EINSTEINBESSO ON THE
MERCURY
PERIHELION
351
where
C,
a, a, b, c,
and d
are
all constants.
The basic idea
now
is to
evaluate
the
angle
0 between
perihelion
and
aphelion
of
an
elliptical orbit,
by
integrating
dcp
in
eq.
(7)
between the values
r1
and
r2
that
r
takes
on
at
perihelion
and
aphelion, respectively.
The
deviation of
this
angle
from
it
gives
the
precession
of
the
perihelion.[34]
Since
r1
and
r2
are
extremal values of
r,
dr/dcj)
will vanish for
r
= r1
and
r
=
r2. Inspection
of
eq.
(7)
shows that
r1
and
r2
therefore
must be roots
of the fourthorder
polynomial
under the
square
root
sign
in
the
equation.
For
elliptical
orbits,
r never
takes
on
the
values of
the
two
other
roots,
which
in
the
manuscript
are
called r'
and r", and
which
are very
small
compared
to
r1
and
r2.[35]
Hence,
the
angle
between
perihelion
and
aphelion
is
found
by
integrating
dcj)
between
r1
and
r2,
the
large
roots
of the fourthorder
polynomial
under the
square
root
sign
in
eq. (7).
With the
help
of
some
contour
integrations,
one
establishes that
(see [p. 11],
[eq.
84])
f
do=C
{1+2(+)(a+r+r)} (8)
All
that
is
left
to do at
this
point
is to
express
the
roots
r1, r2,
r',
and
r"
in terms
of
the
coefficients
a, b, c,
and
d
of the
polynomial
under the
square
root
sign
of
eq. (7) (see
[pp.
1012],
[pp.
1415],
[pp.
2527]
and
[p.
33], especially
[p.
14],
[eqs.
98100] and
[p.
15],
[eqs.
111113]).
When
this is done,
one
finds,
after
some
rewriting (see
[p.
14],
[eq.
101]):
f
do=u(1+5A),
(9)
where
a
is
the
semimajor
axis and
e
the
eccentricity
of
the
elliptical
orbit.
From
eq.
(9)
it
follows that
according
to
the "Entwurf"
theory
the
field
of
a
static
sun
produces
an
advance of
the
perihelion
of
5 A
4 a(
1

e2)
(10)
per
revolution.
A
similar
expression
is
found
in
the
final
version of the
general theory
of
relativity.
The
only
difference
is
that instead of the factor

a
factor
3
occurs.[36]
Hence,
the "Entwurf"
theory predicts
an
effect
5
the size
of the effect
predicted
by
general
relativity
in its final
form. For
Mercury,
this amounts to
a
perihelion
advance
of about
18"
per
century.[37]
[34]This
method
is
also used
in
Einstein 1915d. For
a
modern discussion of
the
method,
see
Moller 1972,
pp.
492497.
[35]See Moller 1972,
pp.
492497.
[36]See
Einstein
1915d,
p.
838,
eq.
13.
The
constant
a
in
the
equation
from the
1915
paper
is
the
same
as
the
constant
A
here.
[37]In Einstein's "Scratch Notebook"
(Vol.
3, Appendix
A, [p.
61]),
an expression
almost
equivalent to
the
one
given
in
eq. (10)
is
given, namely,
\0n3(a/cT')2. When
T'2 is
replaced
by
T2(1

e2),
where
T is
the
period
of revolution of
a planet,
this
again
corresponds
to
of
the
effect
predicted
by
general relativity
in its final
form
(see
Einstein
1915d,
p.
839,
eq. 14).