422

DOC.

43 COSMOLOGICAL CONSIDERATIONS

limit

at

infinity,

the

mean

density

p

must decrease toward

zero more rapidly

than

1/r2 as

the distance

r

from

the

centre

increases.*

In this

sense,

therefore,

the

universe

according

to Newton

is

finite,

although

it

may

possess

an

infinitely

great

total

mass.

From this

it

follows in

the

first

place

that the radiation

emitted

by

the

heavenly

bodies

will,

in

part,

leave

the

Newtonian

system

of the

universe, passing radially

outwards,

to become ineffective and lost

in

the infinite.

May

not

entire

heavenly

bodies

fare

likewise?

It

is

hardly

possible

to

give

a

negative

answer

to

this

question.

For it

follows

from

the

assumption

of

a

finite limit for

Q

at

spatial infinity

that

a

heavenly body

with

finite

kinetic

energy

is able

to

reach

spatial infinity

by overcoming

the Newtonian

forces of

attraction.

By

statistical mechanics this

case

must

occur

from

time to

time,

as

long

as

the total

energy

of the stellar

system-transferred

to

one

single

star-is

great enough

to

send

that star

on

its

journey

to

infinity,

whence

it

never can

return.

We

might try

to avoid

this

peculiar

difficulty

by

assuming

a

very

high

value for

the

limiting potential

at

infinity.

That

would be

a possible way,

if the

value of

the

gravitational

potential

were

not itself

necessarily

conditioned

by

the

heavenly

bodies.

The truth

is

that

we are compelled

to

regard

the

occurrence

of

any

great

differences

of

potential

of

the

gravitational field

as

contradicting

the

facts.

These

differences

must

really

be of

so

low

an

order of

magnitude

that the stellar

velocities

generated by

them

do

not

exceed

the

velocities

actually

observed.

If

we

apply

Boltzmann's law

of

distribution

for

gas

molecules to

the

stars, by

comparing

the stellar

system

with

a

gas

in thermal

equilibrium,

we

find

that the Newtonian

stellar

system

cannot

exist

at all.

For there is

a

finite ratio

of densities

corresponding

to

the

finite difference of

potential

between the

centre

and

spatial infinity.

A

vanishing

of

the

density

at

infinity

thus

implies

a

vanishing

of

the

density

at

the

centre.

*

p

is the

mean density

of

matter, calculated for

a region

which is

large as

compared

with the distance between

neighbouring

fixed

stars,

but small in

comparison

with

the

dimensions

of

the

whole stellar

system.