[ 349 ] 



XXX. On the Derivation of the Lorentz- Einstein 

 Transformation. By Harold Jeffreys, M.A., JJ.Sc* 



IF we have two systems of reference, S and S', and space 

 and time co-ordinates be measured relative to each of 

 them, the usefulness of the Lorentz-Einstein transformation 

 Kes in the fact that it provides a means of determining 

 either set of measurements when the other is given, and the 

 relative motion of S and S' is known. Let the space 

 co-ordinates and the time referred to S be #, t y, z, t, and those 

 referred to S' be a', ?/, z', t' . One datum we have to start 

 with is that provided by the Michelson and Morley experi- 

 ment, recently confirmed by a different method by Majorana f, 

 namely, that the velocity of light is always the same relative 

 to both systems of reference. Now if any object (a particle 

 or a wave-front, for instance) moving with the velocity of 

 light relative to S, be compared with a particular luminous 

 wave-front starting at the same point and moving in the 

 same direction relative to S, we see that the two coincide at 

 all times. They must therefore, by the fundamental 

 principle that makes it possible to refer objects to space-time 

 co-ordinates at all. coincide at all times relative to S'. Thus 

 if c be the velocity of light, it is true, whatever the moving 

 object may be, that 



m<i1<£)~^M%hm+ <£)'-'•« 



Put icdt — du, i c dt' — du', die 1 + dif + dz 2 -f du 2 = ds 2 , 



doo' 2 + dy' 2 + dz' 2 4- du' 2 = ds' 2 . 



Then (1) becomes simply 



ds = implies ds' =0 (2) 



Most writers on the principle of relativity have assumed 

 in deducing the transformation from the fundamental 

 principles that 



ds = ds' ; (3) 



but this is clearly more than the proposition (2) directly 

 warrants, for (2) would hold equally well if ds' were equal 

 to k ds, where k may be any function of a?, y, z, and t. If 

 the two sets of co-ordinates w r ere related, for instance, in the 

 same way as the co-ordinates of a point and its inverse 

 with regard to a four-dimensional hypersphere, I' would be 



* Communicated by the Author. 

 f Physical Review, May 1918. 



Phil. Mag. S. 6. Vol. 38. No. 225. Sept. 1919. 2 B 



