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should not be surprised if such a change does not alter the form of the equations of Lorentz, obviously independent of the choice of axes.

We are thus led to consider a continuous group which we call the Lorentz group and which admit as infinitesimal transformations:

1° the transformation T₀ which is permutable with all others;

2° the three transformations T₁, T₂, T₃;

3° the three rotations [T₁, T₂], [T₂, T₃], [T₃, T₁].

Any transformation of this group can always be decomposed into a transformation of the form:

$x^{\prime }=lx,\quad y^{\prime }=ly,\quad z^{\prime }=lz,\quad t^{\prime }=lt$

and a linear transformation which does not change the quadratic form

$x^{2}+y^{2}+z^{2}-t^{2}\,$

We can still generate our group in another way. Any transformation of the group may be regarded as a transformation of the form:

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

preceded and followed by a suitable rotation.

But for our purposes, we should consider only a part of the transformations of this group; we must assume that l is a function of ε, and it is a question of choosing this function in such a way that this part of the group that I call P still forms a group.

Let's rotate the system 180° around the y-axis, we should find a transformation that will still belong to P. But this amounts to a sign change of x, x', z and z'; we find:

(2)	$x^{\prime }=kl(x-\epsilon t),\quad y^{\prime }=ly,\quad z^{\prime }=lz,\quad t^{\prime }=kl(t-\epsilon x)$

So l does not change when we change ε into -ε.

On the other hand, if P is a group, then the inverse substitution of (1)

(3)	$x^{\prime }={\frac {k}{l}}(x-\epsilon t),\quad y^{\prime }={\frac {y}{l}},\quad z^{\prime }={\frac {z}{l}},\quad t^{\prime }={\frac {k}{l}}(t-\epsilon x),$

must also belong to P; it will therefore be identical with (2), that is to say that

$l={\frac {1}{l}}.$

We must therefore have l = 1.

§ 5. — Langevin waves

Langevin has put the formulas that define the electromagnetic field produced by the motion of a single electron in a particularly elegant form.

Let us remember the equations

(1)	$\square \psi =-\rho ,\quad \square F=-\rho \xi .$