# 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

 $dx'_{a}=\sum(b)\pi_{ba}dx_{b}$ (30)

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)[1]

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

or

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

Further we have for the infinitely small quantities $\xi_{a}$[2] 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)[3]

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

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

 $\chi'_{ab}=\frac{1}{2}\sum(cd)\left(p_{ca}p_{db}-p_{da}p_{cb}\right)\chi_{cd}$ (31)

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

1. Comp. § 7, l. c.
2. For the infinitesimal quantities $x_a$ occurring in (19) we have namely (comp. (30))

$x'_{a}=\sum(b)\pi_{ba}x_{b}$

and taking into consideration (19) and (20), i e.

$\xi_{a}=\sum(b)g_{ab}x_{b},\ x_{a}=\sum(b)\gamma_{ba}\xi_{b}$

and formula (7) l. c, we may write (comp. note 2, p. 758, l. c.)

$\begin{array}{l} \xi'_{a}=\sum(b)g'_{ab}x'_{b}=\sum(bcde)p_{ca}p_{db}\pi_{eb}g_{cd}x_{e}=\\ \\ \qquad=\sum(cd)p_{ca}g_{cd}x_{d}=\sum(cdf)p_{ca}g_{cd}\gamma_{fd}\xi_{f}=\sum(c)p_{ca}\xi_{c} \end{array}$

3. Put $\Xi_{a}^{I}\Xi_{b}^{II}=\vartheta_{ab}$. Then we have

$\vartheta'_{ab}=\Xi_{a}^{'I}\Xi_{b}^{'II}=\sum(cd)p_{ca}p_{db}\Xi_{c}^{I}\Xi_{d}^{II}=\sum(cd)p_{ca}p_{db}\vartheta_{cd}$

and similar formulae for the other three parts of (25).