Page:PoincareDynamiqueJuillet.djvu/23

They then vanish in virtue of equations $\sum f\left(x-x_{1}\right)=\sum \alpha \left(x-x_{1}\right)=0$ and in virtue of equations (6). Yet this is precisely what was demonstrated.

We can also achieve the same result by considerations of homogeneity.

Indeed, ψ, F, G, H are functions of $x-x_{1},\ y-y_{1},\ z-z_{1},\ \xi _{1}={\frac {dx_{1}}{dt_{1}}},\ \eta _{1}={\frac {dy_{1}}{dt_{1}}},\ \zeta _{1}={\frac {dz_{1}}{dt_{1}}}$ being homogeneous of degree -1 with respect to x, y, z, t, x₁, y₁, z₁, t₁ and their differentials.

So the derivatives of ψ, F, G, H with respect to x, y, z, t (and hence also the two fields f, g, h; α, β, γ) will be homogeneous of degree -2 with respect to the same quantities, if we remember also that the relation

$t-t_{1}=r={\sqrt {\sum \left(x-x_{1}\right)^{2}}}$

is homogeneous with respect to these quantities.

But these derivatives depend on these fields of x - x₁, the velocities ${\frac {dx_{1}}{dt_{1}}}$, and the accelerations ${\frac {d^{2}x_{1}}{dt_{1}^{2}}}$; they consist of a term independent of accelerations (velocity wave) and a term linear in respect to accelerations (acceleration waves). But ${\frac {dx_{1}}{dt_{1}}}$ is homogeneous of degree 0 and ${\frac {d^{2}x_{1}}{dt_{1}^{2}}}$ is homogeneous of degree -1; hence it follows that the velocity wave is homogeneous of degree -2 with respect to x - x₁, y - y₁, z - z₁, and the acceleration wave is homogeneous of degree -1. So in a very distant point an acceleration wave is predominant and can therefore be regarded as being assimilated to the total wave. In addition, the law of homogeneity shows that the acceleration wave is similar to itself at a distance and at any point. It is therefore, at any point, similar to the total wave at a remote point. But in a distant point the disturbance can propagate as plane waves, so that the two fields should be equal, mutually perpendicular and perpendicular to the direction of propagation.

I shall refer for more details to a work by Langevin in the Journal de Physique (Year 1905).

§ 6. — Contraction of electrons

Suppose a single electron in rectilinear and uniform motion. From what we have seen, we can, through the Lorentz transformation, reduce the study of the field determined by the electron to the case where the electron is motionless; the Lorentz transformation replaces the real electron in motion by an ideal electron without motion.

Let α, β, γ, f, g, h be the real field; let α', β', γ', f', g', h' be the