Page:Deux Mémoires de Henri Poincaré.djvu/11

 Then we can calculate the integral

$$\int\omega\left(\eta_{1}\right)\dots\omega\left(\eta_{n}\right)d\eta_{1}\dots d\eta_{n}$$

extended to positive values of η such that $$\eta_{1}+\dots+\eta_{p}$$ lies between x and x + dx. Let

$$\varphi$$ is a function that depends on the function ω and we have

$$I=\frac{dh}{(p-1)!}\int_{0}^{h}(h-x)^{p-1}\varphi(x)dx$$

$$I'$$ and $$I$$ are calculated in the same manner, we only need to introduce under the sign of integration the factor x or the factor h - x''. Ultimately, we can write

where the factor C is the same in both cases. We do not have to deal with it because it is sufficient to determine the ratio of X to Y.

Now we obtain the Planck formula - which can be regarded as an expression of reality - if we make on the function ω the following hypothesis, which is consistent with quantum theory.

Let ε be the magnitude of the quantum of energy which is specific to the resonators considered, and denote by δ an infinitely small quantity. The function ω is zero, except in the intervals

$$k\epsilon < \eta < k\epsilon + \delta\,$$

and for each of these intervals the integral $$\int_{k\epsilon}^{k\epsilon+\delta}\omega\ d\eta$$ has the value 1.

These data are sufficient for determining the function $$\varphi$$ and the ratio $$\tfrac{Y}{X}$$ for which we find, as I said before, the value given