Relations in FAnsteins Theory. 603 



the third is 



'" B^B&'aB^ 2 B#aB«%BaV 



j Ay B 2 / a« B \ 



B^aB^v" B#„' 

 B^aB^vB^v 



Av B 3 * , / 



<l -, -n k— log V— </> 



so that (9) gives 



O^ Ai/ aft B /7 A/3 , .-, Kv O , / /ia\ 

 = — Q CI P ^r '' Z. +Z(/ ^r - =r— log V —Q. (10) 



ox? J J B^vB^B^ J B^aB^B^v s 



Comparing (8) and (10), we see that the relations (6) in 

 the new coordinates are identically satisfied at the origin. 

 Being tensor relation^, their validity is unaffected by a 

 transformation o£ coordinates, and they are therefore 

 identically satisfied in the original coordinates at the origin. 

 But the origin is an arbitrary point of the continuum. 

 Hence the relations in question are identically satisfied 

 throughout the continuum. 



These being proved, we may write (2) 



and multiplying by y ov 



Taking the contracted covariant derivative in accordance 

 with (7), we have 



— 07TK 1 ^ = U /jtI/ - ^<J lx - - 

 O <*■ v 



^y,v 2 -n „ 5 



since g^—O, unless u = v. 



Hence, by the identical relations, we have T^„ = 0, and the 

 laws of energy and momentum are deduced as algebraic 

 identities from Einstein's law of gravitation. 



