400 PROCEEDINGS OF THE AMERICAN ACADEMY. 



The total dissipation power of the condenser and reactance is after t 

 seconds, 



■2pr=p+ p' = C^oV"^^' (1 — cos 2co^), watts (41) 



= 7oVe~^^' (1 — cos2'jo^), watts (41a) 



where y = c-^, the oscillation conductance of the circuit. The energy 

 in the condenser t seconds after release, if Wi = mn, Figure 7, = MN, 

 Figure 12, is 



Wc = -^ce-2^« [1 — cos (2 u^t + 2 c^)] joules (42) 

 = Wie-^^t [1 - cos (2 i^t + 2<^)]. joules (43) 



The energy in the reactance t seconds after release is 



Wi= -^ ce-2^< (1 — cos 2 Hit) joules (44) 



= TFje"^-'' (1 — cos 2 (nt). joules (45) 



The total energy of the system t seconds after release is 



w = Wc + Wi = -—- ce ^-t' [1 — cos (2 (ot + <^) cos <^] joules (46) 



= 2 TFje-s^' [1 — cos (2 u>t + <^) cos </>]. joules (47) 

 The semi-system energy Wg at time t is therefore 



Ws = Wie~2^i [1 — cos (2 ojf 4- ^) cos <f}]. joules (48) 

 The initial energy of the system is 



W, = ^ sin^ <A = 2 TF, sin^ <^ = 2 W^ sin« <^ = -^. joules (49) 



The total loss hy dissipation is at any instant 



= -^ {sin' (^ - [1 - cos (2 w# + «^) cos 4>'] e-2^'| . joules (50) 



