﻿512 New Approach to the Theory of Relativity. 



since the observed velocity of O' (see § 9) is v, we must have 

 x =vt . Hence the situation at this instant is represented 

 by fig. 2. 



In position 3, at the instant when the signal has reached 

 A', let the clock at A' read t 1 , and let the station opposite A' 

 at that instant have the abscissa x and the clock-reading t, as 

 shown in fig. 3. 



3. 



■■{?: 



X=X 

 t = t 



In position 4, at the instant when the- return signal has 

 reached 0', let the clock at 0' read t 2 , and let x 2 and t 2 be the 

 abscissa and clock-reading of the station opposite 0' at that 

 instant. Then t 2 '=rt 2 , since the observed rate (§ 10) of the 



Fig. 4 



rU 



x = x 2 =vt 2 



t = t 2 



clock at 0' in passing from position 1 to position 4 is r. Also, 

 t x 2 = vt 2 , since the observed velocity of 0' (§ 9) in passing 

 from position 1 to position 4 is v. Hence the situation at 

 this instant is as represented in fig. 4. 



Now in the interval between fig. 2 and fig. 3, we note that 

 a light-signal has passed along the stationary platform S 

 from x=vt io x=x, with velocity c ; hence, by the definition 

 of observed velocity (§9), 



t=?=. (i) 



Similarly, in the interval between fig. 3 and fig. 4, the 

 return signal has passed from x—x to x—vt 2 with velocity 

 — c ; hence, 



^^=,-c (2) 



t 2 -t 



Further, from the way in which the clock at A' has been 



