96 THE PROBLEM OF PARTITION OF 



the respective numbers of freedoms in these subsystems, 

 and let c x . . . c v be the respective capacities of these 

 subsystems for energy. If P is the equilibrium value of 

 the universal potential, the amounts of energy, E . . . E v , 

 in each subsystem are c x P . . . c v 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 



d.c,,P=0 . . . (1; 



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 accumu- 

 late 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 P being the rate of transmission from the 

 subsystem m, we have, if the r's are constant, 



These two conditions 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 



The three equations give 



c i 



where a and 6 are functions of P alone. The simplest admis- 



