The Equipartitioii Law in a System of Particles. 587 



kind which have coordinates falling in a given range 

 dxdy dzd^xd^ydtyz. To each macroscopic state will corre- 

 spond in general a number of microscopic states, and since 

 we have seen that all microscopic states are equally likely to 

 occur, the probability of the system's being in a given macro- 

 scopic state is proportional to the number of corresponding 

 microscopic states. 



For the system under consideration let a particular 

 macroscopic state be specified by stating that N a , N a , N a ... 

 N 6 , N 6 ,N 6 ,... &c, are the number o£ particles of the 

 corresponding masses m a , m b , &c, which fall in the specified 

 elementary regions dx dy dz d^ x d^y dty z , Nos. la, 2a, 3a, . . . 

 lb, 2b, ?>b, . . . &c. By familiar methods of calculation* it is 

 evident that the number of arrangements by which the 

 particular distribution of particles can be effected, that is, in 

 other words, the number of microscopic states W which 

 -correspond lo the given macroscopic state, is given by the 

 •expression 



w _ |N.|g,|lj... . 



N„ X„ N\, . . . N» X„ X 



and this number W is proportional to the probability that 

 the system will be found in the particular macroscopic state 

 considered. 



If now we assume that each of: the regions 



dxdy dzdtyxd'tyyd'tyz) Nos. la, 2a, 3a, . . . \l>, 2b, 3b, . . . &c. 



is great enough to contain a large number of particles t, 



we may apply the Stirling formula |N= (V /2ttN I —J for 



evaluating jN a , [Nj, &c, and, omitting negligible terms, shall 

 obtain for log \V the result 



/x' n' n" n" x"' n'" \ 

 logW=-K a [ Wa loz Wa + Wa log K + ^ log K + ...) 



- n AxJ^x^n, 1( ^x^ x, lo ^N7 + --J 



&c. 



* See for example Planck, loc. tit. 



t The idea of successive orders of infinitesimals which permit the 

 differential region dxdydz&tyxdtyydtyz to contain a large number of 

 particles is a familiar one in mathematics. 



