To determine the values of α0 and ω0, we can use equations
from which we derive
(15) |
and
(16) |
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
(17) |
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 ; 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.