8 



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 



dp 



div'h^' ■=^p^{m)pmhdwm^ — Q2{a)pahx ^ 'lp^{ahc)pah,c%^''Gab' • (56) 



ÓXa 



§ 39. We shall now consider a second complex 6„, the com- 

 ponents of which are defined bj 



é^o;. ==:- GS {a)<s'''gah + 2^ {a)r'Gah . . . (")7) 

 Taking also the divergency of this complex we find that the 

 difference 



div'h^'o — p 2 {m)pmh divniia 

 has just the value which we can deduce from (56) for the corre- 

 sponding difference 



div'h é'— p2{m)pjnhdivjn^ 

 It is thus seen that 



div'h'S'' — div'ifi'o = pS{m)p,nh{divm'6 - divm^o) 

 and that we have therefore 



div é =. div ^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„h, (viz. those of the third order) are the same in div/i^ a,nd divh^o- 

 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 tlie general validity of equation (58). It is especially to be 

 noticed that if é and o^ 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. ^) 

 For the part of the principal function belonging to it we write 



ƒ 



LdS, 



where L is defined by (35) (1915). From 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. 



