In order to decide this let us fix our attention upon a special Lorentz transformation represented by (10), (11), (12), with a vector v in any direction and of any magnitude q < 1 but different from zero. For a moment we shall not suppose any special relation to hold between the unit of length and the unit of time, so that instead of t, t′, q, we shall write ct, ct′, and q/c, where c represents a certain positive constant, and q is < c. The above mentioned equations are transformed into
r′ṽ = rṽ,
r′v = c(rv - qt)/√(c² - q²),
t′ = (qrv + c²t)/c√(c² - q²)
They denote, as we remember, that r is the space-vector (x, y, z), r′ is the space-vector (x′ y′ z′)
If in these equations, keeping v constant we approach the limit c = ∞, then we obtain from these
r′ṽ = rṽ,
r′v = rv - qt,
t′ = t.
The new equations would now denote the transformation of a spatial co-ordinate system (x, y, z) to another spatial co-ordinate system (x′ y′ z′) with parallel axes, the null point of the second system moving with constant velocity in a straight line, while the time parameter remains unchanged. We can, therefore, say that classical mechanics postulates a covariance of Physical laws for the group of homogeneous linear transformations of the expression