Page:Deux Mémoires de Henri Poincaré.djvu/14

To determine the values of α₀ and ω₀, we can use equations

${\frac {\partial \log \Theta }{\partial \alpha }}=0,\ {\frac {\partial \log \Theta }{\partial \omega }}=0$

from which we derive

(15)	${\frac {\Phi '\left(\alpha _{0}\right)}{\Phi \left(\alpha _{0}\right)}}+\omega _{0}=0$

and

(16)	$\alpha _{0}-{\frac {k}{\beta -\omega _{0}}}=0$

We see from these formulas that α₀ and ω₀ depend on the quantity β, that is to say the total amount of energy h which was communicated to the system; this is a result which was to be expected. Equation (16) tells us further that α₀ will always be real. This quantity determines immediately the average energy of a molecule as it follows from (14) and (16)

$X={\frac {I}{\alpha _{0}}}$

Now we see that the average energy of a molecule is proportional to absolute temperature T. We can write

$\alpha _{0}={\frac {c}{T}}$

where c is a known constant, and equation

(17)	$Y=-{\frac {\Phi '\left(\alpha _{0}\right)}{\Phi \left(\alpha _{0}\right)}}$

which we draw from (13) and (15), gives us the average energy as a function of temperature. We see that this result is independent of the ratio between the numbers n and p.

Suppose now that we know for all temperatures the average energy of a resonator. By (17) we will thus know for all positive values of α the derivative ${\tfrac {d\log \Phi (\alpha )}{d\alpha }}$; we will deduce from them Φ(α) except for a constant factor. Of course, these findings will at first be limited to real values of α, but the function Φ(α) is assumed to be as determined throughout the semi-plane α about which we spoke, when it is given at all points of the real and positive semi-axis.