Electric Currents in Netioorks of Conductors. 243 



motive force are known. We shall proceed to apply it to the 

 solution of some network problems, in which the self and 

 mutual induction of the branches is taken into account to 

 determine the distribution of currents and combined resistance 

 at any instant during the variable state. 



§ 15. Consider, first, the case of a galvanometer with a 

 coefficient of self-induction L and resistance G, and shunted 

 by a shunt of resistance S, but wound so as to have no co- 

 efficient of self-induction, and let the shunt and galvanometer- 

 coils be so far removed that there is no coefficient of mutual 

 induction. This is the ordinary practical case. 



Let a battery be joined up and let the battery and connec- 

 tions have a resistance B and electromotive force B (see fig. 16) . 



We have then a two-mesh network. Call the current in 

 the galvanometer- and shunt mesh y and the current in the 

 shunt and battery mesh x+y. Then the current through 

 the galvonometer is y, the current through the shunt is x, and 

 the current through the battery is oc+y. 



The dissipation function H is 



B^' + S^ + G^H, 

 which may be written 



BTS ^T7 + G+S y3_2S^+yy=H; 

 and the electromagnetic energy is 



Hence, by the general equation, 



dt dy +2 dy ~~ ' 



we have the two cycle equations for the y and x+y cycles, 



d 

 and 



,-Ly + G+Sy— S#+y = 



JB + S x+y - Sy = E, 

 or 



d 



and 



(l|+g),-s,=o, 



By + B + S x = E. . 



