D O C . 4 4 3 G E N E R A L R E L AT I V I T Y A N D M O T I O N 4 3 1 This equation is valid for any variation of the ’s and the ’s, hence also for such a one as can be obtained by mere infinitesimal transformation of the coor- dinate system (transformation variation). For such a transformation, vanishes, because is an invariant and because according to the field equations H vanishes everywhere. Furthermore, we must set , (13) where is an infinitesimal vector (with the derivatives , etc.). According to (9) one has . (14) Taking (13) and (14) into account, one obtains from (12) the equation . (15) Let us rearrange this equation, which is equivalent to the field equations and which forms the basis of our further considerations. The reason for rearranging will become apparent only later. The first term contained in the brackets of equation (15) is first rearranged by extracting the derivative with respect to , so that and appear. Then these derivatives of Γ are expressed by the Γ themselves, by means of the relation , which is valid according to (9) and (10). After simple rearrangement, the first of the three terms of (15) then becomes . (16) The reason for this rearrangement will become clear later. We summarize our result in the following form:[27] , (15a) , (15b) where , (15c) and follows from (15) and (16), and is a linear, homogeneous function of the first and second derivatives of the with respect to the coordinates. ( ) is gμν Γμν α δH H –g --------- - Γμν α δ Γσν α ξ,σ μ – Γμσ α ξ,σ ν – Γμν σ ξ,α σ Γμν σ , α ξσ – ξ,α μν – + = ξσ ξ,σ α τ , α ∂Γμν ∂H gμνδα τ – 1 2 -- - gμτδα ν gντδα) μ + ( + = [p. 8] 0 ∂xα ∂ gμνΓμν τ , α gμαΓμν τ , ν – ( )ξτ gμν( Γτν α ξ,τμ – Γμτ α ξ,τν – Γμν τ ξ,α τ ξ,α μν –+ ) – gμα( + Γτν ν ξ,τμ – Γμτ ν ξ,τν – Γμν τ ξ,ν τ ξ,ν μν –+ ) = xτ Γμν α , α Γμν α , ν H 0= ∂xα ∂ ξσ{ Γμν α g,μν σ – Γμν ν g,μα) σ + ( δσ α( gμνΓμρΓντ τ ρ gμνΓμνΓρτ)}τρ – – ξ,α σ g μνΓμν σ gμσΓμν)ξ,σ ν – ( σ gμνΓμν α gμαΓμν)ν – ( – + ∂xα ∂A α 0= A α tσ αξσ B α += tσ α Γμν α gσ μν – Γμν ν gσ μα) + ( δσ α( gμνΓμρΓντ τ ρ gμνΓμνΓρτ) ρ τ – – = B α ξα tσ α