Page:LorentzGravitation1916.djvu/26

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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.