how these magnitudes are to be regarded as results of physical measurements. image002 Obviously our problem can be exactly formulated in the following manner. What are the values _x′, y′, z′, t′_, of an event with respect to _K′_, when the magnitudes _x, y, z, t_, of the same event with respect to _K_ are given? The relations must be so chosen that the law of the transmission of light in vacuo is satisfied for one and the same ray of light (and of course for every ray) with respect to _K_ and _K′_. For the relative orientation in space of the co-ordinate systems indicated in the diagram (Fig. 2), this problem is solved by means of the equations: image003 _y′_ = _y_ _z′_ = _z_ image004 This system of equations is known as the “Lorentz transformation.”[9] [9] A simple derivation of the Lorentz transformation is given in Appendix I. If in place of the law of transmission of light we had taken as our basis the tacit assumptions of the older mechanics as to the absolute character of times and lengths, then instead of the above we should have obtained the following equations: _x′_ = _x_ – _vt_ _y′_ = _y_ _z′_ = _z_ _t′_ = _t_ This system of equations is often termed the “Galilei transformation.” The Galilei transformation can be obtained from the Lorentz transformation by substituting an infinitely large value for the velocity of light _c_ in the latter transformation. Aided by the following illustration, we can readily see that, in accordance with the Lorentz transformation, the law of the transmission of light _in vacuo_ is satisfied both for the reference-body _K_ and for the reference-body _K′_. A light-signal is sent along the positive _x_-axis, and this light-stimulus advances in accordance with the equation _x_ = _ct_, _i.e._ with the velocity _c_. According to the equations of the Lorentz transformation, this simple relation between _x_ and _t_ involves a relation between _x′_ and _t′_. In point of fact, if we substitute for _x_ the value _ct_ in the first and fourth equations of the Lorentz transformation, we obtain: image005 from which, by division, the expression _x′_ = _ct′_ immediately follows. If referred to the system _K′_, the propagation of light takes place according to this equation. We thus see that the velocity of transmission relative to the reference-body _K′_ is also equal to _c_. The same result is obtained for rays of light advancing in any other direction whatsoever. Of cause this is not surprising, since the equations of the Lorentz transformation were derived conformably to this point of view. XII. THE BEHAVIOUR OF MEASURING-RODS AND CLOCKS IN MOTION