198 PROPAGATION OF ELECTRICITY. 



denominator is the sum of the resistances, or the total resistance of 

 the circuit. We have thus 



211, CASE IN WHICH THE CIRCUIT CONTAINS ELECTROMOTIVE 

 FORCES. It is easy to see what Kirchhoff's laws (208) become, when 

 there are electromotive forces at work. 



The first theorem is not modified. The sum of the quantities of 

 electricity which start from, or terminate at, a point is always zero in 

 the permanent state; for this point can neither be an unlimited 

 centre for the production of electricity, nor a centre of absorption. 



The second theorem must be modified. Suppose that in the 

 preceding circuit, at the points where the various conductors terminate, 

 the metals change, or that these places are also points where the 

 summits of other conductors branch off. The current is not the 

 same throughout the whole circuit ; let / a , i b ..... t\ be the different 

 values of the current in the different conductors between any two 

 successive points of contact or of division. From Ohm's law we 

 shall have 



and consequently 



v'.+n+- +ni-(v.- v a )+(v 6 - vy+ . . . +(v,- vy 



= ( V 5 - v ') + (V. - v '>) + '-+ (V. - V',) 



or 



(6) E-^Vf. 



Thus, in a closed circuit, the sum of the products of the resistance 

 of each conductor by the strength of the corresponding current is equal 

 to the algebraical sum of the electromotive forces of the circuit. 



This sum is zero if the circuit is made up of conductors of the 

 same kind, or of metals at the same temperature, for the latter obey 

 the law of tensions. 



The two relations (5) and (6) give all the equations necessary 



