Physics. — "-On the mass of a inatevial system accoi'cliiu/ to the 

 gravitation theory of Einstein." Bj Dr. G. Nordstrom. (Com- 

 municated by Prof. H. A. Lorentz). 



(Communicated in the meeting of December 29, 1917). 



§ 1. In this paper some formulae will be deduced for the mass 

 of a material system according to Einstein's gravitation theory. The 

 principal purpose of these formulae is to express the mass firstly 

 by a volume integral over the material system and secondly by a 

 surface integral over a surface surrounding the system. 



First 1 shall indicate in this paragraph the general formulae which 

 will be used further on. The following calculations are principally 

 based upon Einstein's paper: "HAMiLTONsches Prinzip und allgemeine 

 Relativitatstheorie" ') (further cited as: Einstein, HAMii-TONSches 

 Prinzip). His article: "Die Grundlage der allgemeinen Relativitats- 

 theorie" ") (further denoted by Einstein, Grundlage) will also be 

 referred to. 



In the first paper Einstein points out that the formulae in his 

 gravitation theory can be deduced from a variation principle of 

 this form : 



'ƒƒƒ> 



{<^S* -f x^'SÏ) dx^ dx^ dx^ dx^ — {), . . . . (1) 



where the first part (?>* of the integrand refers to the gravitation 

 field and the second part x^'^l to the matter (inclusively the electro- 

 magnetic field), x is the gravitation constant, which in Einstein's 

 paper has been put equal to 1. ©* is a function of g^^" and 



gar — ;. 



OXo^ 



'^X is a function of <;-"■' and of several parameters which determine 

 the state of the matter. 



The components i!^ of the stress-energy-tensor for the matter are 

 represented by the following expression (formula (19) Einstein, 

 HAMiLTONsches Prinzip) : 



1) A. Einstein, Bed. Ber. 1916, p. 1111. 



2) A. Einstein, Ann. d. Phys. 49, p. 769, 1916. 



