Electric Currents in Networks of Conductors. 223 



nect all the points together is p — 1. If we add one line more 

 we make a closed circuit somewhere in the system ; that is 

 to say, a portion of space is enclosed and forms a cell cycle 

 or mesh. Every fresh line added then makes a fresh mesh; 

 and hence if there are I lines altogether joining p points, the 

 number of cycles or cells will be & = Z— (j> — 1). Now let 

 such a system of points and lines represent conducting wires 

 joining fixed points, and forming a conducting network. Let 

 a symbol be affixed to each point which represents the elec- 

 trical potential at that point, and also a symbol affixed to 

 each line representing the electrical resistance of the con- 

 ductor represented by it. In such a diagram of conductors 

 the form is a matter of indifference so long as the connections 

 are not disturbed and lines are not made to cross unless the 

 conductors they represent are in contact at that point. 



Consider a network, PI. VI. fig. 1, formed by joining nine 

 points by thirteen conductors. Then there will be 13 — (9 — 1) 

 = 5 cycles or cells. Now let an electromotive force E act in 

 one branch B, and give rise to a distribution of currents in the 

 network. Let a, /3, 7, S, &c. represent the potentials at the 

 points, and A, B, C, D, &c. the electrical resistances of the 

 conductors joining these points, and imagine that round each 

 cycle or circuit an imaginary current flows, all such currents 

 flowing in the same direction. 



A circuit is considered to be circumnavigated positively 

 w 7 hen you walk or go round it so as to keep the boundary on 

 your right hand. Hence, going round an area A in the di- 

 rection of the arrow is positive as regards the inside if you 

 walk inside the boundary-line, and negative as regards ex- 

 ternal space B if you walk in the same direction round the 

 outside. We shall consider a current, then, as positive when 

 it flows round a cycle in the opposite direction to the hands 

 of a watch. Returning then to the network, we consider that 

 round each cycle flows an imaginary current in the positive 

 direction. The real currents in the conductors are the 

 differences of these in adjacent cycles or meshes, and the 

 imaginary currents will necessarily fulfil the condition of con- 

 tinuity, because any point is merely a place through which 

 imaginary currents flow, and at which therefore there can be 

 no accumulation nor disappearance of electricity. 



Let x, y, z, &c. denote these imaginary like-directed cur- 

 rents. Then x — y denotes the real current in the branch I, 

 and similarly x—z that in branch H. Then x, y, z, &c. may 

 be called the cyclic symbols of these areas. The cyclic symbol 

 of external space is taken as zero; hence the real current in 

 branch B is simply x. 



Let an electromotor act on the branch B, bringing into 



R2 



