Page:LorentzGravitation1916.djvu/25

 § 28. Between the differentials of the original coordinates $$x_{a}$$ and the new coordinates $$x'_{a}$$ which we are going to introduce we have the relations

and formulae of the same form (comp. § 10) may be written down for the components of a vector expressed in $$x$$-measure. As the quantities $$\mathrm{q}_{a}$$ constitute a vector and as

$\sqrt{-g'}=p\sqrt{-g}$

we have according to (28)

$\frac{1}{\sqrt{-g'}}w'_{a}=\frac{1}{\sqrt{-g}}\sum(b)\pi_{ba}w_{b}$|undefined

or

$w'_{a}=p\sum(b)\pi_{ba}w_{b}$

Further we have for the infinitely small quantities $$\xi_{a}$$ defined by (19)

$\xi'_{a}=\sum(b)p_{ba}\xi_{b}$

and in agreement with this for the components of a vector expressed in $$\xi$$-units

$\Xi'_{a}=\sum(b)p_{ba}\Xi_{b}$

so that we find from (25)

$\chi'_{ab}=\sum(cd)p_{ca}p_{db}\chi_{cd}$

Interchanging here $$c$$ and $$d$$, we obtain

$\chi'_{ab}=\sum(cd)p_{da}p_{cb}\chi_{dc}=-\sum(cd)p_{da}p_{cb}\chi_{cd}$

and

The quantity between brackets on the right hand side is a second order minor of the determinant $$p$$ and as is well known this minor