(mx, -i ex)² + (my, -i ey)² ≠ 0.

Therefore (ey + i my)/(ex + i ex) is different from +i, and we can therefore define a complex argument (φ + iψ) in such a manner that

tan (φ + iψ)

ey + i my

= ---------------

ex + i mx

If then, by referring back to equations (9), we carry out the transformation (1) through the angle ψ and a subsequent rotation round the Z-axis through the angle φ, we perform a Lorentz-transformation at the end of which my = 0, ey = 0, and therefore m and e shall both coincide with the new Z-axis. Then by means of the invariants m² - e², (me) the final values of these vectors, whether they are of the same or of opposite directions, or whether one of them is equal to zero, would be at once settled.

§ 6. Concept of Time.

By the Lorentz transformation, we are allowed to effect certain changes of the time parameter. In consequence of this fact, it is no longer permissible to speak of the absolute simultaneity of two events. The ordinary idea of simultaneity rather presupposes that six independent parameters, which are evidently required for defining a system of space and time axes, are somehow reduced to three. Since we are accustomed to consider that these limitations represent in a unique way the actual facts very approximately, we maintain that the simultaneity of two events exists of themselves.[[18]] In fact, the following considerations will prove conclusive.

Let a reference system (x, y, z, t) for space time points (events) be somehow known. Now if a space point A (x₀, y₀, z₀) the time t₀ be compared with a space point P (x, y, z) at the time t, and if the difference of time t - t₀, (let t > t₀) be less than the length A P i.e. less than the time required for the propagation of light from A to P, and if q = (t - t₀)/(A P) < 1, then by a special Lorentz transformation, in which A P is taken as the axis, and which has the moment q, we can introduce a time parameter t′, which (see equation 11, 12, § 4) has got the same value t′ = 0 for both space-time points (A, t₀), and (P, t). So the two events can now be comprehended to be simultaneous.