1218 
make hypotheses on this subject. The extrapolation, which offers itself 
most naturally, and which is also tacitly made in classical mechanics, 
is that the values (1) remain unaltered for all space and time 
up to infinity. On the other hand the desire has arisen to have 
constants of integration, or rather boundary-values at infinity, which 
shall be the same in all systems of reference. The values (1) do not 
satisfy this condition. The most desirable and the simplest value 
for the g,, at infinity is evidently zero. EisrteiN has not succeeded 
in finding such a set of boundary values *) and therefore makes the 
hypothesis that the universe is not infinite, but spherical: then no 
boundary conditions are needed, and the difficulty disappears. From 
the point of view of the theory of relativity it appears at first sight 
to be incorrect to say: the world 7s spherical, for it can by a trans- 
formation analogous to a stereographic projection be represented in 
a euclidean space. This is a perfectly legitimate transformation, 
which leaves the different invariants ds?, G ete. unaltered. But even 
this invariability shows that also in the euclidean system of coordi- 
nates the world, in natural measure, remains finite and spherical. 
If this transformation is applied to the g,, which Einstein finds 
for his spherical world, they are transformed to a set of values 
which at infinity degenerate to Lj 
Q--0'0 0 
Ore 1000 
| (24) 
be Rele 
arabe RS U 
It appears, however, that the g,, of ErsreiN’'s spherical world 
[and therefore also their transformed values in the euclidean system 
of reference} do not satisfy the differential equations originally 
adopted by EINSTEIN, viz: 
GET “feet. ee 
Einstein thus finds it necessary to add another term to his equa- 
tions, which then become 
Cif > ih eS Ee dee NN 
Moreover it is found necessary to suppose the whole three-dimen- 
1) le. page 148. It will appear below that Enysrem’s hypothesis is equivalent 
to a determined set of values at infinity, viz: the set (2A). It is, in fact, evident 
that, if the universe measured in natural measure be finite, then, if euclidean coor- 
dinates are introduced the gz» must necessarily be zero at infinity, and inversily 
if the gw, at infinity are zero of a sufficiently high order, then the universe is 
finite in natural measure. = 
ed nd Mk zl 
