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.