96 THE PROBLEM OF PARTITION OF 



the respective numbers of freedoms in these subsystems, 

 and let Cj . . . 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 t , 

 in each subsystem are c^P . . . 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 



? T logX^ P d.C m P=() ... (1; 

 l^t* 



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 cJP being the rate of transmission from the 

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



^r m d.c m P=0 .... (2) 



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 



^d.c m P=0 .... (3) 

 The three equations give 



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



