1219 
sional space to be filled with matter, of which the total mass is so 
enormously great, that compared with it all matter known to us is 
utterly negligible. This hypothetical matter I will call the “world- 
matter’. - 
EINsTEIN only assumes ¢hree-dimensional space to be finite. It is 
in consequence of this assumption that in (2A) g,, remains 1, instead 
of becoming zero with the other g,,. This has suggested the idea ') 
to extend Hinsrern’s hypothesis to the four-dimensional time-space. 
We then find a set of g,, which at infinity degenerate to the values 
0-505 :0 0 
Or 00: 20 
ren 0 ED bee 
070 0 0 
Moreover we find the remarkable result, that now no “world- 
matter’ is required. 
__In order to point out the analogy of the two cases I give the two 
sets of formulae togetner. The formulae A refer to EiNsTrIN’s (three- 
dimensional) hypothesis, the formulae B refer to the assumption here 
introduced (four-dimensional). I shall use the indices 7 and j, when 
they take the values 1, 2, 3 only; w and v take the values from 
1 to 4. Further = is a sum from 1 to 4 and >' from 1 to 3; 
oat A Oy IE Rp: 
I first take the system of reference used by Erste. In case A 
we take 2,= ct, in B I take, for the sake of symmetry ®), «, = ict. 
In both cases A is the radius of the hypersphere. The y,, for the 
two cases are 
A | B 
Li tj Uy, 
EN eR ! NE RRA Mag OFS <5 ct eee 
1 1 Ji d 
Jij Y R?— S'z?; Rek 
ia. 
In order better to show the spherical character I introduce hyper- 
spherical coordinates by the transformations: 
1) The idea to make the four-dimensional world spherical in order to avoid the 
necessity of assigning boundary-conditions, was suggested several months ago by 
Prof. EHRENFEST, in a conversation with the writer. It was, however, at that time 
not further developed. 
2) We can also take x, = ct. Then the four-dimensional world is hyperbolical 
instead of spherical, but the results remain the same. 
78 
Proceedings Royal Acad. Amsterdam, Vol. XIX. 
