855 



We now return to the equation (II) obtained from the variation 

 principle. With regard to (13), (14) and (17) this equation leads to 



(26) i^ = 0. 



Since i' has the character of a current vector, it is not allowed 



to consider variations of the alternntinij part of V-,'^, when we wish 

 to keep the current vector in the equations. In regions where only 

 an electromagnetic field exists and no current, the variation principle 

 remains valid without any restriction. 



Tiie expressions {IV) and (25) onl)- dift'er from those of Einstein 

 by the terms in S,, hence an electromagnetic field is also possible 

 in places with vanishing current vector /'. There the vector S, 

 behaves as a potential vector. 



We can further make the following important remarks: 



1. In the tield equations {IV) Sx does not contribute to ihe sym- 

 metrical part of R'f,j . 



2. When there is no current the displacement is by (///) co7i- 

 formal, the fundamental tensor being diminished with 2(U''Sxg^'' 

 when the pseudoparallel displacement is cLv . 



3. When there is no current and no potential (23) passes into 

 the ordinary equation of the gravitational tield, in the same way 

 as Einstein's equation. 



3. The potentialvector Sy. It is remarkable that here the potential 

 vector S) occurs as unambiguously determinated, not as a vector to 

 which an arbitrary gradient vector may be added. This difficulty 

 disappeai's when we make the supposition that the parameters which 

 define the displacement are not the same for covariant and for 

 contravariant vectors') and thus no longer adopt the invariance of 

 transvection. It is namely possible to alter covariant parameters 

 independently of the transformation of the original variables by 

 changing the measure^) of the covariant vectors. This change of 

 measure 



1) For these displacements cf. tlie above mentioned paper in Math. Zeitschrift 13. 

 ') This change ot measure has nothing to do witli an introduction of a ds. 



