856 



(27) T 'wj. =1 wt 



ill which T is an arbitrary function, leaves the parameters of ttie 

 contravariant displacement unaltered, while the covariani parameters, 



which we will also further denote with r',I, will be transformed 

 in the following way : 



B „ d la T ^ 



(28) •r\.= r'y_. '^Ax. 



Such a change of measure cannot be applied in the same easy 

 wa}' to contravariant vectors, the new components t— i f/j-' being in 

 general no more exact differentials. In this case we would be obliged 

 to consider space-time as a system of non-exact differentials, and it 

 would no more be possible to re|)resent a point by four finite co- 

 ordinates. This case has doubtlessly but little attraction so long as 

 there are other possibilities. 



When we wish lu "loose" the vector 6,, in the above mentioned 



sense, we have only to consider the r',1, as the parameters of the 



covariani displacement and to define the r/^, the parameters of the 

 contravariani displacement, in the following way : 



(29) n,„ = r'l^ ■+ s^ A] =!,[*! — V, ii'.'^ «' + 7. ^I v + V. K^ '>- 



We then have obtained that /V/. is independent of 6'; and thai, 

 when covariani measure is changed, Sx is transformed in the 

 following way : 



d la T 



(30) 'S^ =s,.+ ^. 



OXa 



It is very remarkable that by (23) F' ,',j has just a form that 

 leads to the desired transformation of the potential vector. If f.i. 



nl contained a term with SjA'a, it would not be possible to 

 obtain an equation of the form (30). 



Representing the covariant differential operator determined by 



rlfj. and iS'fj. by V, (HI) is clianged into : 



\lll C.9' = — '/, Aa, I + '/, l« ^ 



Va ff)^ = — V. <)'='/-■ *' — Vi ."«> i" + Vi 'x 9'!' — 2 Sa gxu ■ 



The tensor (7;„ is a quantity variable with transformation of co- 

 variant measure, for its components do not change, while the 



