# Page:PoincareDynamiqueJuillet.djvu/4

measurement. How do we perform our measurements? By transportation, one on the other, of objects regarded as invariable solids, one will answer immediately; but this is not true any more in the current theory, if the Lorentz contraction is admitted. In this theory, two equal lengths are, by definition, two lengths for which light takes the same time to traverse.

Perhaps it would be enough to give up this definition, so that the theory of Lorentz is as completely rejected as it was the system of Ptolemy by the intervention of Copernicus. If that happens one day, it will not prove that the effort made by Lorentz was useless; because Ptolemy, no matter what we think about him, was not useless for Copernicus.

Also I did not hesitate to publish these few partial results, although in this moment even the whole theory seems to be endangered by the discovery of magnetocathodic rays.

## § 1. — Lorentz transformation

Lorentz had adopted a particular system of units, so as to eliminate the factors 4π in the formulas. I'll do the same, plus I choose the units of length and time so that the speed of light is equal to 1. Under these conditions the fundamental formulas become (by calling f, g, h the electric displacement, α, β, γ the magnetic force, F, G and H the vector potential, φ the scalar potential, ρ the electric density, ξ, η, ζ the electron velocity, u, v, w the current):

 (1) ${\displaystyle {\begin{cases}u={\frac {df}{dt}}+\rho \xi ={\frac {d\gamma }{dy}}-{\frac {d\beta }{dz}},\quad \alpha ={\frac {dH}{dy}}-{\frac {dG}{dz}},\ f=-{\frac {dF}{dt}}-{\frac {d\psi }{dx}},\\\\{\frac {d\alpha }{dt}}={\frac {dg}{dz}}-{\frac {dh}{dy}},\quad {\frac {d\rho }{dt}}+\sum {\textstyle {\frac {d\rho \xi }{dx}}=0,\quad \sum {\frac {df}{dx}}=\rho ,\quad {\frac {d\psi }{dt}}+\sum {\frac {dF}{dx}}=0,}\\\\\square =\vartriangle -{\frac {d^{2}}{dt^{2}}}=\sum {\frac {d^{2}}{dx^{2}}}-{\frac {d^{2}}{dt^{2}}},\quad \square \psi =-\rho ,\quad \square F=-\rho \xi .\end{cases}}}$

A material element of volume dx dy dz suffers a mechanical force whose components X dx dy dz, Y dz dx dy, Z dx dy dz are deduced from the formula:

 (2) ${\displaystyle X=\rho f+\rho (\eta \gamma -\zeta \beta )\,}$

These equations are capable of a remarkable transformation discovered by Lorentz and which owes its interest from the fact, that it explains why no experience is suited to show us the absolute motion of the universe. Let:

 (3) ${\displaystyle x^{\prime }=kl(x+\epsilon t),\ t^{\prime }=kl(t+\epsilon x),\ y^{\prime }=ly,\ z^{\prime }=lz}$

l and ε are two arbitrary constants, and

${\displaystyle k={\frac {1}{\sqrt {1-\epsilon ^{2}}}}.}$