﻿If now 



Approach to the Theory of Relativity. 497 



^1=9(^2 + ^0), so that t 2 — t 1 = t 1 —t , 



then the clock at A is said to be synchronous with the standard 

 clock at *. 



It follows as a theorem that two clocks which are syn- 

 chronous with the clock at will be synchronous, by the 

 same definition, with each other. 



Laying out a system of coordinates by light signals. 



5. All the clocks being now properly " regulated/'' and 

 " set " so as to be synchronous with the central clock at 0, 

 the observers proceed to lay out a system of coordinates — 

 that is, to assign to each station a pair of numbers to be 

 called the abscissa and ordinate of that station. 



Since no observer is supposed to move from his own 

 station, the ordinary process of measurement, by carrying a 

 metre-stick about from place to place, is not practicable ; 

 hence the observers resort to a further use of the method 

 of light-signals, as follows. A fixed line OX through the 

 central station being taken as the axis of x, light-signals are 

 sent from to the various stations along this axis. If a 

 signal which starts from when the clock at reads t Q 

 arrives at A when the clock at A reads £ l5 then the difference 

 in clock-readings, multiplied by a constant numerical factor c, 

 is taken as the abscissa of the station A, 



x = c(t 1 -t ) ; 



and the point A is denoted by (#, 0) f . In this way the 

 points (1, 0), (2, 0), <fec. on OX are determined, and simi- 

 larly, the points ( — 1, 0), ( — 2, 0), &c. on the extension 

 of OX through O. 



6. Having thus assigned coordinates to all the stations on 

 the axis of x. the observers now determine — still without any 

 ordinary measurements — the coordinates of points not on 

 the axis of x. Consider any point {in, 0) on the axis. 

 Observers at two points (m-fw, 0) and (m—n, 0), which 

 may be called "equidistant" from (m, 0), send out light- 

 signals, each observer starting his signal when the clock at 

 his station reads t . Any station C at which these two signals 



* If t l = ^(t i +t )-\-e 1 then the clock at A. is e seconds too fast, and 

 must be " set back " accordingly. 



f The arbitrary constant c merely determines the scale on which the 

 map is laid out. 



Phil. Mag. S. 6. Vol. 23. No. 136. April 1912. 2 L 



