230 



where / = ^^ — 1, then C and y] are real and {2B) becomes: 



ds' =z — clr' - R' sin^ -- [dx^' -f sin' ip dxV] -f ms* -~- r' dt\ . (4i?) 

 If in A we also take /■ = R /, then (2 .4) becomes : 



ds= — dr' — R' sin' -- [dip' + si7i' ifj d{)'] 4- c- c/«\ . . . (4^) 



The two systems A and B now differ only in ff^^. For the sake 

 of comparison we add the system C, with 



1 = , (;, :=0 (36') 



in which the line-element is 



ds' =r - dr' — r' [t/if'" -f si7i' ip d»^] -\- c^ dt' . . . (4C) 



Both A and 7i become identical with C for R =z cc. 



If in ^ the origin of coordinates is displaced to a point -/,, ipj, ^,, 

 and in B to a time-space point to,, x,, 4'i- ^*^i, then the line-element 

 conserves the forms (2 A) and (2 B) respectively. These can then 

 again by the same transfoi'mations be altered to (4 A) and (4 B). 

 In A the variable t, which takes no part in the transformation, 

 remains of conrse the same. In B on the other hand the new 

 variable t after the transformation is generally not the same as before, 



I will put, for both systems A and B 



r 



■'' = « 



In the. system B this / is not the same as in {IB), but it is the 

 angle which was called ? above. I will continne to use r as an 

 independent variable, and not ■/. 



2. In the theory of general relativity there is no essential 

 difference between inertia and gravitation. It will, however, be 

 convenient to continne to make this difference. A field in which 

 the line-element can be brought in one of the forms (4 A), (4 B) or 

 (4 C) with the corresponding condition (3.4), (3 /i), or (3 C), will be 

 called a field of pure inertia, without gravitation. If the g^^ deviate 

 from these values we will say that there is gravitation. This is 

 produced by matter, which I call "ordinary" or "gravitating" 

 matter. Its density is (>,. In the systems B and C there is no other 

 matter than this ordinary matter. In the system A the whole of 

 space is filled with matter, which, in the simple case that the line- 

 element is represented by (2.4) or (4^) produces no "gravitation", 

 but only "inertia". This matter I have called "world-matter". Its 

 density is q^. When taken over sufficiently large units of volume 



