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
§ 39. We shall now consider a second complex , the components of which are defined by
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
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 , (viz. those of the third order) are the same in and . 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 and are determined by (52), (53) and (57) and if the function defined in § 32 is taken for , 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. For the part of the principal function belonging to it we write
- This has also been done by de Donder, Zittingsverslag Akad. Amsterdam, 35 (1916), p. 153.