1077 



(2) 



According to its behaviour with respect to transfbrruatioiis Ï is 



a mixed volume tensor, ^153Z a volume scalar, @* is no volume scalar, 



but this quantity is formed from the volume scalar — v — g G 



(w^here G is the total cuvDature of the four-dimensional continuum) 



ÖV"' ... 



by elimination of the second derivative ^ — ^ — by partial mtegration. 



We have^) 



— ©*= y/—gG-2 





(3) 



where '21 is a four-fold vector in the sense given to it in the special 

 theory of relativity of Einstein-Minkowsky. 91 is thus covariant for 

 LoRENTz transformations. The sign of 6* and for ^l has been chosen 

 in such a way that the expression (2) gives the density of energy 

 of the matter witli the right sign. For — 6* and '^ir we have the 

 expressions : 



0>* = l/ — ^ ^ g/'-' g"'-'- ^^/5 



a/3/." 



lit 



J 



To 





ox 



lJ.v':z 



jLtl' 

 O 





(^-9^")^ 



y-vy 



where the Christoffel symbols 



|_ y J 2 ybxji bx 

 are used. According to the equations' 



W=rg 



-'[':]• • 







= — JS" 



1 ^^ .-, ^9'"" 



.'/yd 9v 



dxx 



(4) 



(6) 



(7) 



(8) 



1) Because of equation (3) and as at the limits of the domain of integration 

 all variations are taken equal to zero, the variation principle (1) is equivalent with 

 the variation principle expressed by the following equation 



<f I I j ((—[/^. G + X 9??) dx, dx, dx, dx^ = , . . (la) 



from which equation Einstein originally started. 



2) Einstein, Grondlage, equations (29j and (32 . 



* 



