ELEMENTARY ELECTRICITY AND MAGNETISM. 2? 



in which / is the total current and the subscripts refer to the 

 respective branches. Solving these two equations for 7 X and 7 2 , 

 we have : 



t- 1 (l7a} 



To determine the combined resistance of a number of coterminous 

 branches of a circuit. Let / be the total current flowing in the 

 circuit, E the electromotive force between the branch points, and 

 R v R 2 , R y - the resistances of the respective branches. Then 

 E/R lt EjR 2 , EjR y etc., are the currents in the respective 

 branches, and from KirchhorT's law we have 



/=- + +- 

 R l R 2 R 3 



Now, the combined resistance R of all the branches is defined as 

 that resistance through which the electromotive force E between 

 the branch points may maintain the total current /; so that RT=E 

 or /= EjR. Substituting this value of /in the above equation 

 we have 



_ E_ E 



R" R, + R 2 + R 3 * 

 whence 



R = - - - - (18) 



1+1 + 1+.. . 



^1 R 2 R 3 



20. General solution of a network. Consider a network of 

 wires consisting of the branches 2, j, 4, 5, and 6 connected 

 through a wire I i to a battery of which the electromotive force 

 is E, as shown in Fig. 13. Let the long arrows, arbitrarily 

 chosen, represent the directions in which the currents in the vari- 

 ous wires are to be considered as positive in sign ; should a cur- 

 rent in one of the branches, as found by calculation, be negative 



