If we now set:
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):