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If we now set:

it follows:

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

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):


The equation of the ellipsoid becomes:

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

and has the volume:

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 ξ', η', ζ '; we must have:



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):