7171 



Derivation of Symmetrical Gravitational Fields. 21 



unknown functions y, and these equations are not in 

 general consistent.] It will be convenient to drop the 

 double label, which is now useless, and so w T e write 



(ds) 2 = g 1 (dx 1 ) 2 + g 2 (dx 2 ) 2 + g 3 (dx 3 ) 2 +g i (dx 4 ) 2 . 



Since ^ s = if r zfz s the Biemann-Christoffel symbols 

 take comparatively simple forms. If r, s, t are distinct 

 numbers of the set 1, 2, 3, 4, 



_ 1 (~dg rt "dffst "dfj ri 



[ rs t i = 1 (^,dgst_dgrs\ = Q 

 Lr5 ^ j -2^ + ^ W " U 



{rs, t} = g tk [rs, k] = g u [rs, t] = 0, 



since g tk = if the umbral symbol k is different from t. 



1 7\a 

 {rs, r} = {sr, r} = g rk [rs, k] = g rr [rs, r] = — -£, 



since # rr = — . 

 g r 



Similarly, 1 .^ 1 ^ 



The Riemann four-index symbol of the second kind 

 {pq, rs} is denned by 



+ {pr, l}{ls, q}-{ps, l}{lr, q] . . . 



(Z an umbral symbol). 



Hence, if the four letters p, q, r, s have all different 



numerical values, 



{pq, rs} =0, 



since, for example, in the summation {pr, l}{ls, q} the first 

 factor {pr, 1} of any term vanishes unless l=p or r, in 

 which cases the second factor vanishes. The remaining 

 types are readily found to be 



r i _ JL d 2 ?<? l_'bg 1 dgq ^ L _'dgp'dg 1 



s 2g q ~^xibx p £gq 2 'dxp'dxs fy v g q ~dx s 'bx p 



1 "dffs ~dg 



where q is not intended to be an umbral or summation 



symbol. 



