General Relativity and Einstein's Theory. 395 



Therefore the motion of a body at rest in S', or in a system 

 reciprocal to S', is described relative to any possible reference 

 system by (21), provided ds is expressed in terms of the para- 

 meters X, y, z, I defining the particular reference system relative 

 to which observations are being conducted. Thus the motion 

 of a body at rest in S', or in a system reciprocal to S', is described 

 relative to the accelerated system 6^ by 



hfVh' dx'+k' dp = 0, (22) 



where i Pdx 



-'-f 



. A UkJ n 

 h = — e 

 n 



and I pdx 



Ae 



I ndx 

 a^ n 



That this expression gives for the acceleration the same value 

 (19) already obtained is easily verified. For put 



H= Vk^ + 1^ U\ 



where dx 



il 



Then 



lfHdl = o, (23) 



and the corresponding Lagrangian equation is 

 d idH\_dH _ 

 'dl\ju)~^x~^' 



Differentiating, it is easily found that 



dl ~ h' dx'^ \k 9x h 9x)' ^^"^^ 



Substituting for h and k their values, 



d/ an n dx 



which leads to (19), when it is remembered that 



■' dr 



c 



C 



n ^= ~ . 

 c 



