Page:Relativity (1931).djvu/58

 framework of rods, so that an event which takes place anywhere can be localised with reference to this framework. 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 $$K'$$, the same event would be fixed in respect of space and time by corresponding values $$x'$$, $$y'$$, $$z'$$, $$t'$$, 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