EINSTEINBESSO
ON THE MERCURY PERIHELION
353
a
Taylor
series in
x
(see
[p.
18],
[eq.
124])
and
using
the
spherical symmetry
of
the
problem, one
finds
that
y4i
can
be
written
as (see
[p.
19],
[eq.
129])
yI (it)
=
4^
(oxx)
,
(14)
Cq
R
V
h
where
R
=
X, and
where
S
=

/
dr
p0r4
(see
[p.
19], [eq. 128],
and
[p. 34],
[eq.
205]),
which
up
to
a
constant is
just
the moment
of inertia of
the
sun
in
ordinary
Newtonian mechanics.
At this
point
a
change
of notation
occurs
in the
manuscript,
which
is
only
mentioned
explicitly
on [p.
20].
The coordinates of
the
points at
large
*
distance from
the
sun
at
which
the field is
evaluated
are no
longer
denoted
by
X but
by
x,
the coordinates
that
until this
point
referred
to
points
in the
interior of
the
sun.
A
JJ
(1
(1)
O
(0)
_
. Adding
y4i,
yi4
and,
y44
to
yMV,
one
finds the metric field
y^v
of
a
rotating
sun
to first
order, which
can
then
be
inverted
to
find
g^ to first
order
(see
[p.
19],
[eqs.
131132]).
2b.
Precession
of
the
perihelion
of
an
orbit
in the
field
of
a
rotating
sun
to
first
order
([pp.
2024],
[p. 29], [pp. 3235]).[43]
To find the
perihelion
motion
in the field
of
a
rotating sun,
one
needs
the
EulerLagrange
equations
and the
Hamiltonian for
a
unit
point
mass
moving
slowly
in this field.
For
a
metric
field such
as
that of
a
rotating
sun
to first
order,
the
EulerLagrange
equations
can
be
written
as (see [p. 20], [eq. 137]; [p. 37], [eq. 227]; [p. 47],
[eq.
319];
Einstein 1913c
(Doc. 17),
p.
1261,
eq. 1d)
• • •
x
=
curl
g x x

\
grad
g44,
(15)
where
g
=
(g14,
g24, g34) =
(Stc/r3) (3 x
x).[44]
Now, considering
the
special case
of
an
orbit in
the
plane
of rotation of
the
sun
and
choosing
this
plane
to be the
xyplane
(see
[p.
21]),
the first term
on
the
righthand
side
of
eq.
(15)
becomes[45]
a
S
o K
curl
g
x x
=

(y, x,
0).
(16)
r3
Since both
curl
g
x
x
and
grad
g44
lie in the
xyplane,
it
follows
from
eq. (15)
that
•
•
x
will
also lie
in
the
xyplane.
Hence, the
planet's
orbit
will
never
leave the
plane
of
rotation of
the
sun.
From
eqs. (15)
and
(16) one can
derive
an
expression
for
the
area velocity
f
(see
[p.
21], [eq. 143]):
2/
=
xy

yx
=
C +
(17)
r
[43]The argument
is
completely analogous to the argument
outlined
in
sec.
1b
above.
[44]In
the
manuscript,
a
Gothic
g is used
for
this
quantity.
[45]In
the
manuscript,
a
factor
k
is
tacitly
absorbed into
S at this
point
([p. 20], [eq. 138]).