Closi?ig Years of the Nineteenth Century. 443 



When the original variables are by direct substitution replaced 

 by the new variables in the differential equations, the latter 

 take the form 



div! hi = 0, cur^ h t 



that is to say, the fundamental equations of the aether retain 

 their form unaltered, when the variables are subjected to the 

 transformation which has been specified. 



We are now in a position to show the connexion of this 

 transformation with Fitz Gerald's hypothesis of contraction. 

 Suppose that two material particles are moving along the axis 

 of x with velocity w - c tanh a. From the relation 



v x . sech a 

 v x = c tanh a + = - 



cosh a + v Xi c~ l sinh a ' 



it follows that v Xl is zero for each of the particles, which implies 

 that they are at rest relative to the new axes. Let %i and x\ 

 denote their coordinates with respect to this latter system ; then 

 the coordinates of one particle at the instant ti, referred to the 

 original axes, will be given by the equations 



x = Xi cosh a + ct l sinh o, t t\ cosh a + Xi c~ l sinh a ; 

 and the coordinates of the other particle will be given by 

 x f - x\ cosh o f cti sinh a, t f = t l cosh a + x\ c' 1 sinh a ; 



so that at time t the latter particle will have the coordinate x", 

 where 



x" = of + w (t - t') 



= x\ cosh a + ct l sinh a + (x - x\) sinh 2 a sech a, 

 which gives 



x" x = (a?'i Xi) (1 w'/c 2 )^. 



This equation shows that the distance between the par- 

 ticles in the system of measurement furnished by the original 

 axes, with reference to which the particles were moving with 

 velocity w, bears the ratio (1 - w-/c'-)^ : 1 to their distance in the 



