48 ELEMENTS OF ELECTRICITY AND MAGNETISM. 



lamps or other units connected in parallel is equal to the recipro- 

 cal of the sum of the reciprocals of the respective resistances. 

 The proposition (#) is almost self-evident. Proposition (3) may 

 be established as follows : Let E be the electromotive force be- 

 tween the points A and B where the circuit divides into a 

 number of branches. Then, according to Ohm's Law, we have 



p 



jilt _ /:::\ 



~ R"' 



where R' , R" and R" ' are the resistances of the respective 

 branches, and /', I" and I" ' are the currents flowing in the 

 respective branches. 



Let / bethetotalcurrentflowinginthecircuit(=/ / + /"-|-/ /// ). 

 The combined resistance of the branches is defined as the resist- 

 ance through which the electromotive force E between the 

 branch points would be able to force the total current /. That 

 is, the combined resistance is defined by the equation 



'- H 



in which R c is the combined resistance. Adding equations (i), 

 (ii) and (iii), member by member, and substituting /R c for 

 /' + /" + /"/, we have 





R e ~ R'^ R" ^ R" 

 whence 



i 



A\. = 2 ~~ 



_L _L l 



W ~^~ vft 



R' " R" R' 



