21 



of the second derivatives of those quantities. This latter involves 

 that, if we replace (91) by 



f dQ \ d / aQ \ 

 72 = Qi + Q, — ^ (ab/e) gab,fe — ^(abfe) ~- gaij, 



\(jgab,/e J OXe\Ogab,feJ 



the second and the third term annul each other. Thus 



R=Q,-:E{ahfé) — [- \gab,f .... (92) 



dXe \Ogab,feJ 



If now we detine a complex i> by the equation 



1 

 iue = _ —öh'R, (93) 



2x 



we have 



1 dR 

 (divi^)h = -^~ (94) 



If finally we put 



t' = t + u + ^ 



we infer from (90) and (94) 



div\' = divt (95) 



and from (88), (89), (93) and (92) 



1 I do d / ÖQ \ 



I'kf^ = -\-Q. + :E (ab) ^gab,k - 2 W) — ^^- ]gab,f- 

 2x I Ogub, h OXh \JjgabjhJ 



S (abf) — gab. h + ^ We) ^ '. 9ab. A (96) 



O.Vf\Ogab,hfJ OXe\Ogab,fe/ ) 



and for e =\= h 



^h'= — \^ {ab) gab, h — S (ab/) -— gahj — 



2x ( Ogab,e OXh\Ogab,feJ 



-.2{abf)^(-^-^\gab,i\ (97) 



Formula (95) shows that the quantities i'a^ can be taken just as 

 well as the expressions (88) for the stress-energy-components and we 

 see from (96) and (97) that (hese new expressions contain only the 

 first derivatives of tlie coefficients ^„6; they are homogeneous quadratic 

 functions of these differential coefficients. 



This becomes clear when we remember that Q^ is a function of 

 this kind and that only Q^ contributes something to the second 

 term of (96) and the first of (97) ; further that the derivatives of Q 

 occurring in the following terms contain only the quantities (/ah and 

 not their derivatives. 



^ 55. Einstein's stress-energy-components have a form widely 

 different from that of the above mentioned ones. They are 



