# Page:Deux Mémoires de Henri Poincaré.djvu/3

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the phenomena which take place in a material system can be represented by equations of the same form, the system may be at rest or being animated by a uniform translatory motion, and this equality of form has to be obtained using a suitable substitution of new variables. It was a question of finding transformation formulas, suitable for the independent variables, the coordinates x, y, z and time t, as well as for the various physical magnitudes, speeds, forces, etc, and by showing the invariance of the equations for these transformations.

The formulas that I established for coordinates and time can be put as[1]

 (1) $x'=kl(x+\epsilon t),\ y'=ly,\ z'=lz,\ t'=kl(t+\epsilon x),$

where ε, k, l are constants which are, however, reduced to one. We see immediately that the origin of the new coordinates ($x'=0$) is

$x=-\epsilon t\,$

so the point moves in the system x, y, z, t with speed in the direction of the x-axis. The coefficient k is defined by

$k=\left(1-\epsilon^{2}\right)^{-\frac{1}{2}}$

and ε is a function of l that has the value 1 for ε = 0. I initially left it undetermined, but I found in the course of my calculations, that to obtain the invariance (that I had in mind) we must put l = 1.

These were the considerations published by me in 1904 which gave place to Poincaré to write his paper on the dynamics of the electron, in which he attached my name to the transformation to which I will come to speak. I must notice on this subject that the same transformation was already present in an article of Mr. Voigt published in 1887, and that I did not draw from this artifice all the possible parts. Indeed, for some of the physical quantities which enter the formulas, I did not indicate the transformation which suits best. That was done by Poincaré and then by Mr. Einstein and Minkowski.

To find the "transformations of relativity", as I will call them now, it is sufficient in some cases to describe the phenomena in the

1. I follow here the notations of Poincaré and I choose the units of length and time so that the speed of light is equal to 1.