We shall have cos iψ = 1/[sqrt](1 - q^2), sin iψ = iq/[sqrt](1 - q^2) where -1 < q < 1, and [sqrt](1 - q^2) is always to be taken with the positive sign.
Let us now write x´_{1} = x´, x´_{2} = y´, x´_{3} = z´, x´_{4} = it´ (3)
then the substitution 1) takes the form
x´ = x, y´ = y, z´ = (z - qt)/[sqrt](1 - q^2), t´ = (-qz + t)/[sqrt](1 - q^2), (4)
the coefficients being essentially real.
If now in the above-mentioned rotation round the Z-axis, we replace 1, 2, 3, 4 throughout by 3, 4, 1, 2, and φ by iψ, we at once perceive that simultaneously, new magnitudes ρ´_{1}, ρ´_{2}, ρ´_{3}, ρ´_{4}, where
(ρ´_{1} = ρ_{1}, ρ´_{2} = ρ_{2}, ρ´_{3} = ρ_{3} cos iψ + ρ_{4} sin iψ, ρ´_{4} =
-ρ_{3} sin iψ + ρ_{4} cos iψ),
and [function]´_{1 2} . . . [function]´_{3 4}, where
[function]´_{4 1} = [function]_{4 1} cos iψ + [function]_{1 3} sin iψ, [function]´_{1 3} = -[function]_{4 1} sin iψ + [function]_{1 3}
cos iψ, [function]´_{3 4} = [function]_{3 4}, [function]´_{3 2} = [function]_{3 2} cos iψ + [function]_{4 2} sin iψ, [function]´_{4 2} =
-[function]_{3 2} sin iψ + [function]_{4 2} cos iψ, [function]´_{1 2} = [function]_{1 2}, [function]_{k h} = -[function]´_{k h},
must be introduced. Then the systems of equations in (A) and (B) are transformed into equations (A´), and (B´), the new equations being obtained by simply dashing the old set.
All these equations can be written in purely real figures, and we can then formulate the last result as follows.
If the real transformations 4) are taken, and x´ y´ z´ t´ be takes as a new frame of reference, then we shall have
(5) ρ´ = ρ [(-qu_{z} + 1)/[sqrt](1 - q^2)], ρ´u_{z}´ = ρ[(u_{z} - q)/[sqrt](1 - q^2)],
ρ´u_{x}´ = ρu_{x}, ρ´u_{y}´ = ρu_{y}.
