Page:Elementary Principles in Statistical Mechanics (1902).djvu/63

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

$A'-{\frac {\epsilon '}{\Theta }}+{\frac {\omega _{1}'}{\Omega _{1}}}+{\frac {\omega _{2}'}{\Omega _{2}}}+{\frac {\omega _{3}'}{\Omega _{3}}},$

(99)

where the letters have similar significations, and $\Theta$, $\Omega _{1}$, $\Omega _{2}$, $\Omega _{3}$ 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

$A+A'-{\frac {\epsilon +\epsilon '}{\Theta }}+{\frac {\omega _{1}+\omega _{1}'}{\Omega _{1}}}+{\frac {\omega _{2}+\omega _{2}'}{\Omega _{2}}}+{\frac {\omega _{3}+\omega _{3}'}{\Omega _{3}}},$

(100)

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 infinitesimal conservative forces of attraction or repulsion between particles in different shells, determined by the same law for all the systems, the functions $\omega _{1}+\omega _{1}'$, $\omega _{2}+\omega _{2}'$, and $\omega _{3}+\omega _{3}'$ will remain constants of motion, and a function differing infinitely little from $\epsilon _{1}+\epsilon '$ 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.^[1]

Again, if the relations between the forces and the coördinates 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

1. 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).