﻿Approach to the Theory of Relativity. 507 



In particular, the results stated in Theorems 2, 3, and 4 

 will still hold true if we interchange S and S' in the statements 

 of those theorems — the relations between the two systems 

 being entirely reciprocal. Put briefly, a figure in either 

 system appears to be shortened in the direction of motion 

 when observed from the other system ; and if two synchro- 

 nized clocks on either system are observed from the other 

 system, the one which seems to be in front in space appears 

 to be behind in time. These are the famous paradoxes of the 

 Theory of Relativity , which are often cited as proof of the 

 assertion that the Theory of Relativity is incompatible with 

 our ordinary ideas of time and space, but which here appear 

 as necessary consequences of perfectly natural and reasonable 

 conventions for setting clocks and laying out coordinates. 



24. In order to emphasize this result still further, and to show how 

 completely the aether drops out of the calculation, we shall now consider 

 two moving platforms, S' and S", and write out explicitly the trans- 

 formation equations that hold between them. The method will be the 

 obvious one of writing the transformation equations connecting each of 

 the systems S' and S" with the stationary system S, and then eliminating 

 S. 



Let v', z/' = the velocities of 0' and O" with respect to S, and put 



v' v' 1 



— =tanh V and — =tanh "V 



c c 



1 1 



so that k'= ^i_ ( „ 7c) =«=coshV' and *" = ^/ x __ (p » /c j f = cosh V". 



Further, let r', r"=the observed rates of the clocks at O' and O" with 

 respect to S, and put 



r' 

 l' = ,=. — , ., NO and l" = 



These are the quantities which enter the equations connecting S' and 

 S" with S, by Theorem 1. 5 



Now let v = the observed velocity of S" with respect to S', and 

 v 

 put - =tanh V, so that 



= cosh V 



s/l-ivjcf 

 also let )-= the observed rate of the clock at O" with respect to S' 



and put 1= - ■ . 



\/l-(v/c) 2 



Then a perfectly straightforward, though somewhat long, calculati 

 gives us the following theorem 



