Page:PoincareDynamiqueJuillet.djvu/5

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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):

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

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:

(4) \rho^{\prime}=\frac{k}{l^{3}}(\rho+\epsilon\rho\xi).

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

\xi^{\prime}=\frac{dx^{\prime}}{dt^{\prime}}=\frac{d(x+\epsilon t)}{d(t+\epsilon x)}=\frac{\xi+\epsilon}{1+\epsilon\xi},

\eta^{\prime}=\frac{dy^{\prime}}{dt^{\prime}}=\frac{dy}{kd(t+\epsilon x)}=\frac{\eta}{k(1+\epsilon\xi)},\quad\zeta^{\prime}=\frac{\zeta}{k(1+\epsilon\xi)},

where:

4bis \rho^{\prime}\xi^{\prime}=\frac{k}{l^{3}}(\rho\xi+\epsilon\rho),\quad\rho^{\prime}\eta^{\prime}=\frac{1}{l^{3}}\rho\eta,\quad\rho^{\prime}\zeta^{\prime}=\frac{1}{l^{3}}\rho\zeta

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

Lorentz 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.