﻿Approach to the Theory of Relativity. 499 



are required, each with his own clock. Observer A, at 

 ( ,r i> y\)-> notes that the aeroplane passes his station when his 

 clock reads t± ; observer B, at (&* 2 , y 2 \ notes that it passes 

 his station when his clock reads t 2 ; then the ratio o£ 

 "distance" divided by "difference in clock-readings'" 

 (supposing this ratio to be constant for any two observers 

 in the line of flight) is called the observed velocity of the 

 aeroplane with respect to the platform : 



u _ \/_ (g| ~xi) 2 + (3/2—3/1) 

 t 2 — t 1 



2 



* 



In particular, the velocity of a light-signal, as determined 

 in this way, is equal to the arbitrarily chosen constant c, and 

 is the same in all directions. 



(That this is true for light travelling along the axis of x, 

 or along a line perpendicular to that axis, is obvious from 

 the way in which the coordinates of points on those lines 

 were assigned ; that it is true also for light travelling in 

 oblique directions follows readily from the definition of 

 distance in § 8.) 



Definition of observed rate of a moving clock. 



10. Let us now suppose an aviator, flying across the 

 platform, carries a clock with him, and let 



t 1 = the reading of the clock at A when the aviator 



passes A ; 

 t 2 = the reading of the clock at B when the aviator 

 passes B ; 

 and T' = the number of seconds ticked off by the clock on 

 the aeroplane during the flight from A to B. 



Then the ratio 



T' 



r = 



t 2 — ti 



(if this ratio proves to be constant) is called the observed 

 rate at which the clock on the aeroplane is running with 

 respect to the clocks on the platform ; that is, the clock on 

 the aeroplane is said to be running r times as fast as the 

 clocks on the platform. 



* The extension of this definition to cases where the velocity is not 

 constant is effected in the usual manner, by the methods of the differential 

 calculus. 



2 L 2 



