DOC. 14
EINSTEIN
AND
BESSO
MANUSCRIPT
385
[p. 12]
(Einstein)
[61][P. 12]
and
[p. 13] are on
two sides
of
one
sheet.
[62]When
r' and
r"
are
neglected,
the form of the
integral
I
in
[eq.
88] corresponds to
the
ß/r2
term
in
the
integral
I
in
[eq.
82]
on [p.
11].
A reconstruction
of the derivation of
[eq.
86] on [p. 11]
(see note
58)
gives
ß
=
3/8A2
(as
can
be
read off
directly
from
[eq.
60] on [p.
9]; see
also
[p. 15], [eq. 110]
for
n
=
3/8).
This factor
shows
up
in
[eq.
90].
It
would
seem,
therefore,
that
the
point
of the calculation
on
[p.
12]
is
to
find
higher-order
contributions
to
the
angle
between
perihelion
and
aphelion,
now
that the lowest-order contribution does
not
seem
to
give
any
deviation from
n
(see
note 59).
However,
as
becomes clear
on [p. 14]
(see
note 70),
a
more
careful evaluation of the lowest-order contribution does
give a
deviation from
n.
Any
effect
coming
from the
ß/r2
term will be
a
factor in
the order of A/r smaller than this deviation
and hence
be
completely negligible.
Perhaps
this is
why
the calculation
on [p.
12]
was
deleted
in its entirety.
[63]A contour integration
is
performed
to
evaluate the
integral
I
in
[eq. 88],
using
the
same
two contours
that
were
used
to
evaluate the
integrals
in
[eq. 72]
and
[eq.
73] on
[p. 10]
(see
note
48).
Once
again,
the
only
contribution
to
the
integral
over
the second
contour
comes
from the
small circle around the
pole
at
r
=
0. Hence,
2I
is
written
as
I0. I0
is
equal
to
2ni times the
residue
in
r
=
0.
The residue
is
found
by
computing
the coefficient of
r2 in
the
Taylor
expansion
of
1//-(r
-
r1)(r
-
r2),
which becomes the coefficient of
1/r in
the Laurent series for the
integrand.
The
resulting expression
for
I
is
given
in
[eq. 89].
[64]Inserting
ß
=
3/8A2
(see
note
62)
and
multiplying
by
c0B/Jc20
-
E2
(cf. [eq. 86] on [p.
11]),
one
arrives
at
[eq. 90]
for the contribution of
the
ß/r2
term in
[eq.
82] on
[p.
11]
to
the
angle
between
perihelion
and
aphelion.
The factor 1/y/r1r2
is
absorbed
into
the
expression
between
square
brackets.
In
the second line of
[eq.
90],
the
numerator
should have
c0
instead of
c20.