1 7 8 D O C . 1 5 8 O N R I E M A N N C U R V A T U R E T E N S O R (13) . From (4), (12) and (13) we get (14) . For the following, it is convenient to choose a local coordinate system for which Then, instead of (8), one obtains the sim- ple equation (8a) . In these coordinates, because of equations (11) and (14) and (7), etc., hold. The other components of are obtained from this by permutation of the indices. Then equation (6) is verified for the above components and local coor- dinate system chosen hence it is also generally valid. From (6) it is easy to conclude that the conditions and are equivalent. Therefore, the law of the pure gravitational field in the sense of equations (2a) is determined by the condition that the asymmetrical[14] component of the Riemann curvature tensor, formed according to Rainich, vanishes. The general system of equations (2a), (3) can also be obtained from a quite anal- ogous consideration. We use the electromagnetic tensor ( ) to define the tensor (15) , whose symmetry properties are the same as those of the Riemann curvature tensor. Furthermore, let us introduce the ெelectromagnetically extended tensor of curvature” Sik lm , S ik lm , = Aik lm , A ik lm , –= Sik lm , 1 2 -- - Rik, lm R ik lm , + ( ) = Aik lm , 1 2 -- - Rik, lm R ik lm , – ( ) = gμν δμν = δμμ 1, δμν 0 if μ ν). ≠, = = ( Δik αβ δikαβ = R12 34 , R 12 34 , – R12 34 , R34 12 , – 0, = = R12 23 , R 12 23 , – R12 23 , R34 14 , – R12 23 , R14 43 , + R13 G13, = = = = R12 21 , R 12 21 , – R 12 21 , R34 43 , – R11 R22 1 2 -- - R –+ G11 G22 + = = = Siklm Aik lm , 0= Gim 0 Rim 1 4 -- - gimR – = = ϕik Eik lm , 2 3 -- - ϕikϕlm 1 2 --( - ϕilϕkm ϕimϕkl) – + = [p. 103]