DOC. 14
EINSTEIN AND BESSO MANUSCRIPT
441
[p.
40]
(Einstein)
[180][p.
40] is
the
verso
of
[p.
39].
[181]On
[p.
40],
the time interval between
perihelion
and
aphelion
is
computed
the
same
way
the
angle
between
perihelion
and
aphelion
is
computed
elsewhere
in
the
manuscript
(see
the
editorial
note,
"The EinsteinBesso
Manuscript
on
the Motion of
the
Perihelion of
Mercury,"
sec.
II.1b). The calculation
is
done
in
the
context
of Newton's
theory.
It starts
from the
area
law
and Newtonian
energy
conservation, both
written
in
polar
coordinates
([eqs. 254255];
C is
the
area
law constant,
a
=
2KM, and
ß is
twice
the
total
energy
of
a
particle
of
unit
mass
in
the
field
of the
sun).
Eliminating
d(j)
(rather
than
dt)
from these
equations,
one
obtains
a
relation
([eq.
256])
between
dt
and
dr
(rather
than
a
relation between
d(p
and dr).
Integrating
this
relation
between
perihelion
and
aphelion
of
some planetary
orbit,
one
finds
an
expression
for half the
period
T
of
that orbit
([eq.
261]).
[182]To
find
the time interval
T/2
between
perihelion
and
aphelion,
the
righthand side
of
[eq.
256]
has
to be
integrated
from
a
to
b,
the values of
r
at perihelion
and
aphelion, respectively.
The
quadratic
form under
the
square
root
sign
of
[eq. 256] can
be written
as
(r

a)(r

b)
with
a
+ b
=
ot/ß
and
ab
=
C2/ß
(note
that
ß
is
negative
for
planetary orbits).
T/2
can
then
be
written
as
I/vß,
where I
is
the
integral on
the lefthand
side
of
[eq. 257].
[Eq.
257]
contains
some
errors.
First,
the
integral
I
is
half
the
integral over
the
contour
K in
the
figure
to
the
right
of
[eq.
257].
A factor
1/2
should
accordingly be
inserted after
all
three
equality
signs.
Second,
a
minus
sign
is
omitted under
the
square
root
sign
after
the first
equality
sign.
Another
minus
sign
has
to be
inserted after the third
equality
sign
(taking
the
positive
rather than the
negative
root
of

1
at this point
leads
to
a
negative
result for
T).
The
integral over
the
contour
K
can
be
replaced
by
the
integral over a
circle in the
complex plane
oriented clockwise and with
infinite radius. At
infinity,
the
integrand can
be
written
as
1/i(1
+
a+b/2z)
(see
the
last
step
in
[eq.
257]).
The
integral
of this last
expression over
a
circle oriented counterclockwise
is
equal
to
a+b/22n
(see
[eq.
258]).
Hence,
2I
=
(a
+
b)ix (see [eqs.
259260]; (a
+ b)n should
be
replaced
by
(a
+
b)n
in
both
equations). Inserting this
result
into
T/2
=
I/+Jß
while
substituting
ot/ß for
a
+
b,
one
arrives
at
T
=
na/(ß)3/2
(see [eqs.
261262];
ß should
be
ß
in both
equations).
[183]The
calculations
at
the foot of
[p.
40],
below the horizontal
line,
briefly
summarize the
central
part
of the numerical calculations
on
[p. 26]
and
[p. 28]
for
the
perihelion
advance
in
the
field
of
a
static
sun
according
to
the "Entwurf"
theory (see notes
124
and
128).
[Eq.
263]
gives
an
expression
for the advance
in
fractions of
n
per
half
a
revolution
(see
[eq. 177]
and
[eq.
180] on
[p.
26]).
There
is
a
factor
1/c0
omitted in the
expression
between
square
brackets.
When numbers
are
inserted, however,
this factor
is
taken
into
account.
For the
interpretation
of
[eqs.
264266],
see
notes
120122. Note that
1

e2
is set
equal to
1
in
evaluating
the
perihelion
advance.