Page:PoincareDynamiqueJuillet.djvu/20

U', V', W' is what become of U, V, W, when we replace t₁ by t'₁; since t'₁ is the same for all molecules, we have:

$dx_{1}^{\prime }=dx_{0}\left(1+{\frac {dU^{\prime }}{dx_{0}}}\right)+dy_{0}{\frac {dU^{\prime }}{dy_{0}}}+dz_{0}{\frac {dU^{\prime }}{dz_{0}}}$

and therefore

$d\tau _{1}^{\prime }=d\tau _{0}\mid 1+{\frac {dU^{\prime }}{dx_{0}}},\quad {\frac {dU^{\prime }}{dy_{0}}},\quad {\frac {dU^{\prime }}{dz_{0}}}\mid$

by setting

$d\tau _{1}^{\prime }=dx_{1}^{\prime }dy_{1}^{\prime }dz_{1}^{\prime }.$

But the element of electric charge is

$d\mu _{1}=\rho _{1}d\tau _{1}^{\prime }$

and moreover for the molecule considered, we have t₁ = t'₁, and therefore ${\tfrac {dU^{\prime }}{dx_{0}}}={\tfrac {dU}{dx_{0}}}$ etc..; we can write:

$d\mu _{1}=\rho _{1}d\tau _{0}\mid 1+{\frac {dU}{dx_{0}}},\quad {\frac {dU}{dy_{0}}},\quad {\frac {dU}{dz_{0}}}\mid$

so that equation (3) becomes:

$\rho _{1}d\tau _{1}(1+\omega )=d\mu _{1}\,$

and equations (2):

$\psi ={\frac {1}{4\pi }}\int {\frac {d\mu _{1}}{r(1+\omega )}},\quad F={\frac {1}{4\pi }}\int {\frac {\xi _{1}d\mu _{1}}{r(1+\omega )}}.$

If we are dealing with a single electron, our integrals are reduced to a single element, provided we consider only the points x, y, x which are sufficiently remote so that r and ω have substantially the same value for all points of the electron. The potentials ψ, F, G, H depend on the position of the electron and also its velocity, because not only ξ₁, η₁, ζ₁ show up in the numerator of F, G, H, but the radial component ω shows up in the denominator. It is of course its position and its velocity at the instant t₁.

The partial derivatives of φ, F, G, H with respect to t, x, y, z (and therefore the electric and magnetic fields) will also depend on its acceleration. Moreover, they depend linearly, since the acceleration in these derivatives is introduced as a result of a single differentiation.

Langevin was thus led to distinguish the electric and magnetic field terms which do not depend on the acceleration (this is what he calls the velocity wave) and those that are proportional to acceleration (that is what he calls the acceleration wave).

The calculation of these two waves is facilitated by the Lorentz transformation. Indeed, we can apply this transformation to the system, so that the velocity of the single electron under consideration becomes zero. We will use for the x-axis the direction of the velocity before the transformation, so that, at the instant t₁,

$\eta _{1}=\zeta _{1}=0\,$