345, 346] Kirchhoff's Laws 301 



between different paths, and it may be important to determine how the 

 electricity will pass through a network of conductors containing junctions. 



The first principle to be used is that, since the currents are supposed 

 steady, there can be no accumulation of electricity at any point, so that the 

 sum of all the currents which enter any junction must be equal to the sum 

 of all the currents which leave it. Or, if we introduce the convention that 

 currents flowing into a junction are to be counted as positive, while those 

 leaving it are to be reckoned negative, then we may state the principle in 

 the form : 



The algebraic sum of the currents at any junction must be zero. 



Again, let the various junctions be denoted by A, B, G..., and let their 

 potentials be J^, V& V c ____ Let R AB ^ e the resistance of any single con- 

 ductor connecting two junctions A and B, and let C AB be the current flowing 

 through it from A to B. Let us select any path through the network of 

 conductors, such as to start from a junction and bring us back to the starting 

 point, say ABC...NA. Then on applying Ohm's law to the separate con- 

 ductors of which this path is formed, we obtain ( 341) 



R 



BC> 



By addition we obtain 2CR = ................................. (269), 



where the summation is taken over all the conductors which form the closed 

 circuit. 



In this investigation it has been assumed that there are no discontinuities 

 of potential, and therefore no batteries, in the selected circuit. If dis- 

 continuities occur, a slight modification will have to be made. We shall 

 treat points at which discontinuities occur as junctions, and if A is a junction 

 of this kind, the potentials at A on the two sides of the surface of separation 

 between the two conductors will be denoted by V A and V A . 



Then, by Ohm's Law, we obtain for the falls of potential in the different 

 conductors of the circuit, 



V G R 



AB> 



and by addition of these equations 



The left-hand member is simply the sum of all the discontinuities of 

 potential met in passing round the circuit, each being measured with its 

 proper sign. It is therefore equal to the sum of the electromotive forces of 

 all the batteries in the circuit, these also being measured with their proper 

 signs. 



