﻿506 Prof. E. V. Huntington on a New 



The inverse transformation equations. 



21. Let v = the observed velocity of 0' as computed by 

 two observers on $, and v — the observed velocity of as 

 computed by two observers on S'. 



Then we find : — 



Lemma 1. The observed velocities v and v will always be 

 equal and of opposite sign : 



v == —v, 



whatever the relative rates of the central clocks at and 0'. 



Further, let r — the observed rate of the clock at 0' with 

 respect to S, and r' = the observed rate of the clock at 

 with respect to S' (see § 10). 



Then :— 



Lemma 2. The numbers r and r' are connected by the 

 relation 



v 2 



rr' = l ^ 



c 2 



22. By solving the equations in Theorem 1 for x. y, and t, 

 and using the results of these lemmas, we obtain the following- 

 theorem : 



Theorem 6. The " inverse transformations" by means of 

 which, when the coordinates x\ y f , and the time t' at any point 

 on S' are known, the coordinates x, y, and the time t of the 

 opposite point on S may be determined, are as follows : 



x=l'k{x'-v't'), y=Vy', 



t = Vk(t'~ V ^x% 



where 



1 , ** 



v'= — i\ k= — ,. - and I 



\/l-(vlcf VI -(v\cf 



These equations are of exactly the same form as the 

 equations in Theorem 1, if we interchange the accented and 

 unaccented letters. 



Experiments in which no observer leaves his own station. 



23. These equations in Theorem 6 show that if the observers 

 on S r are not allowed to leave their several stations, the 

 system S ; does obey the Principle of Relativity ; for the 

 symmetry of these equations shows that, if we confine 

 ourselves to the kind of observations so far considered, the 

 observers on W have as much right as the observers on S to 

 suppose that their platform is at rest in the aether. 



