﻿The Identical Relations in Einstein s Theory, 381 

 Now, 



(B^/ A p ) r = (IV/) r A p + BpvS A Pl r ; 



so after cancellation we get 



(B vff r p + B ( rr/4-Br V /)A /t p 



The expression in brackets on the left vanishes identically. 

 Since Ap is arbitrary, the expression in brackets on the right 

 must also vanish. Q.E.D. 

 The identity 



B vaT P + B aTV P + B TV a<> = 



follows at once from the well-known identical relations 

 between the Riemann symbols. The three-term identity is 

 usually stated in the form 



(/uLTav) -f (/jlcvt) -f (/jlvtct) = 0, 



or in the modern notation 



Bfxpcrr~\~ ±>[xtv(t ~\~ tjjxarv == 5 



here B^cr denotes greB txva 6 = (firav), Since B [irv(T = 'B 1/(Tf j ir . 

 and Bh(ttv = BcffivTi 



= B pV (JT 4" Bpa/JLT T j5<T/J.V7 



We assume that the determinant \g^v | = g does not vanish 

 in the region under consideration ; therefore the expression 

 in brackets must vanish. 



This identity can also be proved by observing that the 

 expression 



(Av, or — Aj^, to) + (Aff ) tv — -A-ff, vt) 4" (A r, vff A-t, <jv) 



vanishes if A v is the derivative of a scalar; for in that case 



A Vt crr — A.cr,vT, Aa t Tv == At, <tv> and A^yu = Aj/,rcr. 



If we contract (1) by putting r = p, we get 



