﻿Approach to the Theory of Relativity. 501 



exactly the same method as that already described for the 

 stationary platform S, provided only that v<c*. 



The method of proof will be as follows : we shall first 

 suppose the coordinates on S' to be theoretically assigned by- 

 means of certain transformation equations ; we shall then 

 show that the result of this assignment is precisely the same 

 as if the observers had used the method of light-signals 

 described above. Since the theoretical method of assign- 

 ment is easily seen to be possible and permanent, it follows 

 that the method of light-signals is also possible and 

 permanent. 



We suppose for convenience that the axis O'X' on S' is 

 sliding along OX in the positive direction ; and we suppose 

 that at the instant when 0' is opposite ; the clocks at and 

 0' read zero seconds. 



We shall not, however, place any restriction on the 

 observed rate, r, of the clock at 0' |- 



The transformation equations. 



13. We now state the following theorem, which is funda- 

 mental for the whole discussion. 



Theorem 1. If the coordinates x , y' and the clock-reading 

 t' at each station on the moving platform S' are assigned, at 

 each instant, in terms of the coordinates x : y and the clock- 

 reading t at the point on S which is opposite that station at that 

 instant, according to the following " transformation equations 



)) 



x' = lk(x—vt), y' = ly, t' = lklt— - 2 w\ 



V C ! 



where 



k= — , , — -- and 1 = 



then the result will be precisely the same as if the observers on 

 S' had adjusted their clocks and laid, out their coordinate system 



* This restriction does not mean that "no velocity greater than the 

 velocity of light is possible," since an " observed velocity " as here 

 defined may have any value from to go ; it means simply that if a plat- 

 form S' were moving through the pother with a velocity greater than the 

 velocity of light, then the " method of light-signals " here described 

 could not be successfully employed by observers on that platform. 



t This feature of our treatment is believed to be new. Einstein, for 

 example, assumes that the two sets of clocks are " of the same physical 

 construction." 



