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If we take other coordinates the right hand side of this equation is transformed according to a formula which can be found easily. Hence we can also write down the transformation formula for the left hand side. It is as follows

div'_{h}\mathfrak{s}'=p\sum(m)p_{mh}div_{m}\mathfrak{s}-Q\sum(a)p_{ah}\frac{\partial p}{\partial x_{a}}+2p\sum(abc)p_{ah,c}\mathfrak{g}^{bc}G_{ab} (56)

§ 39. We shall now consider a second complex \mathfrak{s}_{0}, the components of which are defined by

\mathfrak{s}_{0h}^{e}=-G\sum(a)\mathfrak{g}^{ae}g_{ah}+2\sum(a)\mathfrak{g}^{ae}G_{ah} (57)

Taking also the divergency of this complex we find that the difference


has just the value which we can deduce from (56) for the corresponding difference


It is thus seen that


and that we have therefore

div\mathfrak{s}=div\mathfrak{s}_{0} (58)

for all systems of coordinates as soon as this is the case for one system.

Now a direct calculation starting from (52), (53) and (57) teaches us that the terms with the highest derivatives of the quantities g_{ab}, (viz. those of the third order) are the same in div_{h}\mathfrak{s} and div_{h}\mathfrak{s}_{0}. Further it is evident that in the system of coordinates introduced in § 37 these terms with the third derivatives are the only ones. This proves the general validity of equation (58). It is especially to be noticed that if \mathfrak{s} and \mathfrak{s}_{0} are determined by (52), (53) and (57) and if the function defined in § 32 is taken for G, the relation is an identity.

§ 40. We shall now derive the differential equations for the gravitation field, first for the case of an electromagnetic system.[1] For the part of the principal function belonging to it we write


where \mathrm{L} is defined by (35) (1915). From \mathrm{L} we can derive the stresses, the momenta, the energy-current and the energy of the

  1. This has also been done by de Donder, Zittingsverslag Akad. Amsterdam, 35 (1916), p. 153.