852 



(4) ^;> = «;^-' {g-^) f V.(-l)(^-^^5.)-V.(n-l).,S„ = 



in wliich V* is Ihe oovariaiit differential operator belonging to J)ft. 

 We suppose tlmt the determinant /?' = R' does not vanish. Hence 



there exists an inverse quantity /■'■": 



(5) R r = J— ; r ff „, =: r 7?',^ — A, . 



When Fu) and <tj,, are the antisynimetrical and the symmetrical 

 part of Rf,, : 



(6) F'u. = /?'-„,] ; G'^i = K'f^.) ') 



and when the word function fy =^ H^^ — R (scalar density) is a 

 still unknown function of Gy, and F.j,, we then have the variation 

 equation : 



( 7) rf / .ip dr = /»■'/' dR'f,, dT = 0'), 



in which 



(8a) r'"' = v"" V—R = iff'"' + /"■") V ~R' 



When we substitute into (7) the value of i4), we get for n = 4 



(9, =ƒ>"■" ''^|^R;>-77(^-| - i') + 2rf(^^ - ^'" « )^V,^(5>S.) j . 



an equation that, R,,, being independent of .S, , is equivalent with 

 the two equations 



(10) dA:^\-A':{^*ii t/'^-P^/'') + v./" -p. /''-'/, S«Ó = 



(1 1) d s. I v.: ƒ ^^ - p,.f' - V, (V,: «''•" - p, ."^ - V, s. ^''^ 1=0, 



*) In this paper i;,. m>^, means '/^ <", ^''u + "„ ^•)- 



') We use the variation symbol d in place of J to prevent confusion \vith the 

 symbol J of the covariant differentiation. 



