GENERAL PROPERTIES OF A NETWORK OF CONDUCTORS. 337 



intensities of the currents may be determined by Kirchhoff's 

 equations. 



The form of the equations, from the algebraical point of view, 

 enables us to establish several important properties ; but we shall 

 rather attempt to deduce these properties from considerations 

 drawn from the nature of the phenomena. 



Suppose that the network contains n conductors, and let m be 

 the number of summits that is to say, points at which at least 

 three conductors terminate. 



The condition 



(18) ^V- 



applied to the summits will give rise to m - i distinct equations. 

 For consider the two extremities A and A' of a conductor, and 

 apply this law successively to all the summits met in going from 

 the point A to the point A' by a path external to the conductor AA' ; 

 we shall thus obtain a series of different equations, for each of 

 them contains at least one new current; but these equations imply 

 the condition that the sum of the currents which traverse any 

 given plane P, cutting the entire system of conductors, including 

 the first, is equal to zero; this in particular is the case for the sum 

 of the currents which terminate at A', so that the equation relative 

 to this point is already implicitly contained in the preceding ones. 



Let / be the minimum number of conductors which must be 

 removed in order to suppress every closed circuit. These p con- 

 ductors form what we shall call a system of necessary wires, and 

 which may in general be chosen in several different ways. 



The condition relative to closed circuits, 



(19) 2V-)-o, 



gives rise to p different equations. In fact, we shall first show 

 that the addition of a wire in any given network only introduces 

 a new equation of the second order. 



Let A and A' be two points already connected by several paths, 

 V and V their potentials. The difference of potential V-V is 



equal to any given expression 2(2'^-^), 2(/ 2 r 2 -* 2 ) relative 



to the different paths which we may follow in going from A to A'. 

 If we join these two points by a new conductor 7*, containing an 

 electromotive force e, and traversed by the current /, we shall also 

 have 



V-V' = tr-e = ?(t\r l -e l ) = ?(t\ 2 r. 2 -eJ = ...... 



VOL. II. Z 



