Page:PoincareDynamiqueJuillet.djvu/24

field after the Lorentz transformation, so the ideal field α', f' corresponds to the case where the electron is motionless; we have:

$\alpha ^{\prime }=\beta ^{\prime }=\gamma ^{\prime }=0,\quad f^{\prime }=-{\frac {d\psi ^{\prime }}{dx^{\prime }}},\quad g^{\prime }=-{\frac {d\psi ^{\prime }}{dy^{\prime }}},\quad h^{\prime }=-{\frac {d\psi ^{\prime }}{dz^{\prime }}};$

and the actual field (in virtue of the formulas 9 of § 1):

(1)	${\begin{cases}\alpha =0,\quad \beta =\epsilon h,\quad \gamma =-\epsilon g\\\\f=l^{2}f^{\prime },\quad g=kl^{2}g^{\prime },\ h=kl^{2}h^{\prime }.\end{cases}}$

We now determine the total energy due to the motion of the electron, the corresponding action, and the electromagnetic momentum, in order to calculate the electromagnetic mass of the electron. For a distant point, it suffices to consider the electron as reduced to a single point; we are thus brought back to the formulas (4) of the preceding § which generally can be appropriate. But here they do not suffice, because the energy is mainly located in the ether parts nearest to the electron.

On this subject we can make several hypotheses.

According to that of Abraham, the electrons are spherical and not deformable.

So when we apply the Lorentz transformation when the real electron is spherical, the electron becomes a perfect ellipsoid. The equation of this ellipsoid is based on § 1:

$k^{2}\left(x^{\prime }-\epsilon t^{\prime }-\xi t^{\prime }+\epsilon \xi x^{\prime }\right)^{2}+\left(y^{\prime }-\eta kt^{\prime }+\eta k\epsilon x^{\prime }\right)^{2}$ $+\left(z^{\prime }-\zeta kt^{\prime }+\zeta k\epsilon x^{\prime }\right)^{2}=l^{2}r^{2}$

But here we have:

$\xi +\epsilon =\eta =\zeta =0,\quad 1+\epsilon \xi =1-\epsilon ^{2}={\frac {1}{k^{2}}},$

so that the equation of the ellipsoid becomes:

${\frac {x^{\prime 2}}{k^{2}}}+y^{\prime 2}+z^{\prime 2}=l^{2}r^{2}.$

If the radius of the real electron is r, the axes of the ideal electron would therefore be:

$klr,\ lr,\ lr.$

In Lorentz's hypothesis, however, the moving electrons are deformed, so that the real electron would become a ellipsoid, while the ideal electron is still always a perfect sphere of radius r; the axes of the real electron will then be:

${\frac {r}{lk}},\quad {\frac {r}{l}},\quad {\frac {r}{l}}.$

We denote by

$A={\frac {1}{2}}\int f^{2}d\tau$

the longitudinal electric energy; by

$B={\frac {1}{2}}\int \left(g^{2}+h^{2}\right)d\tau$