108 Dr. L. Silberstein on General Relativity 



and, therefore, ^ = 1, which is precisely the case of the 

 element (1). Thus, that line-element characterizes a space- 

 time of constant curvature, the value of k=l/R 2 being one- 

 sixth of If, the above differential invariant. Q. E. D. 



Such being the case, we know beforehand that the 

 theorem (5) will hold. Yet it may be well to verify it by 

 calculating directly some at least of the Riemann symbols 

 corresponding to the tensor (2), in part also to make the 

 reader more familiar with the handling of these remarkable 

 symbols. Now, their general definition may be put into 

 the form 



«=^{!}-&{a} + {! 



to be summed as usual. Thus, with the normal form (6), 

 and therefore, with the values (10), we find, for instance, 

 remembering that g 1 = const. = — 1, 



?H 8 «HtI (14) 



(2112)=- A + ± (M, 



das ±g 2 \dxj 



which becomes in the case of (2), with x x — r, (2112) = 



— sin 2 ^. Most symbols vanish, as for ex. all (iiii), and 



more generally all (ijkk), — this in virtue of the general 

 property (ijhk) = — (ijkh) ; again, many symbols that in 



general would not vanish do so in our case owing to ^ — =0, 



and so on. Finally we find, f. ex., another non-vanishing 

 symbol, 



and so on. Thus we have 



(2112)= i^); (3113)= Ifo), 



as it should be; for, by (5), (2112)= ™(#i 2 2 — #22#n)> aud 



in our system gi 2 = 0, ffn=— 1; and so on. Having thus 

 verified (5) in a pair of examples, we can safely apply that 

 theorem to the tensor gij transformed from (2) to any system 

 U{. If we use, for instance, normal coordinates, i. e. such 

 that ds 2 = 'Zgudui 2 i then all Riemaun symbols vanish with 



