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

(8)	$\int \omega \left(\eta _{1}\right)\dots \omega \left(\eta _{n}\right)d\eta _{1}\dots d\eta _{n}=\varphi (x)dx$

$\varphi$ 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

(9)	$nY=C\int _{0}^{h}x(h-x)^{p-1}\varphi (x)dx$

(10)	$pX=C\int _{0}^{h}(h-x)^{p}\varphi (x)dx$

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^[1]. 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

1. This is the first theory of Planck, in which it is assumed that the energy of a resonator can only have values 0, ε, 2ε, 3ε, etc..