§ 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

- ↑ Comp. § 7, l. c.
- ↑ 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.)

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

- ↑ 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).