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

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

§ 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

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

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

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))

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