JLTXMAJnc' THBORKM ON THB AVERAOB DISTRIBUTION 



- 



Knee Mr one of the variable* may be taken for q n , the law of distribution 

 of ndoe. of the kinetic energy is the same for all the variables. The mean 

 nkhM of the kinetic eneigy corresponding to any variable is 



Kss \(E-V) = - T (52). 



n *i 



The maximum value is T=nK (53). 



The mean value of If is 



When n is very large, the expression (51) approximates to 



].,-"-* dk ................................. (55). 



Recapitulation. 



The result of our investigation may therefore be stated as follows : 

 (a) We begin by considering a set of material systems which satisfy the 

 general equations of dynamics (2) and (3), and the equation of energy (1). If 

 in these systems the distribution of configurations satisfies equation (45), and 

 the distribution of motion satisfies equation (51), these equations will continue 

 to be satisfied during the subsequent motion of the system. One result of 

 equation (51), to which we shall have to refer, is that the average kinetic 

 energy corresponding to any one of the variables is the same for every one 

 of the variables of the system. 



(ft) We now turn our attention to a system of real bodies enclosed in 

 a rigid vessel impervious to matter and to heat. We know by experiment that 

 in such a system the temperature cannot remain steady in every part unless 

 the temperature of every part of the system is the same, and that this condition 

 is necessary in whatever manner the configuration of the system may be varied 

 by altering the position and mean density of the portions of sensible size into 

 which we are able to divide it. 



Now if the system of real bodies is a material system which satisfies the 

 equations of dynamics, and if equations (45) and (51) are also satisfied, the 

 condition of the system will, as we have shewn, (a), be steady in every respect, 



