KENNELLY. — OSCILLATING-CURRENT CIRCUITS. 401 



This is the total expenditure of energy in Pr up to time t. At any 



T 



complete energy cycle, when cos (2w^ + (^) = cos ^, and t = m—, 



Wa = tJ^c [sin^ «^ (1 - e'^^'"?)] = W^ (1 -e"^'"?^. joules (51) 



2 



The energy dissipated in the first energy cycle, when t = 7^/2, 



u\ = TFo (1 - e --^r). joules (52) 



The dissipation in successive energy cycles 1st, 2d, 3rd, 4th, etc., is 



Wi, Wi€~'-'^, %'ie~2-^r, wie~^-''^, etc. joules (53) 

 The total ultimate energy dissipated is 



W^ = wi{l + e-^T + e-2.r + . . .) joules (54) 



~ joules (55) 



1 — €-^2- 



"With the values in the case considered of 11^= 12%!, JJ^= 1000, 

 c = 4 X 10-^ a> = 1224.75, <^ = 50° 46', ^ = 1000, T= 0.00513, 

 Wq = 2, we have, for the attenuation factor of one power period or 

 semi-period of p. d., e"-"^ = 0.005,987. The energy dissipated by Pr 

 in the first energy cycle is thus 2 X 0.994,01 = 1.988,02 joules. The 

 second cycle dissipates 0.005,987 X 1.988,02 = 0.0119 joule. Each suc- 

 cessive cycle dissipates 0.5987 per cent of the amount dissipated in the 

 last preceding cycle. It is thus evident that in a damped oscillatory 

 discharge, a relatively large fraction of the energy is rejected from the 

 system in the first half-cycle of voltage or current, i. e., the first com- 

 plete energy cycle. 



Logarithmic Decrement. 

 If V be any vector oscillating-quantity of the tjrpe 



V = Fo€-^' sin (o^t + <^), Ph. Q. (56) 



such as a voltage, current, or force. 



Then the rotating vector of this quantity FqC— ''e^""', in passing from 

 one assigned position to another, in a time t^ seconds, decreases fi-om 

 the first to the second value by the exponential, or damping factor, 

 r-"i. The exponent, -t^i, of the damping may be defined as the Nape- 



VOL. XLVI. — 26 



