Page:Deux Mémoires de Henri Poincaré.djvu/13

and Poincaré again transforms them by substitutions

$x=n\omega ,\ h=n\beta ,\ p=nk$

which give

${\begin{array}{l}nY={\frac {Cn^{p+1}}{2i\pi }}\int _{0}^{\beta }\int _{(L)}{\frac {\omega }{\beta -\omega }}\Theta ^{n}d\omega \ d\alpha \\\\pX={\frac {Cn^{p+1}}{2i\pi }}\int _{0}^{\beta }\int _{(L)}\Theta ^{n}d\omega \ d\alpha \end{array}}$

he posed

$\Theta =\Phi (\alpha )e^{\alpha \omega }(\beta -\omega )^{k}$

Note that ω is nothing else than the average energy of a single resonator for the case

$\eta _{1}+\dots +\eta _{n}=x$

that β is the value that would be ω if all available energy h was in the resonator, and that k is the ratio between the number of molecules and the resonators.

When, in the applications of the probability theory to the molecular theories, we seek the state of a system that presents the maximum of probability, we always find that, thanks to the immense number of the molecules, this maximum is so pronounced that one can neglect the probability of all the states which deviate appreciably from the most probable state. In the case which occupies us, there is something similar.

Let us admit with Poincaré that, for values given of h and β, the function Θ has a maximum for α = α₀, ω = ω₀ and passes through the point α₀, the place of the maximum, the line l whose distance α₀ in the beginning could be selected at will. As the exponent n is very high, the maximum of Θⁿ is extremely pronounced and the only elements of the integrals which we have to take into account, are those who are in the immediate vicinity of α₀ and of ω₀. That immediately gives us for the sought ratio

${\frac {nY}{pX}}={\frac {\omega _{0}}{\beta -\omega _{0}}}$

and, by virtue of the equation

$nY=pX=h=n\beta$

(13)	$Y=\omega 0$

(14)	$X={\frac {\beta -\omega _{0}}{k}}$