DOC. 24 PERIHELION
MOTION
OF
MERCURY 115
where
A
and
B
signify
the
constants
of
the
energy law,
and
where
O
=
a/2r
(8a)
u2 =
dr2+r2do2/ds2
is
granted.
We have
now
to
evaluate the
equations
to
the
next
order.
The last of the
equations
(7) yields,
then, together
with
equa
tion
(6b),
d2xx
oVr,41
dxadxA
dgt+dxt

^
2
T4
ds ds
9
ds2 ds ds
or,
correct to
the first
order,
dx4
.
a
ds
r
(9)
We
now
turn to
the
first
of
the
three
equations
(7).
The
right
side
yields:
a)
for the index combination
a = r =
4
n44
dxA
ds
or considering equations
(6c)
and
(9),
correct to
the
second order,
ccx,f
_

2r
\
b)
for the
index
combination
a
#
4, t #
4 (which
alone
still
needs
to
be considered),
upon
considering
the
product
(dxa/ds)(dxr/ds),
using equation (8)
to first
order, correct to
the second
order,
[14]
OCX
3 xaxx\
dxa dxx
2 r2
J
ds ds
The summation
gives
__
3(dr'~2
 IaI
2\ds)I
Using
this
value
we
obtain
for
the
equation
of motion the
form, correct to
the second
order,
d2x,
ctx,f _
II
(7b)
I 2r3\
r
l
\asJJ
•
I.,
which
together
with
equation
(9)
determines the
motion
of
the
mass point.
Moreover,
it
should be observed
that
equations
(7b)
and
(9)
for the
case
of circular motion
give no
deviation
from
Kepler's
three
laws.
From
equation (7b) follows,
above
all,
the
exact validity
of the
equation
r2do/ds
=
B, (10)
where
B is
a
constant. The
law
of
areas
is
therefore
valid
to
second order if
we use
the
"proper
time" of the
planet
to
measure
time. In order to determine
the secular
rotation of
the orbital
ellipse
from
equation
(7b),
we
substitute
the
terms
of the first order in the
parentheses
most
advantageously by
means
of
equation
(10)
and the first
of
the
equations
(8),
through
which
procedure
the
terms
of second order
on
the
right
side
are
not altered. The
parentheses
take
on
the form
2A
Finally,
if
we
choose
s^(1

2A) as
the time
variable,
and
if
we redesignate
it
as s, we
have,
with
a
somewhat different
meaning
of
the constant B;
d2xv
dO
i[
B2
(7c)
ds2
QXy
1
+
[15]
In order
to
determine the
equation
of
the orbit,
we now
pro
ceed
exactly
as
in the Newtonian
case.
From
equation
(7c) we
obtain first
rfp
+
r2#2
ds2
=
2A
2D.
If
we
eliminate ds from this
equation
with the
help
of
equation
(10), we
obtain
(dx\2
2A a
Vw
? +
BiX~
x2 +
ax3, (11)
where
we
denote
by x
the
quantity
1/r.
This
equation differs
from the
corresponding one
in Newtonian
theory only
in
the last term
on
the
right
side.
The
angle
described
by
the radius
vector
between the
perihelion
and
the
aphelion
is
consequently
given
by
the
elliptic integral
dx
As
I,
2A
U 2
+ccx3
B2+
BZXX