Page:Elementary Principles in Statistical Mechanics (1902).djvu/64

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 function of the partial energies, say

$A-{\frac {\epsilon _{1}}{\Theta _{1}}}\ldots -{\frac {\epsilon _{n}}{\Theta _{n}}}.$

(101)

Let us suppose that we have also a second ensemble composed of systems in which the forces are linear functions of the coördinates, 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

$A'-{\frac {\epsilon _{1}'}{\Theta _{1}'}}\ldots -{\frac {\epsilon _{m}'}{\Theta _{m}'}}.$

(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

$A+A'-{\frac {\epsilon _{1}}{\Theta _{1}}}\ldots -{\frac {\epsilon _{n}}{\Theta _{n}}}-{\frac {\epsilon _{1}'}{\Theta _{1}'}}\ldots -{\frac {\epsilon _{m}'}{\Theta _{m}'}},$

(103)

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 coördinates, 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 different periods, the new types of normal vibration will differ infinitesimally 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