KENNELLY. — OSCILLATING-CURRENT CIRCUITS. 415 



has wider applications, in these respects, than the polar-coordinate 

 vector-diagram. 



Analytically, we have the following relations : — 



The fundamental differential equation for quantity q is satisfied by 



q = .1 €-(--") « + Be~('+^)i coulombs (77) 



where A and B are integration constants, while -^ and O follow from 

 the construction of the triangle OPQ of Figure 17. Choosing the con- 

 stants consistently with the discharge of a condenser initially charged 

 to potential Uq volts, the discharging p. d. after t seconds is 



u = C7o cot "A ^"^' sinli (^^-t + O'l'^^) volts (78) 



= lJ,e-'Hmh{nt + gd-^if) volts (79) 



form which q follows by the relation q = u/s = uc coulombs. U^ is 

 the initial vector value of the discharging p. d. by Figures 17 and 18. 

 The instantaneous current i is 



i = /o^"'" sinh Vtt amperes (80) 



where /« = Ujm = IJjz^. amperes (81) 



The current i will therefore be a maximum when 



tanh n# = fir = sin t/', numeric (82) 



or Ut = ^</"V- hyp. radians (83) 



The emf. of self-induction in the circuit at any instant is 



e= U^ cot ^ €--^« sinh {Qt — gd'^if/) volts (84) 



= Lr.r^' sinh (O;^ - gd-^il/). volts (85) 



The apparent resistance of the circuit u/i is 



Z= p + ICl coth nt. ohms (86) 



That is, the apparent resistance of the circuit, judging from the dis- 

 charging p. d. and the discharging current, commences at cc and tends 

 rapidly to the limit (p + IQ) ohms. 



The instantaneous power of the condenser in the circuit is 



p = UJo cot ij/ e-2x< sinh nt • sinh (nt + ^fi?~V) watts (87) 

 = Uj^i-~'i sinh nt ■ sinh (nt + gd-^if). watts (88) 



