4 0 D O C . 1 7 G R A V I T A T I O N A N D E L E C T R I C I T Y where stands for a certain tensor density. It is easy to prove that in connection with (7) the equations (5) are equivalent to the equations . (10) By lowering the upper indices and by taking into account the relations , where stands for a covariant tensor,[5] we obtain , (10a) where is a covariant vector.[6] This set of equations, together with the two indi- cated above, (7) and , (4) are the result of the variation procedure in its simplest form. The striking thing about this result is the occurrence of a vector in addition to the tensor and the quantities .[7] In order for this result to be consistent with the hitherto known laws of gravitation and electricity, where the symmetric component of the is conceived of as a metric tensor and the antisymmetric one as an electromag- netic field, we have to assume that vanishes. We shall do so in the following. However, for future considerations (e.g., the problem of the electron), one should bear in mind that the Hamiltonian principle does not offer any indication for the vanishing of the . Setting the to zero leads to an overdetermination of the field in that for 16 + 64 variables we have 16 + 64 + 4 algebraically mutually inde- pendent differential equations.[8] §2. The Pure Gravitational Field as a Special Case. Let the ’s be symmetric. Equations (7) are fulfilled identically. By exchang- ing μ and ν in (10a) and subtracting, we then obtain in an easily comprehensible form . (11) If the part of Γ antisymmetric in the two last indices is denoted Δ, then (11) assumes the form f μ ∂gμν ∂xα ---------- gβνΓβα μ gμβΓαβ ν gμνΓαβ β – δα ν fμ + + + 0 = [p. 416] gμν gμν- –g --------- gμν –g = = gμν ∂xα ∂gμν – gσνΓμα σ gμσΓαν σ gμνφα gμαφν + + + + 0 = φν ∂gνα ∂xα ----------- ∂gαν ∂xα ----------- – 0 = 0 Rμν α ∂Γμν· ∂xα -----------¹ -– © § Γμβ α Γαν β α ∂Γμν ∂xν' ----------- - Γμν α Γαβ β – + + = = φτ gμν) ( Γμν α gμν φτ φτ φτ gμν Γν μα , Γμ αν , Γμ, να – Γν αμ , – + 0 =