DOC. 14 EINSTEIN AND
BESSO MANUSCRIPT
383
[p. 11]
(Einstein)
[53][Pp.
10-11]
are on one
side of
a
sheet
double the
size
of
the
other sheets of
the
manuscript
(see
note
45
for further
details).
The
page
number
in
the
top right
corner
of
[p.
11]
is in
Besso's
hand.
[54][Eq. 81]
is
obtained
by
inserting
[eq.
79]
and
[eq.
80]
into
[eq.
69].
[55]The
general
form of the
numerator in
the
integrand
of
[eq.
68]
on [p.
9]
is
now
taken
into
account
(see
note
47).
This
means
that
a
term
a/r
has
to be
added
to
the
numerators in
the
integrands
of
[eqs.
69-71]
on [p.
10],
which
is
done
in
[eq. 82];
the
term
ß/r2, which
is
initially
added
as
well, is neglected in
the calculation that follows
(see, however,
note
62).
As
a
consequence,
the
constant multiplying
the
second
part
of the
integral
in
[eq.
71]-with the
integrand
1/r2y/(r
-
r1)(r2
-
r)
(see
note
47)-changes
from
1/2(r'
+ r")
to
a
+
1/2(r'
+ r").
The notation
a'
is
introduced for
this
new
constant
(see
[eq. 83]).
In
this
way,
the
integral
I
in
[eq.
82]
turns into
the
first line
of
[eq. 84].
[56]With
the
help
of
[eqs.
72-73]
on
[p.
10], one
arrives
at
[eq. 84]
for the
integral
I. The
erroneous
factor 2
(see notes
48 and
49)
is
deleted
(see
note
60).
[57]These five
lines
are
in
Besso's
hand.
They
give
the relation between
roots
and coefficients
of
a
fourth-order
polynomial. Only terms containing
the
product
of the
small roots
r'
and
r"
are neglected,
whereas
in
[eqs.
79-80]
on [p.
10],
terms with
just
one
of the small
roots
were
neglected
as
well.
[58]The
left-hand side of
[eq. 86]
is
essentially equivalent to
the
right-hand
side
of
[eq.
68]
on
[p. 9]
for the
angle
between
perihelion
and
aphelion.
Einstein
apparently
derived the
equation
Besso derived
on [pp.
8-9]
for
himself,
probably to
check the results of
the
contour integrations
on [p.
10].
It
seems very likely
that Einstein started from Besso's
equations
[eq.
60]
and
[eq.
64]
on
[p.
9].
However,
the
constants
F
(defined in
[eq.
61]
on [p.
9])
and
e
(defined in
[eq.
63]
on
[p.
9]) are
not
introduced here.
Moreover,
the
polynomial
under
the
square
root
sign
in
[eq.
86]
has
a
constant
term,
whereas the
polynomial
in
[eq.
68]
does
not (one
gets
this term
by
keeping
the
A2/r2-term
in
[eq. 60]
which
was
dropped
in
[eq. 61]).
Since this coefficient
is equal
to
the
product
of
all
four
roots
of the
polynomial
and thus
proportional
to
r'r",
it is
negligible.
The
slight
differences between other coefficients of the
two polynomials
are
negligible
as
well.
Finally,
there
is
a sign error
in
[eq. 86].
The factor
7/4c20 -
3/8E2
in
the coefficient of
r2 in
the
polynomial
should have the
opposite
sign.
[59]The
integral
in
[eq.
86]
is
evaluated with the
help
of
[eq.
84]
in its
uncorrected form
(see
note 56).
Hence,
2ix
should
be
n
in
[eq.
86].
With
the
help
of
[eq.
77] on
[p.
10],
r'
+
r"
in
[eq.
86]
is
replaced
by
-d/c.
Reading
off
d
and
c
(the
coefficients of
r
and
r2)
from the
polynomial
on
the left-hand side of
[eq. 86],
and
neglecting
all but
the
leading
terms,
d/c
is set
equal
to
-2A.
So,
the factor between
square
brackets
on
the
right-hand
side
of
[eq. 86]
vanishes.
Using
[eq.
80]
on [p.
10]
and the
same
crude
approximation
for
c as
above,
one
finds
[eq.
87].
When
[eq.
87]
is
inserted for
^Jr1r2
in
[eq.
86],
the
integral-in
this crude
approximation-is
seen
to be
equal to
2tt. Since the
integral
gives
the
angle
between
perihelion
and
aphelion,
this
obviously
should
be
tz
.
A
more
careful evaluation of
the
angle
between
perihelion
and
aphelion, showing
a
deviation from the Newtonian value
n,
is
given on [p. 14] (see
note
70).
[60]From
the result found
in
[eq.
86] (see note 59),
it is
clear that there
is
a
factor
1/2
omitted
in
front of the
integral.
The mistake
can
be
traced
to
[eqs.
72-73]
on [p. 10]
(see
note
48).
[Eq.
89]
on [p. 12]
suggests
that Einstein had
no
trouble
locating
the
origin
of
this
mistake.