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

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