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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


so that (31) becomes


According to (27) this becomes


for which we may write


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


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


§ 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.