355-357] Generation of Heat 311 



maximum or a minimum, is also arrived at by the solution of au equal 

 number of equations. If it is possible to discover a function of the unknown 

 quantities such that the two systems of equations become identical, i.e. if 

 the equations which express that the function is a maximum or a minimum 

 are the same as those which contain the solution of the physical problem 

 then we may say that the solution of the problem is contained in the single 

 statement that the function in question is a maximum or a minimum. 



Examples of functions which serve this purpose are not hard to find. In 

 | 189, we proved that when an electrostatic system is in equilibrium, its 

 potential energy is a minimum. Thus the solution of any electrostatic 

 problem is contained in the single statement that the function which ex- 

 presses the potential energy is a minimum. Or, again, the solution of any 

 dynamical problem is contained in the statement that the "action" is a 

 minimum, while in thermodynamics the equilibrium state of any system 

 can be expressed by the condition that the "entropy" shall be a maximum. 

 It will now be shewn that the function which expresses the total rate of 

 generation of heat plays a similar role in the theory of steady electric 

 currents. 



357. THEOREM. When a steady current flows through a network of 

 conductors in which no discontinuities of potential occur (and which, therefore, 

 contains no batteries), the currents are distributed in such a way that the rate of 

 generation of heat in the network is a minimum, subject only to the conditions 

 imposed by Kirchhoff's first law; and conversely. 



To prove this, let us select any closed circuit PQR ... P in the network, 

 and let the currents and resistances in the sections PQ, QR,... be # 15 # 2 

 and R lt jR 2 > ____ Let the currents and resistances in those sections of the net- 

 work which are not included in this closed circuit be denoted by x a , x b , ... 

 and R a , RI, .... Then the total rate of production of heat is 



R0 8 + 2JW ........................... (285). 



A different arrangement of currents, and one moreover which does not 

 violate Kirchhoff's first law, can be obtained in imagination by supposing all 

 the currents in the circuit PQR ... P increased by the same amount e. The 

 total rate of production of heat is now 



and this exceeds the actual rate of production of heat, as given by expression 



(285), by 



^(2^6 + e 2 ) ........................... (286). 



Now if the original distribution of currents is that which actually occurs 

 in nature, then 



ZR^ = 0, 



by Kirchhoff's second law. Thus the rate of production of heat, under the 



