350 Dr. Harold Jeffreys on Derivation of tlxe 



a function of #, y, z, and £, so that the available data are 

 not sufficient to prove the constancy of k*. With the 

 further assumption that 



x\ y ', 0', and £' are linear functions of x, y, z, and t, (4) 



we see at once that k must be a constant ; but there is no 

 more reason for this assumption than for the other. 



If, however, instead of (4) we make the assumption that 

 any object moving uniformly with regard to S is also 

 moving uniformly with regard to S', *(5) 



it is possible to infer from this and (2) both (3) and (4). 



For believing (5) there is a good a priori reason, afforded 



by the relativistic modification of Newton's first law of 



motion. This asserts that for an "isolated" particle referred 



to axes in an isolated non-rotating body the differential 



coefficients of the measured space co-ordinates with regard 



to the measured time are constants. Thus for any such 



. , dx du dz dx' du' dz' ,, 



particle -=-. -f-. -7-, yt? -¥i -> tt are all constants. 

 v dt' dt' dt' dt' ' dt' ' dt' 



By comparing any uniformly moving object with an isolated 



particle, we therefore see that whenever — , -7- , -~ are 

 -j 1 -j i -j \ ill at at 



all constant, -jr, -Jp~, -777 are all constant. Now if this 



ds 

 be so, -j- is a constant, for it is equal to 



{«'-(S)'-(S)'-(S'J'- 



., ds' . ,. t ,1 dx dy dz 

 Also 7 , is constant, and thus -,-, -f- , --. 

 dt ds ds ds ' 



du 

 , -7- are all 

 ds 



constant. 



TT d 2 x d 2 i/ d 2 z d 2 u _ 



Hence rf? = 5? == ^ = ^ =0 - 





The argument is reversible and therefore 



leads to the 



proposition : 



If d?x _ d 2 y _ d 2 z _ d 2 u 

 ds 2 ds 2 ds 2 ds 2 





then d 2 x' d 2 y' d 2 z f dht' 

 ds' 2 ~ ds' 2 ~ ds' 2 ~ ds 12 



• • • (6) 



Thus (5) is equivalent to (6). 





* E. Cunningham, { The Principle of Relativity,' p 89, or Proc. Lond. 

 Math. Soe. 1909.. 



