A transformation which is accomplished by means of (10), (11), (12) with the condition 0 < q < 1 will be called a special Lorentz-transformation. We shall call v the vector, the direction of v the axis, and the magnitude of v the moment of this transformation.
If further ρ′ and the vectors u′, e′, m′, in the system (x′ y′ z′) are so defined that,
(13) ρ′ = ρ[(-quv + 1)/√(1 - q²)],
ρ′u′v = ρ(uv - q)/√(1 - q²),
ρ′uṽ = ρ′uv,
further
(14) (e′ + im′)ṽ = ((e + im) - i[v, (e + im])']ṽ)/√(1 - q²).
(15) (e′ + im′)v = (e + im) - i[u, (e + im)]v.
Then it follows that the equations I), II), III), IV) are transformed into the corresponding system with dashes.
The solution of the equations (10), (11), (12) leads to