666 Prof. YV. Peddie on the 



Let there be altogether v subsystems, let N t . . . . N„ be the 

 numbers of freedoms in each subsystem, and let c x . . . . c v be 

 the capacities of the subsystems for energy. If P be the 

 equilibrium value of the universal potential, the amounts 

 of energy, Ej . . . . E„, in each subsystem are CjP, . . . . c„P 

 respectively. As in Boltzmann's treatment, the equilibrium 

 state is the most probable state ; and so, following Planck's 

 modification of that treatment, the probability being estimated 

 by the number of ways in which cP units of energy can be 

 contained in N freedoms, we obtain, as the condition of 

 equilibrium, the equation 



v "N" 4-r P 



s "log im+ : r -d.c m v=o (i) 



Now, in the condition of statistical equilibrium, as in the 

 approach to it, there is constant transmission of energy from 

 one subsystem to another ; and the energy tends to accu- 

 mulate in those subsystems from which the rate of transmission 

 is slowest. Hence the total rate of transmission tends to a 

 minimum. So r m c m V being the rate of transmission from 

 the subsystem ?n, we have, if the r's are constant, 



2«"r„B.c m P = 0. ...... (2) 



i 



These two equations imply nothing more regarding the 

 potential P than that it is statistically uniform throughout 

 the total system. It might be slowly varying with time. If 

 we further add the condition of conservation of energy, we 



get 



£»d.c m P = . (3) 



i 



The three equations give 



N OT 



E m = ^ P = e «+6r ) r_ 1 ' T • • • • ( 4 ) 



where a and h are functions of P alone. The simplest 

 admissible conditions are a = u¥~ i , b = ySP -1 , where a and /3 

 are absolute constants, in which case (4) becomes 



E»->»r==^g^> • ... • (4') 



€ P ' — 1 -. - > 



