﻿Partition of Energy and Newtonian Mechanics. 387 



in which it can exist in the systems characteristic of the 

 state A r being the same for them all. 



The procedure here adopted is more general than that 

 adopted by Jeans inasmuch as it is not assumed that equal 

 volumes in partial spaces corresponding to different types of 

 coordinate are equally probable with each other. 



Now consider a few generalities regarding a more compli- 

 cated system comprising a large number of separate types 



of coordinates. Let A : , A 2 , be characteristics of different 



parts of the system, such that the coordinates involved in 

 the specification of any one characteristic are not involved in 

 any of the others and are in addition all of one specified 



type. Let W b W 2 , be the respective probabilities, 



calculated on the predetermined scale, that in any random 

 choice of a system from all those possible the respective parts 



shall possess the characteristics A l5 A 2 , A complete 



system obtained at a single random choice may possess two 

 or more of these characteristics simultaneously, and the 

 probability that it possesses them all is of the form 



w = w lv w 2 ,w 3 



If we now put 



S = k log W, 



then we know that S is proportional to Boltzmann's measure 

 of the entropy of the system with the specified character- 

 istics, probabilities now being measured on the basis 

 provided by the generalized space as described above. 



Now let E/, E 2 ';> ••• De the energies of those parts of the 

 system with which the properties A 1? A 2 , ... are associated 

 and let E be the total energy given by 



E = E 1 ' + E 2 ' + 



The total entropy S is then given by 



S =2MogW,.. 



Now the characteristics A b A 2 , may be chosen so as 



to determine the partition of energy. To be precise let any 

 characteristic property A,, be satisfied if the corresponding 

 energy E/ lies between E,. — \e r and E,. + ^e,.. Let it be 

 assumed as a property of the system that if left to itself it 

 will assume a steady state in which the energy is divided in 

 a definite manner, namely one in which E,/ becomes equal 

 to the corresponding E,. at least to within the small range e . 

 Then W must be equal to unity for these values of E/. E_ • 



and this is its maximum value. Tt follows that S is also 



2 02 



