DOC. 43

COSMOLOGICAL CONSIDERATIONS

431

universal

constant;

and Hamilton's

principle,

of

course,

guarantees

the

validity

of laws of

conservation. It

will be

shown

in

§

5

that

field

equation (13a)

is

compatible

with

our conjectures

on

field

and

matter.

§

5.

Calculation

and

Result

Since all

points

of

our

continuum

are

on an

equal footing,

it

is sufficient

to

carry through

the calculation

for

one

point,

e.g.

for

one

of

the

two

points

with the

co-ordinates

x1

=

x2

=

x3

=

x4

=

0.

Then

for

the

guv

(13a) we

have to

insert the

values

-1

0

0

0

0 -1

0 0

0

0 -1

0

0 0 0

1

wherever

they appear

differentiated

only once or

not at

all.

We thus obtain

in

the

first

place

Guv

= d/dx1[uv,1]

+

d/dx2[uv,2]

+

d/dx3[uv,3]

+

d2log

-g/dxudxv.

From this

we

readily

discover, taking

(7), (8),

and

(13)

into

account,

that

all

equations

(13a)

are

satisfied if

the

two

relations

-2/R+x=-kp/2, -x=

kp/2,

or

x

=

kp/2

=

1/R .

.

. .

(14)

are

fulfilled.

Thus the

newly

introduced universal

constant

X

defines

both the

mean

density

of

distribution

p

which

can

remain

in

equilibrium

and

also

the radius

R

and the

volume

2n2R3

of

spherical

space.

The total

mass

M of the

universe,

accord-

ing

to

our

view,

is

finite,

and

is

in fact

M

=

p

.

2n2R3

=

4n2R/k

= n232/k3p

.

.

(15)

Thus the theoretical

view of

the actual

universe,

if

it

is

in

correspondence

with

our

reasoning,

is

the

following.

The

[15]