KENNELLY. — OSCILLATING-CURRENT CIRCUITS. 403 



circuit as shown by an ideally perfect voltmeter would be 707.1 volts. 

 The question is what would such an instrument show in the presence 

 of a known damping 1 The reading of the instrument would evidently 

 fall as the damping continued, and, in most practical cases, would Ml 

 very rapidly ; so that the inquiry must be limited to a certain definite 

 instant during the charge or discharge ; or at least to a certain definite 

 interval of time within the process. At any instant we have 



^2= P7e-2-'sin^o)#. (Ph. Q.)2 (61) 



The time integral of this square from time t = Q to t = cc is 



r^'dt = ^(-^}i=^ (I- cos' cl.). (Ph. Q.)^ sees. (62) 



But after the lapse of a time ti seconds, including m complete periods 

 of the oscillating quantity, its value will have become 



'Vmr = T>-^'"^ sin o^t = Foe-^'i sin uit, Ph. Q. (63) 



where T is the period of the oscillation, and m is any positive integer ; 

 so that 



f\^dt = ^(1- cos'^ <^) (1 - e-2-<i). (Ph. Q.y sees. (64) 

 t/ 4 ■'' 



The mean of this square during the interval is 



7 A''^^ = iir (1 - cos^ cf>) (1 - e-2^^0, (Pb. Q)^ (65) 



h Jo 4-''ri 



and the root of this mean square during the interval is 



hJo V2 \ 2 J-h J 



«1 



y^f'^'i ( . , ./sinh-L^A -r>, ^ , . 



But Fo/\/2 is the r. m. s. value of the undamped oscillating quantity, 



and V ^e'' 2 1 '^ 'I would be the r. m. s. value of the oscillating quantity 

 at the mid-interval if it were to continue thereat throughout un- 



