Page:PoincareDynamiqueJuillet.djvu/21

and we will take ε = -ξ, so that

$\xi '_{1}=\eta '_{1}=\zeta '_{1}=0\,$

We can therefore reduce the computation of the two waves to the case where the electron velocity is zero. Let's start with the velocity wave, we first note that this wave is the same as if the electron motion was uniform.

If the electron velocity is zero, then:

$\omega =0,\quad F=G=H=0,\quad \psi ={\frac {\mu _{1}}{4\pi r}},$

μ₁ is the electrical charge of the electron. The speed was reduced to zero by the Lorentz transformation, we have now:

$F^{\prime }=G^{\prime }=H^{\prime }=0,\quad \psi ^{\prime }={\frac {\mu _{1}}{4\pi r^{\prime }}},$

r' is the distance from point x', y', z' at point x'₁, y'₁, z'₁, and therefore:

$\alpha ^{\prime }=\beta ^{\prime }=\gamma ^{\prime }=0,$ $f^{\prime }={\frac {\mu _{1}\left(x^{\prime }-x_{1}^{\prime }\right)}{4\pi r^{\prime 3}}}\quad g^{\prime }={\frac {\mu _{1}\left(y^{\prime }-y_{1}^{\prime }\right)}{4\pi r^{\prime 3}}},\quad h^{\prime }={\frac {\mu _{1}\left(z^{\prime }-z_{1}^{\prime }\right)}{4\pi r^{\prime 3}}}.$

Now let us carry out the reverse Lorentz transformation to find the true field corresponding to the velocity ε, 0, 0. We find, with reference to equations (9) and (3) of § 1:

(4)

${\begin{cases}&\alpha =0,\quad \beta =\epsilon h,\quad \gamma =-\epsilon g,\\\\&f={\frac {\mu _{1}kl^{3}}{4\pi r^{\prime 3}}}\left(x+\epsilon t-x_{1}-\epsilon t_{1}\right),\quad g={\frac {\mu _{1}kl^{3}}{4\pi r^{\prime 3}}}\left(y-y_{1}\right),\quad h={\frac {\mu _{1}kl^{3}}{4\pi r^{\prime 3}}}\left(z-z_{1}\right).\end{cases}}$

We see that the magnetic field is perpendicular to the x-axis (direction of velocity) and the electric field, and the electric field is directed to the point:

(5)	$x_{1}+\epsilon (t_{1}-t),\ y_{1},\ z_{1}.$

If the electron continues to move in a rectilinear and uniform way with the velocity it had at the instant t₁, that is to say, with the velocity -ε, 0, 0, the point (5) would be the one occupied at the instant t.

Taking the acceleration wave, we can, through the Lorentz transformation, reduce its determination to the case of zero velocity. This is the case if we imagine an electron whose oscillation amplitude is very small, but very fast, so that the displacements and velocities are much smaller, but the accelerations are finished. We thus come back to the field that has been studied in the famous work by Hertz entitled Die Kräfte elektrischer Schwingungen nach der Maxwell'schen Theorie, and that for a point at great distance. In these conditions:

I° Both electric and magnetic fields are equal.

2° They are perpendicular to each other.

3° They are perpendicular to the normal of the spherical wave, that is to say to the sphere whose center is the point x₁, y₁, z₁.