Page:PoincareDynamiqueJuillet.djvu/18

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

We are thus led to consider a continuous group which we call the  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:

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:

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)

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. — waves
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