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§ 28. Between the differentials of the original coordinates x_{a} and the new coordinates x'_{a} which we are going to introduce we have the relations

dx'_{a}=\sum(b)\pi_{ba}dx_{b} (30)

and formulae of the same form (comp. § 10) may be written down for the components of a vector expressed in x-measure. As the quantities \mathrm{q}_{a} constitute a vector and as


we have according to (28)[1]




Further we have for the infinitely small quantities \xi_{a}[2] defined by (19)


and in agreement with this for the components of a vector expressed in \xi-units


so that we find from (25)[3]


Interchanging here c and d, we obtain



\chi'_{ab}=\frac{1}{2}\sum(cd)\left(p_{ca}p_{db}-p_{da}p_{cb}\right)\chi_{cd} (31)

The quantity between brackets on the right hand side is a second order minor of the determinant p and as is well known this minor

  1. Comp. § 7, l. c.
  2. For the infinitesimal quantities x_a occurring in (19) we have namely (comp. (30))


    and taking into consideration (19) and (20), i e.

    \xi_{a}=\sum(b)g_{ab}x_{b},\ x_{a}=\sum(b)\gamma_{ba}\xi_{b}

    and formula (7) l. c, we may write (comp. note 2, p. 758, l. c.)


  3. Put \Xi_{a}^{I}\Xi_{b}^{II}=\vartheta_{ab}. Then we have


    and similar formulae for the other three parts of (25).