Electric Currents in Networks of Conductors. 251 

 Solving it we have 



m 2 + 



^Gm G?_ _G 2 G+S 

 L, + 4L 1 2 ~4L 1 2 SCL,' 



or 



/ , G \ , V / G 2 S 2 C 2 -4G+SL 1 SC 



r + 2lJ = ± 2Lm. 



Hence, for the roots to be imaginary, 



4L X G + S SC must be greater than G 2 S 2 C 2 , 

 or 



4L* G , GS 

 C > G + S* 



If this relation holds good, then the discharge is oscillatory 

 in the condenser; and accordingly we see that to prevent 

 electrical oscillation in the galvanometer circuit, the product 

 of resistance of the galvanometer and combined resistance of 

 galvanometer and shunt must be equal to or greater than 

 four times the self-induction of the galvanometer divided by 

 the capacity of the condenser. 



We may write the solution of the quadratic above, 



m= 



/G + 



2L 1 ± V L i 



where 



= -a± V-l/3, 



— IT, and **= V T^X _ 4Lf 



fG + S 



S 



G 2 



LxC. 



4L 2 



/G + S 



G 2 



and accordingly when fi is real, that is when 



G + S 



8 . G 2 

 is > 



LxO x ° ^4Lf 

 we have/ for solution of equation, 



q=Ae- ai cos fit + fie~ at sin fit. 

 When £=0, g = Q = the original charge of the condenser, and 

 ^| = when *=0; 



therefore Q=A and Q-=B: 



fi 



and q = Q*-«* /cos fit + | sin fit). 



