Astronomy. — “On the possibility of statistical equilibrium of the 
universe’. By Prof. W. pr Sitter. 
(Communicated at the meeting of Nov. 27, 1920). 
EinsreiN has on several occasions expressed the opinion that the 
existence of a finite amount of matter in the universe must neces- 
sarily lead to the adoption of a finite three-dimensional space. In 
his inaugural address at Leiden ') he says: 
“Wir können aber auf Grund der relativistischen Gravitations- 
“sleichungen behaupten, dass eine Abweichung vom euklidischen 
“Verhalten bei Raumen von kosmischer Grössenordnung dann vor- 
“handen sein muss, wenn eine noch so kleine positive mittlere 
“Dichte der Materie in der Welt existiert. In diesem Falle muss die 
“Welt notwendig räumlich geschlossen und von endlicher Grösse 
“sein, wobei ihre Grösse durch den Wert jener mittleren Dichte 
“bestimmt wird.” 
It appears to me that this statement cannot be accepted unreser- 
vedly. The gravitational field-equations are: 
Ge (CS =e. ae ae 
If we suppose all matter to be at rest and free from any strain 
or internal forces, then the tensor 7’,, has the value 
Dist 0, au other T=), Sb caer eee 
o being the density in natural measure. We can pute =o, +0, 
where the average value of o, is zero; y, is then the average den- 
sity. If we neglect o, the equations (1) are satisfied by the g,, 
. 
implied by the line-element: 
ds? = — dr? — R? sin? 5 dp? -+ sin? wp d6*|] + ce dt, . . (34) 
if we take 
2 1 
40, = Ta" ye = (EINSTEIN) (27... u) 
or by those of the line-element : 
RE gld” SH sin? wy dO] + cos? zede, . (3B) 
with 
1) Aether und Relativitätstheorie. Berlin, Julius Springer, 1920, p. 13. 
