312 Steady Currents in Linear Conductors [OH. ix 



new imaginary distribution of currents exceeds that in the actual distribu- 

 tion by e 2 2.Ri, an essentially positive quantity. 



The most general alteration which can be supposed made to the original 

 system of currents, consistently with Kirchhoff's first law remaining satisfied, 

 will consist in superposing upon this system a number of currents flowing 

 in closed circuits in the network. One such current is typified by the 

 current e, already discussed. If we have any number of such currents, the 

 resulting increase in the rate of heat production 



= 2^(^ + 6 + e + e" + ...) 2 - 2JW, 



where e, e', e''... are the additional currents flowing through the resistance 

 RI. As before this expression 



(e + e' + e" +...) + 2ft (e 4- e' + e" + ...) 2 



by Kirchhoff's second law. This is an essentially positive quantity, so that 

 any alteration in the distribution of the currents increases the rate of heat- 

 production. In other words, the original distribution was that in which the 

 rate was a minimum. 



To prove the converse it is sufficient to notice that if the rate of heat- 

 production is given to be a minimum, then expression (286) must vanish as 

 far as the first power of e, so that we have 



21^ = 0, 



and of course similar equations for all other possible closed circuits. These, 

 however, are known to be the equations which determine the actual dis- 

 tribution. 



358. THEOREM. When a system of steady currents flows through a net- 

 work of conductors of resistances R 1 ,R 2) ..., containing batteries of electromotive 

 forces E l) E 2 , ... , the currents o^, # 2 , ... are distributed in such a way that the 

 function 



^Rx*-Z$Ex ........................... (287), 



is a minimum, subject to the conditions imposed by Kirchhoff's first law ; and 

 conversely. 



As before, we can imagine the most general variation possible to consist 

 of the superposition of small currents e, e', e", ... flowing in closed circuits. 

 The increase in the function (287) produced by this variation is 



= 2e. (^Rx - $E) + 2e' (...) + ... 



+ 2(e-f e'+...) 2 ........................... (288). 



If the system of currents x, x, ... is the natural system, then the first line 



