General Relativity and Einstein's Theory. 389 



Equating these two expressions, 



and addiner them 



^'^-W+^P^vj— ^ • (^) 



Now 



C-dt^^ ■=■ (dx-^ — vdt-i)^ -\- dy-^^ -\- rf^i", 

 whence, solving for dt^ and dt^, 



dt^- dt = -^^Jx^, (3) 



2)/ C'' — V * 

 ^A+^^.= - ^._^.' ^^, (4) 



where ?7i is the component of v perpendicular to dr^. Substitut- 

 ing in (i) and equating to zero the coefficients of dx\, dy-^ and 

 dz-^, it follows that 



dt' _ V 9t' 



9x~~~C- 'St ' 

 dt' 



The time interval dt' between two points as measured in S', 

 when the space and time intervals between these two points are 

 dx, dy, dz and dt in S, is evidently 



^ , 5/' ,5/' , 3t' , , 9t' ,^ 



dx dy dz dt ' 



since f depends only on x, y, z, and t. 



Substituting for the differential coefficients the values found 

 above, 



From (2) and (4) it follows that 

 dr\ = C'dt^ 



C'3t' J Vv 



