Page:The principle of relativity (1920).djvu/43

 system, and we thus obtain the co-ordinates (x, y, z) for the stationary system, and (ξ, η, ζ) for the moving system. Let the time t be determined for each point of the stationary system (which are provided with clocks) by means of the clocks which are placed in the stationary system, with the help of light-signals as described in § 1. Let also the time τ of the moving system be determined for each point of the moving system (in which there are clocks which are at rest relative to the moving system), by means of the method of light signals between these points (in which there are clocks) in the manner described in § 1.

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 i 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 τ_{0} from the origin of the system k along the X-axis towards x´ and let it be reflected from that place at time τ_{1} towards the origin of moving co-ordinates and let it arrive there at time τ_{2}; then we must have

1/2(τ_{0} + τ_{2}) = τ_{1}