2yJ ^ 



QdS, 

 where 



In the integral dS, the element of the field-tigiire, is expressed in 

 .i'-units. The integration has to be extended over the domain within a 

 certain closed surface <j; x is a positive constant. 



§ 33. When we pass from the system of coordinates x,, x^ to 



another, the value of 6^ proves to remain unaltered; it is a scalar quantity. 

 This may be verified by first proving that the quantities {ik, hi) 

 form a covariant tensor of the fourth order ^). Next, (^^0 being a 

 contravariant tensor of the second order ^), we can deduce from (40) 

 that {Gha) is a covariant tensor of the same order ^). According to 

 (•41) G is then a scalar. The same is true^) for Q d S. 



We remark that gia^gai'") and gabje = gah,ef- We shall suppose 

 Q to be written in such a way (hat its form is not altered by 

 interchanging g^a and gah or gabje and gab,ef' If originally this condi- 

 tion is not fulfilled it is easy to pass to a "symmetrical" form of 

 this kind. 



It is clear that Q may also be expressed in the quantities ^^* and 

 their first and second derivatives and in the same way in the ()"'''s 

 and first and second derivatives of these quantities. 



If the necessary substitutions are executed with due care, these 

 new forms of Q will also be symmetrical. 



§ 34. We shall first express the quantity Q in the ^„^'s and their 



1) This means that the transformation formulae for these quantities have the form 



{ik.lm) = 2 {abce) p„ipbkPciPem {ab,ce) 

 See for the notations used here and for some others to be used later on my 

 communication in Zittingsverslag Akad Amsterdam 23 (1915), p. 1073 (translated 

 in Proceedings Amsterdam 19 (1916), p. 751). In referring to the equations and 

 the articles of this paper I shall add the indication 1915. 



2) Namely: 



^'^'— ^(a6):T«/,jr6/.(7«^ 

 The symbol (g^^) denotes the complex of all the quantities gkl, 



3) Namely : 



G'im = 2 (ab) paipbm Gab- 



*) On account of the relation 



1* 



