Page:Deux Mémoires de Henri Poincaré.djvu/10

 Let us consider a system made up of n resonators of Planck and p molecules, n and p being very great numbers; let us suppose that all the resonators are equal between them and that it is the same for the molecules. Let us indicate by $$\xi_{1},\dots,\xi_{p}$$ the energies of the molecules and by $$\eta_{1},\dots,\eta_{n}$$ those of the resonators; each one of these variables will be able to take all the positive values.

Poincaré showed first that the probability so that the quantities of energy are between the limits $$\xi_{1}$$ and $$\xi_{1}+d\xi_{1},\dots,\xi_{p}$$, and $$\xi_{p}+d\xi_{p}$$, $$\eta_{1}$$ and $$\eta_{1}+d\eta_{1},\dots,\eta_{n} $$, $$\eta_{n}$$ and $$\eta_{n}+d\eta_{n}$$, can be represented by

$$\omega\left(\eta_{1}\right)\dots\omega\left(\eta_{n}\right)d\eta_{1}\dots d\eta_{n}d\xi_{1}\dots d\xi_{p}$$

where ω is a function for which we can make different hypotheses.

Once we know this function we can tell how much energy h will be distributed over the molecules and resonators. For this purpose, we can imagine a space of p + n dimensions, $$\xi_{1},\dots,\xi_{\rho},\eta_{1},\dots,\eta_{n}$$, the infinitely thin layer S, in which the total energy

$$\xi_{1}+\dots+\xi_{p}+\eta_{1}+\dots+\eta_{n}$$

lies between h and an infinitely close value h + dh. The three integrals will be calculated

$$\begin{array}{l} I=\int\omega\left(\eta_{1}\right)\dots\omega\left(\eta_{n}\right)d\eta_{1}\dots d\eta_{n}d\xi_{1}\dots d\xi_{p}\\ \\I'=\int x\omega\left(\eta_{1}\right)\dots\omega\left(\eta_{n}\right)d\eta_{1}\dots d\eta_{n}d\xi_{1}\dots d\xi_{p}\\ \\I''=\int(h-x)\omega\left(\eta_{1}\right)\dots\omega\left(\eta_{n}\right)d\eta_{1}\dots d\eta_{n}d\xi_{1}\dots d\xi_{p}\end{array}$$

$$\left(x=\eta_{1}+\dots+\eta_{n}\right)$$

extended to the layer S, and we have $$\tfrac{I'}{I}$$ for the energy that the resonators take, and $$\tfrac{I''}{I}$$ for that of all the molecules. Therefore, if Y is the mean energy of a resonator, and X is that of a molecule,

$$nYI=I',\ pXI=I''$$

To calculate the integral I, we may first give fixed values to variables $$\eta_{1},\dots,\eta_{n}$$ and consequently to their sum x, and extend the integration over ξ for all positive values of these variables, for which the sum $$\xi_{1}+\dots+\xi_{p}$$ is between h - x and h - x + dh. This gives us

$$\int d\xi_{1}\dots d\xi_{p}=\frac{1}{(p-1)!}(h-x)^{p-1}dh$$