

 to Si by an impressed emf of 1000 volts, the other elements being with- 

 out charge. The initial electric energy of the system W^ = 10 joules. 

 To find the oscillation of the system, we form the equivalent single- 

 element system (Figure 1) with s = 25 X 10* darafs, ^ = 0.1 henry, 

 r = 200 ohms, and impress 1000 volts initially on the condenser if. 

 This will take a charge of 0.004 coulomb, and an electric energy of 

 2 joules. These are the oscillation-charge and oscillation-energy of the 

 composite system considered. The oscillations of the system are the 

 same as those indicated in Figures 7 to 15. After the oscillations have 

 subsided, there will be a residual energy of 8 joules in the system, neg- 

 lecting dielectric leakage, 0.016 coulomb at 800 volts in.^^i, —0.004 cou- 

 lomb at —560 volts, in .'^2, and —0.004 coulomb at —240 volts in Ss. 



NON-OSCILLATORY CoNDENSER DISCHARGES. 



Although the non-oscillatory discharge of a condenser lies outside 

 of the title of this paper, an outline of the case may be admitted, not 

 only in order to complete the discussion, but also to present therein 

 certain important analogies to the oscillating-current discharge. 



If in Figure 12, the semi-circuit resistance p is increased until it is 

 equal to Zq, the resistanceless impedance, both the reactance j7w, the angle 

 </), and the angular velocity w disappear ; so that the discharge becomes 

 non-oscillatory and aperiodic. If p is increased beyond this aperiodic 

 value, the discharge continues to be non-oscillatory, but becomes what 

 may be called ultraperiodic. We may first consider the ultraperiodic 

 case. 



In Figure 1 7 let op be the exponential time-factor of an ultraperiodic 

 circuit. About op as diameter, construct the semicircle oqp. With 

 center o, and radius oq = Wo, the resistanceless angular velocity, in- 

 tersect the semicircle in q. Then the chord pq is the non-oscillatory 



