RESISTANCE OF A MULTIPLE CONDUCTOR. 1 95 



V 1} V 2 V n , be the potentials at the summits A v A 2 A n , and 



let t\, / 2 i n be the currents reckoned positively when the circuit 



is traversed in a certain direction. These strengths are not equal, 

 for at the various summits there may be other conductors which 

 bring or carry away currents. 

 We shall have successively : 



For the first conductor i^r^ = V x - V 2 , 

 For the second conductor 2 2 r 2 = V 2 - V 8 , 



For the n th conductor i n r n = V n -V r 



Adding these equations together, all the potentials disappear, and 

 we have finally 



from which 



(4) ^i>=0. 



The two expressions (3) and (4) are known as Kirchhofs laws. 



Fig. 52- 



209. RESISTANCE OF A MULTIPLE CONDUCTOR. As an appli- 

 cation of these theorems, let us consider the case in which the 

 circuit divides into multiple arcs, between two points A and B 

 (Fig. 52). Let I be the current in the undivided part in front of 

 A and beyond B, r v r 2 . . . r n the resistances of the conductors, 

 *i, I 2 . . i n the respective currents, and lastly let R be the resistance 

 of the single circuit which would be equivalent to the multiple circuit 

 between the same two points. We shall have 



O 2 



