﻿502 Prof. E, V. Huntington on a New 



by the method of light-signals explained above for platform 8, 

 with the same choice of the constant c*. 



Here v = the observed velocity of the point 0' with respect 

 to the platform S, and r = the observed rate of the clock 

 at 0'. 



In order to establish this theorem we must show :■ — 



(1) that the coordinates x\ y assigned to any given station 



will be permanent that is, independent of the time t ; 



and further, that if x \ y', and t- are assigned in the maimer 

 indicated, then 



(2) the clock at 0' will be found to run at a uniform rate 

 according to the test laid down in § 2 ; 



(3) each clock on S' will be found to be synchronous with the 



clock at 0' according to the test laid down in § § 3-4 ; 

 and 



(4) if a light-signal, started from 0' when the clock at 0' 

 reads tj, arrives at any station A' = (a/, y') when the 

 clock at A! reads t{ , then the " distance " \/ V 2 -f y' 2 from 

 f to A' is c times the u difference in clock-readings " 

 H — *o • 



The verification of the truth of these statements is merely 

 a matter of direct computation from the transformation equa- 

 tion s, which may well be left to the reader \. 



* The expression for k in the above formulas can be simplified by the 

 use of hyperbolic functions. Thus, if we determine a quantity V so that 



then 



- = tanh V, 

 c 



cosh V. 



Further, by a particular assumption in regard to the rate at which the 

 clock at O' is running, the value of I can be reduced to unity. But for 

 the theorems which we shall give, this assumption is not necessary, 

 and will not be made. 



The notation, k, I, here used, agrees with the notation of Lorentz 

 rather than with that of Einstein, who uses $ in place of k, and has 1=1. 



t Thus, if a given station on S' is opposite A- t wheu the clock A T reads 

 t r and, later, is opposite A 2 when the clock at A 2 reads t 2 , then the u t" 

 in the formula for x will have increased by t 2 —t„ while (since the 

 station in question is passing platform S with velocity v) the " x " in that 

 formula will have increased by v (t 2 — t x ) ; hence the value of x—vt will 

 remain unchanged. 



\ The method by which the transformation equations themselves are 

 obtained will be explained in the appendix. 



