HAVE ANALOGOUS PROPERTIES. 39 



In like manner let us imagine a second ensemble formed by 

 distributing in phase the system of particles in the other shell 

 according to the index 



where the letters have similar significations, and O, O x , O 2 , 11 3 

 the same values as in the preceding formula. Each of the 

 two ensembles will evidently be in statistical equilibrium, and 

 therefore also the ensemble of compound systems obtained by 

 combining each system of the first ensemble with each of the 

 second. In this third ensemble the index of probability will be 



k + ^-!^ + SL^ + 2d^ + a3L-, (ioo) 



vy i/j 1/2 *a 



where the four numerators represent functions of phase which 

 are constants of motion for the compound systems. 



Now if we add in each system of this third ensemble infini- 

 tesimal conservative forces of attraction or repulsion between 

 particles in different shells, determined by the same law for 

 all the systems, the functions o^ + &>', &> 2 + o> 2 ', and &> 3 + w 3 ' 

 will remain constants of motion, and a function differing in- 

 finitely little from e l + e will be a constant of motion. It 

 would therefore require only an infinitesimal change in the 

 distribution in phase of the ensemble of compound systems to 

 make it a case of statistical equilibrium. These properties are 

 entirely analogous to those of canonical ensembles.* 



Again, if the relations between the forces and the coordinates 

 can be expressed by linear equations, there will be certain 

 " normal " types of vibration of which the actual motion may 

 be regarded as composed, and the whole energy may be divided 



* It would not be possible to omit the term relating to energy in the above 

 indices, since without this term the condition expressed by equation (89) 

 cannot be satisfied. 



The consideration of the above case of statistical equilibrium may be 

 made the foundation of the theory of the thermodynamic equilibrium of 

 rotating bodies, a subject which has been treated by Maxwell in his memoir 

 " On Boltzmann's theorem on the average distribution of energy in a system 

 of material points." Cambr. Phil. Trans., vol. XII, p. 547, (1878). 



