1090 



We here have found a generalization for formula (41) which 

 also holds when ^,.4 =|= 0. It must still be remarked that ^,4 occurs 

 in the expression for ï^ — g (see the last formula (48)). 



The more general formulae for (43), (45) can easily be obtained 

 in the same way as above. 



^r = ~=— ........ (43a) 



V—g dr 



V —g dr 



In this § and in the preceding one we have confined our discussion 

 to bodies with spherical symmetry. If we have a body of finite 

 dimensions, which does not possess spherical symmetry, the corre- 

 sponding gravitation field is different from that belonging to a body 

 of the same mass but with spherical symmetry. We see however, 

 that the greater the distance from the body in question becomes, 

 the more the two fields must become equal. Therefore we can 

 de^ne the mass m of a finite material system of arbitrary form by 

 the formula 



~i=*ïj,„. r^^^l . (50) 



g drj cz r = ^\\/—g dr J 



In the last expression we have introduced — p"^ = gpp analogous 

 to the notation in formula (27). In order that formulae (50) may 

 have a definite meaning, the limit on the right-hand side must of 

 course have the same value for any direction in which we move 

 towards the infinite. Formula (50) supposes therefore the system of 

 coordinates to be chosen in such a way that at an infinite distance 

 the field possesses spherical symmetry. 



For the case we are considering formula (43a) gives 





dw^\ 

 9 ^"-y 



Urn r^(^r= lim I ), (51) 



and as formula (21) in § 2 is also valid for a stationary field, which 

 has no spherical symmetry, this equation gives together with (50) 



E = c'm, (52) 



as is demanded by the theory of relativity. Thus we have shown 

 that the calculation of the mass of a stationary system by means 

 of formula (50) from the field at points at a great distance and the 

 calculation of the mass by means of formula (52) from' the total 

 energy at rest give the same result also for bodies without spherical 



