348

EINSTEIN-BESSO ON THE MERCURY PERIHELION

equinox along

the

ecliptic

to the

ascending node,

and from there

along

the

orbit

to the

perihelion.[24]

Using

the

distinctions drawn

above-static/rotating,[25]

first order/second

order,

and

perihelion/nodes-three

main strands of calculation

can

be

discerned

in the

manuscript.

These three strands

are

listed below

together

with the

numbers of

the

pages

on

which

they

occur.

Advance of

the

perihelion

in the field

of

a

static

sun

to

second order

([pp.

1-17],

[pp.

25-30],

[p.

33], [p.

40]).

Retrogression

of

the

perihelion

in the field

of

a rotating sun

to first order

([pp.

18-24],

[p. 29], [pp.

32-35]).

Retrogression

of

the

nodes

in the field

of

a

rotating

sun

to first

order

([p. 31],

[pp.

41-42],

[pp.

45-49]).

A brief

outline of

these

central three strands of calculation

follows,

cross-referenced

with

the

equation

numbers

in

square

brackets that

were

inserted

in

the

transcription.[26]

1a.

Field

of

a

static

sun

to

second order

([pp.

1-7],[p. 13],[pp.

16-17]). For

the

case

of

a

static

sun,

in

first-order

approximation

and in

Cartesian coordinates

(x,

y,

z, t),

the

44-component

is the

only

nontrivial

component

of

the

"Entwurf"

field

equations

=

/c(0MV

+ ?V)

(see

[p. 1], [eq.

2]):

(1) K

-Aa

y44=

--,

(1)

co

where

A

is

the

Laplacian,

k

is

the

gravitational

constant

from

the

"Entwurf"

theory

(related to the

Newtonian

gravitational

constant K

through

k

=

(8n/c^)K; see [p.

1],

[eq. 3], [p. 13], [eq.

95],

and

[p.

52]),

p0

is the

mass

density

of

the

sun,

and

c0

is

the

speed

of

light

in vacuo.[27]

The solution of

this

equation is

y4\

=

(A/c2r),

where

A

=

(k

M/4tc)

([p. 1], [eq.

6]),

M

being

the

mass

of

the

sun.

This

quantity

is

chosen

as

the

expansion parameter

for

the

power

series

expansion

of the

field.

Adding

y^

to

y^v,

the

standard

diagonal

Minkowski

metric, the field

y^v

of

the

sun

to

first

order

([p.

1],

[eq.

4])

is

found. This contravariant metric

field

can

easily

be

inverted

to find the

covariant metric

field

g^v

to first

order

([p. 1], [eq.

5]).

Collecting

all terms

in

the

"Entwurf"

field

equations containing

either

one

factor

yßV

[24]See, e.g.,

Clemence

1947,

p.

363;

Smart

1953,

pp.

21-22.

[25]This

distinction

is

made

in

terms

of

case

I

vs.

case

II at several points in the manuscript

(see

[p.

25],

[p.

26],

[p.

28],

and

[pp.

32-35]).

[26]Equation

numbers

in this

editorial

note

will be

referred

to

as "eq. (1),"

etc.,

equation

numbers inserted

in the

transcription

will be

referred

to

as

"[eq.

1],"

etc.

[27]On

[p.

8]

of

the

manuscript,

c0

is

defined

as

the

speed

of

light at

infinite distance from

the

sun

("die

Lichtgeschwindigkeit

für r-Abstand

der

Sonnne-unendlich"),

on [p.

16]

as

the

speed

of

light

to first

approximation

in

a

coordinate

system

in

which the

sun

is at rest

("[die]

Lichtgeschwindigkeit

im

Sonnensystem

in

erster Annäherung").