Page:Deux Mémoires de Henri Poincaré.djvu/14

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

from which we derive

and

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)

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

where c is a known constant, and equation

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.