Page:Elementary Principles in Statistical Mechanics (1902).djvu/62

statistical equilibrium which have been described are peculiar to it, or whether other distributions may have analogous properties.

Let $\eta '$ and $\eta ''$ be the indices of probability in two independent ensembles which are each in statistical equilibrium, then $\eta '+\eta ''$ will be the index in the ensemble obtained by combining 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 $\eta '+\eta ''$ will be a constant of motion. Now when infinitesimal forces are added to the compound systems, if $\eta '+\eta ''$ 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, $\eta '+\eta ''$ 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 contained 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

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

(98)

where $\epsilon$ denotes the energy of the system, and $\omega _{1}$, $\omega _{2}$, $\omega _{3}$, its three moments of momentum, and the other letters constants.