357-360] General Theory 313 



of this expression vanishes by Kirchhoff's second law (cf. equations (270)), 

 and the increase in heat-production is the essentially positive quantity 



shewing that the original value of function (287) must have been a minimum. 



Conversely, if the original value of function (287) was given to be a 

 minimum, then expression (288) must vanish as far as first powers of e, e', ..., 

 so that we must have 



^Rx = E, etc., 



shewing that the currents x, x, ... must be the natural system of currents. 



359. THEOREM. If two points A, B are connected by a network of con- 

 ductors, a decrease in the resistance of any one of these conductors will decrease 

 (or, in special cases, leave unaltered) the equivalent resistance from A to B. 



Let x be the current flowing from A to B, R the equivalent resistance of 

 the network, and V A V B the fall of potential. The generation of heat per 

 unit time represents the energy set free by x units moving through a 

 potential-difference V A V B . Thus the rate of generation of heat is 



ffc-Tft 



or, since V A V B = Rx, the rate of generation of heat will be Rx*. 



Let the resistance of any single conductor in the network be supposed 

 decreased from R l to R^, and let x l be the current originally flowing through 

 the network. If we imagine the currents to remain unaltered in spite of the 

 change in the resistance of this conductor, then there will be a decrease in 

 the rate of heat-production equal to (R - R^) x?. The currents now flowing 

 are not the natural currents, but if we allow the current entering the net- 

 work to distribute itself in the natural way, there is, by 357, a further 

 decrease in the rate of heat-production. Thus a decrease in the resistance of 

 the single conductor has resulted in a decrease in the natural rate of heat- 

 production. 



If R, R are the equivalent resistances before and after the change, the 

 two rates of heat-production are Rx z and R'x\ We have proved that 

 R'a? < Rx 2 , so that R' < R, proving the theorem. 



GENERAL THEORY OF A NETWORK. 



360. In addition to depending on the resistances of the conductors, the 

 flow of currents through a network depends on the order in which the con- 

 ductors are connected together, but not on the geometrical shapes, positions 

 or distances of the conductors. Thus we can obtain the most general case of 

 flow through any network by considering a number of points 1, 2, ... n, con- 

 nected in pairs by conductors of general resistances which may be denoted by 

 R 12 , jRas, .... If, in any special problem, any two points P, Q are not joined 



