356

EINSTEIN-BESSO ON

THE

MERCURY PERIHELION

With the

help

of Newton's second

law,

Fz

and

Fy

in

eq. (21) can

be

replaced

by z

and

y,

respectively.

For

z

and

y one can

then substitute

eq.

(15),

the

Euler-Lagrange equations

for

a

unit

point

mass

in

the metric

field

of

a

rotating

sun

(see

[p.

47],

[eq.

322]).

In

this

way,

the

integrand

in

eq.

(21)

can

be

expressed

in

(x, y,

z)

and

(x, y,

z).

These

coordinates and their time derivatives

are

then written

in terms

of

r, i,

0, 0,

0,

and

0

• •

(see

[p.

46],

[pp.

48-49]). Assuming

that

0 is

negligible compared

to 0,

one can

then

evaluate the

integral

in

eq. (21).

The end result

is

(see [p.

49],

[eq. 329])

Sok

8LX

=

sin

i

cos

0

n.

(22)

r

From

eqs.

(20)

and

(22)

it

follows that the

precession

80

of

the

nodes

per

revolution

is

given by

([p.

49],

[eq.

331])

86

=-J (23)

2

fr

From

eq. (23) one sees

that the rotation of the

sun

produces

a

retrogression

of the

nodes.

On

[pp.

41-42],

[p.

47],

and

[p.

49],

the

expression

for

50

in

eq.

(23)

is

evaluated

for

Mercury, Venus,

and Mars. The end results for

Mercury

and

Venus

given

on [p.

41]

and

[p.

49] are

-2.3"

and

-0.9"

per century, respectively.

Due

to

some

trivial

arithmetical

errors,

these results

are a

factor

in

the order of

103

too

large.[53]

When this

is

corrected

for,

the

effect of

the

sun's rotation

on

the nodes of

a

planetary

orbit

is

seen

to be

of the

same

order of

magnitude

as

its

effect

on

the orbit's

perihelion

calculated

earlier

in

the

manuscript.

Both effects

are

completely negligible.

On

[p.

41],

the results

for

Mercury

and

Venus

are

listed

along

with

the numbers

given

by

Newcomb[54]

for

the

discrepancy

between

(Newtonian)

theory

and observation for the motion of

nodes.

The

discrepancies

given by

Newcomb

are

5.0" and

10.2"

per century

for

Mercury

and

Venus, respectively.

So,

the effect of the sun's rotation

in

the "Entwurf"

theory only

seems

to

make the

discrepancies slightly

greater.

III

With the

exception

of

one

page

on

the

Nordström

theory

([p.

53]),

one

page

on

special

relativity

([p.

39]),

and

some

auxiliary

calculations

in

Newtonian

mechanics, all

calcu-

lations

in

the

manuscript

are

done

in

the

context

of the "Entwurf"

theory.

This

argues

for

dating

the

manuscript

to

a

period

between

May 1913,

when Einstein

and

Grossmann

finished

the "Entwurf"

paper,[55]

and November

1915,

when Einstein abandoned the

[53]See

[p.

49],

note

231.

[54]Newcomb 1895,

p.

109.

For

more

details,

see [p.

41],

note

188.

[55]A

reference

to

the

"Entwurf"

separatum

occurs

in

what

appears to

be the

earliest

part

of

the

manuscript,

on [p.

8],

thus

providing

clear evidence that

it

was

written after the

paper was

finished.