40 OTHER DISTRIBUTIONS 



into parts relating separately to vibrations of these different 

 types. These partial energies will be constants of motion, 

 and if such a system is distributed according to an index 

 which is any function of the partial energies, the ensemble will 

 be in statistical equilibrium. Let the index be a linear func- 

 tion of the partial energies, say 



Let us suppose that we have also a second ensemble com- 

 posed of systems in which the forces are linear functions of 

 the coordinates, and distributed in phase according to an index 

 which is a linear function of the partial energies relating to 

 the normal types of vibration, say 



^~i?'*'~if (102) 



Since the two ensembles are both in statistical equilibrium, 

 the ensemble formed by combining each system of the first 

 with each system of the second will also be in statistical 

 equilibrium. Its distribution in phase will be represented by 

 the index 



and the partial energies represented by the numerators in the 

 formula will be constants of motion of the compound systems 

 which form this third ensemble. 



Now if we add to these compound systems infinitesimal 

 forces acting between the component systems and subject to 

 the same general law as those already existing, viz., that they 

 are conservative and linear functions of the coordinates, there 

 will still be n + m types of normal vibration, and n + m 

 partial energies which are independent constants of motion. 

 If all the original n + m normal types of vibration have differ- 

 ent periods, the new types of normal vibration will differ infini- 

 tesimally from the old, and the new partial energies, which are 

 constants of motion, will be nearly the same functions of 

 phase as the old. Therefore the distribution in phase of the 



