(Ions, electrons), then these equations form the electromagnetic basis of Lorentz's electrodynamics and optics for moving bodies.
If these equations which hold in the system K, are transformed to the system k with the aid of the transformation-equations given in § 3 and § 6, then we obtain the equations:—
1/c[ρ´u_{ξ} + [part]X´/[part]τ] = [part]N´/[part]η - [part]M´/[part]ζ, [part]L´/[part]τ = [part]Y´/[part]ζ - [part]Z´/[part]η,
1/c[ρ´u_{η} + [part]Y´/[part]τ] = [part]L´/[part]ζ - [part]N´/[part]ξ, [part]M´/[part]τ = [part]Z´/[part]ξ- [part]X´/[part]ζ,
1/c[ρ´u_{ζ} + [part]Z´/[part]τ] = [part]M´/[part]ξ - [part]L´/[part]η, [part]N´/[part]τ = [part]X´/[part]η - [part]Y´/[part]ξ,
where
(u_{x} - v)/(1 - u_{x}v/c) = u_{ξ},
u_{y}/(β(1 - vu_{x}/c^2)) = u_{η}, ρ´ = [part]X´/[part]ξ + [part]Y´/[part]η + [part]Z´/[part]ξ]
= β(1 - vu_{x}/c^2)ρ,
u_{z}/(β(1 - vu_{x}/c^2)) = u_{ζ},
Since the vector (u_{ξ}, u_{η}, u_{ζ}) is nothing but the velocity of the electrical mass measured in the system k, as can be easily seen from the addition-theorem of velocities in § 4—so it is hereby shown, that by taking
