238 



K, i.e. within the sun. Tiiere the density ') is q =z ()^ -^ q^. In the 

 sjstem B, we have (j = q„ and this is zero except for r < r. 

 The line-element then has the form 



c?.s' = — adr" - b [rfifj* 4 si/?» if> rf^'] 4- fc^dt'' , 

 and in a stationary state rt, 6, ƒ are functions of r only. The equations 

 become somewhat simpler if we introduce 



/ -- l(/ a , 711 := Ig b . JÏ = Ut f . 



If differential coefficients with respect to r are indicated by accents 

 we find 



G,, = m" + i n" + 1 m' {m' — /') + i- n' (n' -/'), 

 '^G.,= — ~ + ^2 ^n" + i m' (2 7n' + /i' — /'), 



— ^ G^,, = i ;/' + I n' (2 7n' + n' - Z'), 



In order to write down the equations (1) we must luiow the values 

 of T,j.^. If all matter is at rest, and if there is no pressure or stress 

 in it, these are: T,, = i/,,q, all other T,,, = 0. These values I call 

 T^J. If we adopt these, then the equations (1) become, after a simple 

 reduction 



n" -f 11 (7?i' 4- 1 n' _ i /') — a [kq — 2 A), ... (10) 

 7?i" 4- 2 "*' ("*' — n' — I') z=z — anQ, . . . . . (11) 



-^ 4- im'(;2'4- i7n')= -a;.. . (12) 



It is easily verified that these are satisfied if we take Q=:Qa, 

 and for g,,,, we take the values corresponding to one of the forms 

 (4^), (4^), or (4(7) of the line-element, with the conditions (SJ), 

 (SB), or (36') respectively. Similarly for (6.4), (6^) and (SA), {SB), 

 if the accents in (10), (11), (12) denote diff'erential coefficients with 

 respect to r, or h respectively. Consequently in the field of pure 

 inertia we have T,,..= Ty.,\ i.e. by the action of inertia alone there 

 are produced no pressures or stresses in the world-matter. 



1) This, of course, is not strictly in accordance with Einstein's hypothesis, by which 

 the condensation of the world-matter in the sun sliould be compensated by a 

 rarefying, or entire absence, of it elsewhere. The mass of the sun however is 

 extremely small compared with the total mass in a unit of volume of such extent 

 as must be taken in order to treat the density of the world-matter as constant. 

 Therefore, if we neglect the compensation, the mass present in the unit of volume 

 containing the sun is only very little in excess of that present in the other units. 

 In the real physical world such small deviations from perfect homogeneity must 

 always be considered as possible, and they must produce only small differences in 

 the gravitational field. 



