BOLTXXAXN'S THBORKM ON THE AVERAGE DISTRIBUTION 



If the distribution of the N systems in the different phases is such that 

 tit* number in a given phase does not vary with the time, the distribution 

 m mvl to be steady. The condition of this is that C must be constant for 

 all pbJMea belonging to the same path. It will require further investigation to 

 determine whether or not this path necessarily includes all phases consistent 

 with the equation of energy. 



If, however, we assume that the original distribution of the systems according 

 to the different phases is such that C is constant for all phases consistent 

 with the equation of energy, and zero for all phases which that equation shows 

 to be impossible, then the law of distribution will not change with the time, 

 and C will be an absolute constant. 



We have therefore found one solution of the problem of finding a steady 

 distribution. Whether there may be other solutions remains to be investigated. 



Let N(l) denote the number of systems in which q 1 is between 6, and 

 6, + </6,, 9, between 6 t and 6, +db,, and so on, and q n between 6 n and b n + db n , 

 the momenta not being specified otherwise than by their being consistent with 

 the equation of energy, then 



N(b)=t...tN(a l , l}d^...da n (29), 



the integration being extended to all values of the momenta consistent with 

 the equation of energy. 



To simplify the integration let us suppose the variables transformed so that 

 the kinetic energy is expressed in terms of the squares of the component momenta, 



T=l(p 1 a 1 ' + n t a;+...+n n a n *) (30), 



where a,... a. are the transformed momenta, and /*,.../* are functions of the 

 co-ordinates, which we may call moments of mobility, and which, in the case 

 of material points, are the reciprocals of the masses. 



Now let us assume 



.(32), 

 (33), 



.(34). 



