If we now set:
![{\displaystyle \square ^{\prime }=\sum {\frac {d^{2}}{dx^{\prime 2}}}-{\frac {d^{2}}{dt^{\prime 2}}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b14a1b3df9ca4a5b96263aa900372dff532b8284)
it follows:
![{\displaystyle \square ^{\prime }=\square l^{-2}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/164d94b1a7819ffc39bba1c87bf18b49f25a524c)
Consider a sphere entrained with the electron in a uniform translational motion, and
![{\displaystyle (x-\xi t)^{2}+(y-\eta t)^{2}+(z-\zeta t)^{2}=r^{2},\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/018aad36f45b0b49923446b64d5d42b2e3fa8f4c)
is the equation of that moving sphere whose volume is
.
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)
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The equation of the ellipsoid becomes:
![{\displaystyle 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}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f72f0a274f05bdcf37e24e190e419e208e381e95)
This ellipsoid moves in uniform motion; for t' = 0, it reduces to
![{\displaystyle k^{2}x^{\prime 2}(1+\xi \epsilon )^{2}+(y^{\prime }+\eta k\epsilon x^{\prime })^{2}+(z^{\prime }+\zeta k\epsilon x^{\prime })^{2}=l^{2}r^{2},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cf34fddcc4709b39a8027a5f2e14116af456ffb7)
and has the volume:
![{\displaystyle {\frac {4}{3}}\pi r^{3}{\frac {l^{3}}{k(1+\xi \epsilon )}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/00d3451b053358d1a83c4f98829450d5d16dbaf5)
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)
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Those are the new velocities ξ', η', ζ '; we must have:
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where:
4bis
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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):
![{\displaystyle \rho ^{\prime }={\frac {1}{kl^{3}}}\rho ,\quad \xi ^{\prime }=k^{2}(\xi +\epsilon ),\quad \eta ^{\prime }=k\eta ,\quad \zeta ^{\prime }=k\zeta .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d63c86542cc505a7e5ffd4c16528c8fec5ebbbac)