32 



because other functions of the coordinates occur in it, but which never- 

 theless no observation will be able to discern from it, the indeüniteness 

 which is a necessary consequence of tlie covariancj of the field 

 equations, again presenting itself. 



What has been said shows that the total gravitation energy 

 in this new system will have the same value as in the 

 original one, as has been found already in § 60 with the restrictions 

 then introduced. 



§ 63. If t' were a tensor, we should have for all substitutions 

 the transformation formulae given at the end of § 40. In reality 

 this is not the case now, but from (96) and (97) we can still 

 deduce that those formulae hold for linear substitutions. They 

 may likewise be applied to the stress-energy -components of the 

 matter or of an electromagnetic system. Hence, if Xa'' represents the 

 total stress-energy -components, i. e. quantities in which the corres- 

 ponding components for the gravitation field, the matter and the 

 electromagnetic field are taken together, we have for any linear 

 transformation 



1'c^ = — = S {kl) pkc ^Ib'^h^ ■ . • . (121) 



We shall apply this to the case of a relativity transformation, 

 which can be represented by the equations 



x\ — «.^•J -|- bcx^, ;v\ = A'„ x\ = .^3, .v\ =r a.v^ + - x^, (122) 



c 



with the relation 



a'--b' = l (123) 



In doing so we shall assume that the system, when described in 

 the rectangular coordinates a.\,.i\,,i\ and with respect to the time o-'^, 

 is in a stationary state and at rest. 



Then we derive from (97) ') 



^) We have g^ = g^i = g^^, = 0, while all the other quantities gab are independent 

 of Xi. Thus we can say that the quantities gab and ga/>,c are equal to zero when 

 among their indices the number 4 occurs an odd number of times. The same may 



be said of 9'«'', ö'«*iS ^ (accordmg to (116)), 3 — f 5 and also of pro- 



Ogab, cd OXk \Ogab,cdJ 



ducts of two or more of such quantities. As in the last two terms of (97) the 

 indices a, b and f occur twice, these terms will vanish when only one of the 

 indices e and h has the value 4. 



As to the first terra of (97) we remark tliat, according to the formulae of § 32, 

 each of the indices a, b and e occurs only once in the differential coefficient of 

 Q with respect to gai>,e, while other indices are repeated. As to the number of 



