?  In other words :  Can we conceive of a relation between place and time of the individual events relative to both reference-bodies, such that every ray of light possesses the velocity of transmission c relative to the embankment and relative to the train ? This question leads to a quite definite positive answer, and to a perfectly definite transformation law for the space-time magnitudes of an event when changing over from one body of reference to another.

Before we deal with this, we shall introduce the following incidental consideration.  Up to the present we have only considered events taking place along the embankment, which had mathematically to assume the function of a straight line.  In the manner indicated in Section 2 we can imagine this reference-body supplemented laterally and in a vertical direction by means of a framework of rods, so that an event which takes place anywhere can be localised with reference to this framework.  Fig. 2 Similarly, we can imagine the train travelling with the velocity v to be continued across the whole of space, so that every event, no matter how far off it may be, could also be localised with respect to the second framework.  Without committing any fundamental error, we can disregard the fact that in reality these frameworks would continually interfere with each other, owing to the impenetrability of solid bodies.  In every such framework we imagine three surfaces perpendicular to each other marked out, and designated as " co-ordinate planes " (” co-ordinate system “).  A co-ordinate system K then corresponds to the embankment, and a co-ordinate system K’ to the train.  An event, wherever it may have taken place, would be fixed in space with respect to K by the three perpendiculars x, y, z on the co-ordinate planes, and with regard to time by a time value t.  Relative to K1, the same event would be fixed in respect of space and time by corresponding values x1, y1, z1, t1, which of course are not identical with x, y, z, t.  It has already been set forth in detail how these magnitudes are to be regarded as results of physical measurements.

Obviously our problem can be exactly formulated in the following manner.  What are the values x1, y1, z1, t1, of an event with respect to K1, 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 K1.  For the relative orientation in space of the co-ordinate systems indicated in the diagram ([7]Fig. 2), this problem is solved by means of the equations : 

eq. 1:  file eq01.gif

                                y1 = y
                                z1 = z

eq. 2:  file eq02.gif

This system of equations is known as the " Lorentz transformation.” *

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: