38 OTHER DISTRIBUTIONS 



statistical equilibrium which have been described are peculiar 

 to it, or whether other distributions may have analogous 

 properties. 



Let rj r and 77" be the indices of probability in two independ- 

 ent ensembles which are each in statistical equilibrium, then 

 rf _j_ y w ni De the index in the ensemble obtained by combin- 

 ing each system of the first ensemble with each system of the 

 second. This third ensemble will of course be in statistical 

 equilibrium, and the function of phase vf + if 1 will be a con- 

 stant of motion. Now when infinitesimal forces are added to 

 the compound systems, if r/ + rf 1 or a function differing 

 infinitesimally from this is still a constant of motion, it must 

 be on account of the nature of the forces added, or if their action 

 is not entirely specified, on account of conditions to which 

 they are subject. Thus, in the case already considered, 

 V + ??" is a function of the energy of the compound system, 

 and the infinitesimal forces added are subject to the law of 

 conservation of energy. 



Another natural supposition in regard to the added forces 

 is that they should be such as not to affect the moments of 

 momentum of the compound system. To get a case in which 

 moments of momentum of the compound system shall be 

 constants of motion, we may imagine material particles con- 

 tained in two concentric spherical shells, being prevented from 

 passing the surfaces bounding the shells by repulsions acting 

 always in lines passing through the common centre of the 

 shells. Then, if there are no forces acting between particles in 

 different shells, the mass of particles in each shell will have, 

 besides its energy, the moments of momentum about three 

 axes through the centre as constants of motion. 



Now let us imagine an ensemble formed by distributing in 

 phase the system of particles in one shell according to the 

 index of probability 



^-I+|+S+S' (98) 



where e denotes the energy of the system, and j , o> 2 , &> 3 , its 

 three moments of momentum, and the other letters constants. 



