# Page:LorentzGravitation1916.djvu/26

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

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

so that (31) becomes

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

According to (27) this becomes

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

for which we may write

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

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

${\displaystyle \psi _{dc}=-\psi _{cd}}$

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

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

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

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

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

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

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

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

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

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

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

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

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

1. Comp. (28) l. c.