10 



and if for ög"^ the value (49) is substituted, this term becomes 



i ^ {ah) Tab cf^«* — i ^ {ahcd) g^'- g,d Tab (^r^, 

 or if ill the latter summation a, h is interchanged with c, d and if 

 the quantity 



T = :S (cd) g'^d T,d (64) 



is introduced, 



1 ^ (ab) {Tab - è gab 2") ör^. 

 Finall}', putting equal to zero the coefficient of each ö^"^ we 

 find from (62) the differential equation required 



G„b= — x{Tab — kgabT) (65) 



This is of the same form as Einstein's field equations, but to see 

 that the formulae really correspond to each other it remains to 

 show that the quantities Tab f^nd J^* defined by (63), f59) and (60) 

 are connected by Einstein's formulae 



Zl = V^:E{a)g<^bT„, (66) 



We must have therefore 



2 ^(a) ^-- { — -^ = — L + ^{a) y^*ac ipa'c' ... (67) 



and for h == c 



2^(a)^«if— -)= 2:(a)i|.-„6,p„.e- (68) 



§ 42. This can be tested in the following way. The function L 

 (comp. § 9, 1915) is a homogeneous quadratic function of the tpaó's 

 and when differentiated with respect to these variables it gives the 

 quantities i^ai>- It may therefore also be regarded as a homogeneous 

 quadratic function of the ip„6. From (35), (29) and (32)^), 1915 we 

 find therefore 



^=\^^ ^{pqrs){gP'g'i' -gi'-g'^y^pq^rs ■ ■ • (ö9) 



Now we can also differentiate with respect to the ^'^*'s, while not 



the xpab's but the quantities \pab are kept constant, and we have e.g. 



According to (69) one part of the latter differential coefficient is 



1) The quantities y^^ in that equation are the same as those which are now 

 denoted by gf"^. 



