To every value of (x, y, z, t) which fully determines the position and time of an event in the stationary system, there correspond a system of values (ξ, η, ζ, τ); now the problem is to find out the system of equations connecting these magnitudes.
Primarily it is clear that on account of the property of homogeneity which we ascribe to time and space, the equations must be linear.
If we put x′ = x - vt, then it is clear that at a point relatively at rest in the system k, we have a system of values (x′ y z) which are independent of time. Now let us find out τ as a function of (x′, y, z, t). For this purpose we have to express in equations the fact that τ is not other than the time given by the clocks which are at rest in the system k which must be made synchronous in the manner described in [§ 1].
Let a ray of light be sent at time τ₀ from the origin of the system k along the X-axis towards x′ and let it be reflected from that place at time τ₁ towards the origin of moving co-ordinates and let it arrive there at time τ₂; then we must have
½ (τ0 + τ2) = τ1
If we now introduce the condition that τ is a function of co-ordinates, and apply the principle of constancy of the velocity of light in the stationary system, we have
It is to be noticed that instead of the origin of co-ordinates, we could select some other point as the exit point for rays of light, and therefore the above equation holds for all values of (x′, y, z, t,).
A similar conception, being applied to the y- and z-axis gives us, when we take into consideration the fact that light when viewed from the stationary system, is always propagated along those axes with the velocity √(c² - v²), we have the questions
∂ ∂τ