(16) rv = (r′v + qt′)/√(1 - q²),
rṽ = r′ṽ,
t = (qr′v + t′)/√(1 - q²),
Now we shall make a very important observation about the vectors u and u′. We can again introduce the indices 1, 2, 3, 4, so that we write (x₁′, x₂′, x₃′, x₄′) instead of (x′, y′, z′, it′) and ρ₁′, ρ₂′, ρ₃′, ρ₄′ instead of (ρ′u′{x′}, ρ′u′{y′}, ρ′u′{z′}, iρ′).
Like the rotation round the Z-axis, the transformation (4), and more generally the transformations (10), (11), (12), are also linear transformations with the determinant + 1, so that
(17) x₁² + x₂² + x₃² + x₄² i. e. x² + y² + z² - t²,
is transformed into
x₁′² + x₂′² + x₃′² + x₄′² i. e. x′² + y′² + z′² - t′².
On the basis of the equations (13), (14), we shall have (ρ₁² + ρ₂² + ρ₃² + ρ₄²) = ρ²(1 - ux², -uy², -uz²) = ρ²(1 - u²) transformed into ρ²(1 - u²) or in other words,
(18) ρ√(1 - u²)