the Determinant of the matrix is +1, all co-efficients without the index 4 occurring once are real, while a₄₁, a₄₂, a₄₃, are purely imaginary, but a₄₄ is real and > 0, and x₁² + x₂² + x₃² + x₄² transforms into x₁′² + x₂′² + x₃′² + x₄′². The operation shall be called a general Lorentz transformation.
(This notation, which is due to Dr. C. E. Cullis of the Calcutta University, has been used throughout instead of Minkowski’s notation, x₁ = a₁₁x₁′ + a₁₂x₂′+ a₁₃x₃′+ a₁₄x₄′.)
If we put x₁′ = x′, x₂′ = y′, x₃′ = z′, x₄′ = it′, then immediately there occurs a homogeneous linear transformation of (x, y, z, t) to (x′, y′, z′, t′) with essentially real co-efficients, whereby the aggregate -x² - y² - z² + t² transforms into -x′² - y′² - z′² + t′², and to every such system of values x, y, z, t with a positive t, for which this aggregate > 0, there always corresponds a positive t’; this last is quite evident from the continuity of the aggregate x, y, z, t.
The last vertical column of co-efficients has to fulfil the condition 22) a₁₄² + a₂₄² + a₃₄² + a₄₄² = 1.
If a₁₄ = a₂₄ = a₃₄ = 0, then a₄₄ = 1, and the Lorentz transformation reduces to a simple rotation of the spatial co-ordinate system round the world-point.
If a₁₄, a₂₄, a₃₄ are not all zero, and if we put a₁₄ : a₂₄ : a₃₄ : a₄₄ = vx : vy : vz : i
q = √(vx² + vy² +vz²) < 1.
On the other hand, with every set of values of a₁₄, a₂₄, a₃₄, a₄₄ which in this way fulfil the condition 22) with real values of vx, vy, vz, we can construct the special Lorentz transformation (16) with (a₁₄, a₂₄, a₃₄, a₄₄) as the last vertical column,—and then every Lorentz-transformation with the same last vertical column (a₁₄, a₂₄, a₃₄, a₄₄) can be supposed to be composed of the special Lorentz-transformation, and a rotation of the spatial co-ordinate system round the null-point.
The totality of all Lorentz-Transformations forms a group. Under a space-time vector of the 1st kind shall be understood a system of four magnitudes (ρ₁, ρ₂, ρ₃, ρ₄) with the condition that in case of a Lorentz-transformation it is to be replaced by the set (ρ₁′, ρ₂′, ρ₃′, ρ₄′), where these are the values of (x₁′, x₂′, x₃′, x₄′), obtained by substituting (ρ₁, ρ₂, ρ₃, ρ₄) for (x₁, x₂, x₃, x₄) in the expression (21).
Besides the time-space vector of the 1st kind (x₁, x₂, x₃, x₄) we shall also make use of another space-time vector of the first kind (y₁, y₂, y₃, y₄), and let us form the linear combination