# Page:LorentzGravitation1916.djvu/26

is related to a similar minor of the determinant of the coefficients $\pi_{ab}$. If $a'b'$ corresponds to $ab$ in the way mentioned in § 25, and $c'd'$ in the same way to $cd$, we have

$p_{ca}d_{db}-p_{da}p_{ch}=p\left(\pi_{c'a'}\pi_{d'b'}-\pi_{d'a'}\pi_{c'b'}\right)$

so that (31) becomes

$\chi'_{ab}=\frac{1}{2}p\sum(cd)\left(\pi_{c'a'}\pi_{d'b'}-\pi_{d'a'}\pi_{c'b'}\right)\chi_{cd}$

According to (27) this becomes

$\psi'_{a'b'}=\frac{1}{2}p\sum(cd)\left(\pi_{c'a'}\pi_{d'b'}-\pi_{d'a'}\pi_{c'b'}\right)\psi_{c'd'}$

for which we may write

$\psi'_{ab}=\frac{1}{2}p\sum(cd)\left(\pi_{ca}\pi_{db}-\pi_{da}\pi_{cb}\right)\psi_{cd}$

Interchanging $c$ and $d$ in the second of the two parts into which the sum on the right hand side can be decomposed, and taking into consideration that

$\psi_{dc}=-\psi_{cd}$

as is evident from (26) and (27), we find[1]

$\psi'_{ab}=p\sum(cd)\pi_{ca}\pi_{db}\psi_{cd}$

§ 29. Finally it can be proved that if equation (10) holds for one system of coordinates $x_{1},\dots x_{4}$, it will also be true for every other system $x'_{1},\dots x'_{4}$, so that

 $\int\left\{ \left[\mathrm{R}_{e}\cdot\mathrm{N}\right]+\left[\mathrm{R}_{h}\cdot\mathrm{N}\right]\right\} _{x'}d\sigma=i\int\{\mathrm{q}\}_{x'}d\Omega$ (32)

To show this we shall first assume that the extension $\Omega$, which is understood to be the same in the two cases, is the element $\left(dx_{1},\dots dx_{4}\right)$.

For the four equations taken together in (10) we may then write

 $\int u_{1}d\sigma=v_{1}d\Omega,\dots\int u_{4}d\sigma=v_{4}d\Omega$ (33)

and in the same way for the four equations (32)

 $\int u'_{1}d\sigma=v'_{1}d\Omega,\dots\int u'_{4}d\sigma=v'_{4}d\Omega$ (34)

We have now to deduce these last equations from (33). In doing so we must keep in mind that $u_{1},\dots u_{4}$ are the $x$-components and $u'_{1},\dots u'_{4}$ the $x$-components of one definite vector and that the same may be said of $v_{1},\dots v_{4}$ and $v'_{1},\dots v'_{4}$.

Hence, at a definite point (comp. (30))

 $v'_{a}=\sum(b)\pi_{ba}v_{b}$ (35)

We shall particularly denote by $\pi_{ba}$ the values of these quantities belonging to the angle $P$ from which the edges $dx_{1},\dots dx_{4}$ issue

1. Comp. (28) l. c.