BOLTZMANX 8 THEOREM ON THE AVERAGE DISTRIBUTION* 



have to deal with in terms of the latter set of variables. Then since by 



dE 



'-J>....(19). 



Similarly we find for the final state of the system 



dq t rfg.rfp, dp n = dq t dq n dp dp n dE (20). 



The left-hand members of these equations have been proved equal, and in 

 the right-hand members dE is the same at the beginning and end of the 

 motion. Dividing out dE we find 



1 1' 

 dq t ' dq.'dp,' dp* = dq t dq n dp,\ dp n ' (21). 



This equation is applicable to the case in which the total energy is supposed 

 not to vary from one particular instant of the motion to another, and in which, 

 therefore, the 2n variables are no longer independent, but, being subject to the 

 equation of energy, are reduced to 2 1. 



Statistical Specification. 



We have hitherto, in speaking of a phase of the motion of the system, 

 supposed it to be defined by the values of the n co-ordinates and the n momenta. 

 We shall call the phase so defined the phase (pq). We shall now adopt a 

 wider definition by saying that the system is in the phase (a,6) whenever the 

 values of the co-ordinates are such that 5, is between 6, and &, + <$>,, q t between 

 l>, and b t +db t , and so on; also p t between a, and a t + da lt and so on. The 

 limits of the first component of momentum,/),, are not specified, because the value 

 t" /i, is not independent of the other variables, being given in terms of E 

 and the other 2n 1 variables in virtue of the equation of energy. 



The quantities a, b are of the same kind as p and q respectively, only 

 they are not supposed to vary on account of the motion of the system. In 

 the statistical method of investigation, we do not follow the system during 

 ito motion, but we fix our attention on a particular phase, and ascertain whether 



