Page:Deux Mémoires de Henri Poincaré.djvu/12

 by Planck's theory. I did not stop at these calculations and I pass immediately to the principal question, whether the discontinuities that I just mentioned must necessarily be admitted.

I will reproduce the reasoning of Poincaré, but I will at first say that in the formulas that we will encounter, α indicates a complex variable of which the real part $$\alpha_r$$ is always positive. In the representation we will limit ourselves to the half of plane α characterized by $$\alpha_r>0$$, and in integrations in respect to α we will follow a straight line l perpendicular to the axis of real α, and prolonged indefinitely on the two sides. The values of the integrals will be independent of the length of the distance $$\alpha_r>0$$ of this line at the origin of α.

Poincaré introduced an auxiliary function that defines the equation

and demonstrated that the function ω and the derived function $$\varphi$$ can be be expressed by using Φ.

We obtain at first, by inverting (11)

For a similar formula for $$\varphi(x)$$ we notice that in equation (11) we can replace η by any of the variables $$\eta_{1},\dots,\eta_{n}$$. Multiplying the n equations which we obtained, we find

$$\left[\Phi(\alpha)\right]^{n}=\int_{0}^{\infty}\dots\int_{0}^{\infty}\omega\left(\eta_{1}\right)\dots\omega\left(\eta_{n}\right)e^{-\alpha x}d\eta_{1}\dots d\eta_{n}$$

or, by virtue of the formula (8)

$$\left[\Phi(\alpha)\right]^{n}=\int_{0}^{\infty}\varphi(x)^{-\alpha x}dx$$

and by inversion

$$\varphi(x)=\frac{1}{2\pi i}\int_{(L)}\left[\Phi(\alpha)\right]^{n}e^{\alpha x}d\alpha$$

The formulas (9) and (10) now become

$$\begin{array}{l} nY=\frac{C}{2i\pi}\int_{0}^{h}\int_{(L)}x(h-x)^{p-1}[\Phi(\alpha)]^{n}e^{\alpha x}dx\ d\alpha\\ \\pX=\frac{C}{2i\pi}\int_{0}^{h}\int_{(L)}(h-x)^{p}[\Phi(\alpha)]^{n}e^{\alpha x}dx\ d\alpha\end{array}$$