Page:PoincareDynamiqueJuillet.djvu/5

 If we now set:

$$\square^{\prime}=\sum\frac{d^{2}}{dx^{\prime2}}-\frac{d^{2}}{dt^{\prime2}},$$

it follows:

$$\square^{\prime}=\square l^{-2}.$$

Consider a sphere entrained with the electron in a uniform translational motion, and

$$(x-\xi t)^{2}+(y-\eta t)^{2}+(z-\zeta t)^{2}=r^{2},\,$$

is the equation of that moving sphere whose volume is $$\tfrac{4}{3}\pi r^{2}$$.

The transformation will change it into an ellipsoid, and it is easy to find the equation. It is easily deduced because of equations (3):

The equation of the ellipsoid becomes:

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

This ellipsoid moves in uniform motion; for t' = 0, it reduces to

$$k^{2}x^{\prime2}(1+\xi\epsilon)^{2}+(y^{\prime}+\eta k\epsilon x^{\prime})^{2}+(z^{\prime}+\zeta k\epsilon x^{\prime})^{2}=l^{2}r^{2},$$

and has the volume:

$$\frac{4}{3}\pi r^{3}\frac{l^{3}}{k(1+\xi\epsilon)}.$$

If we want that the charge of an electron is not altered by the transformation, and when we call ρ' the new electrical density, it follows:

Those are the new velocities &xi;', η', ζ '; we must have:

where:

Here I should mention for the first time a discrepancy with.

poses (with different notations) (loco citato, page 813, formulas 7 and 8):

$$\rho^{\prime}=\frac{1}{kl^{3}}\rho,\quad\xi^{\prime}=k^{2}(\xi+\epsilon),\quad\eta^{\prime}=k\eta,\quad\zeta^{\prime}=k\zeta.$$