DOC. 14 EINSTEIN AND BESSO MANUSCRIPT
389
[p.
14]
(Besso)
[68]With the
help
of
[eq. 77] on [p. 10], [eq. 98]
for
r'
+
r"
can
be
read off
from
the
polynomial
under the
square root sign
in
[eq.
68]
on [p.
9].
Only
the
leading
term in
the coefficient of r2 is
taken into
account.
[69][Eq.
99]
and
[eq.
100] express
two
more
relations
between roots
and coefficients of the
polynomial
under the
square
root sign in
[eq. 68] on [p. 9].
The
sum
r1
+
r2
of the
two large
roots
is
equal
to
the coefficient of
r3
minus
r'
+
r".
Reading
off the coefficient of
r3
from
[eq.
68]
and
using
[eq. 98]
for
r'
+
r",
one
finds
[eq.
99]
for
r1
+
r2.
The
product r1r2
is
equal
to
minus the coefficient of
r2
minus
(r1
+ r2)(r' + r").
Reading
off
the
coefficient of
r2
from
[eq.
68]
and
using
[eqs.
9899]
for
(r1
+ r2)(r' + r"),
one
finds
[eq. 100]
for
r1r2.
[70]With
the
help
of the
final
result
[eq.
84]
of the
contour
integrations on
[pp.
1011]
and
the relations between
roots
and coefficients in
[eqs.
98100],
the
integral
in
[eq. 68]
on
[p. 9]
for the
angle
between
perihelion
and aphelion
can
be
computed.
The
result
is
given
in
various
equivalent
forms
in
[eq.
101].
From
the
second
expression on
the second line of
[eq. 101],
it
follows that the
field
of
a
static
sun produces a perihelion
advance of
5/16
(Ac0/F)2 in
fractions
of
n
per
half
a
revolution. On
[p. 26]
and
[p. 28],
the
numerical
value
of
this
expression
is
computed
for
Mercury
and converted
to
seconds of
arc
per century.
The
expression on
the third
line
of
[eq.
101] clearly
shows that the
prediction
of the "Entwurf"
theory
is
5/12
times the
prediction
of the
general theory
of
relativity
in its
final
form
(see
the
editorial
note,
"The
EinsteinBesso
Manuscript
on
the Motion of
the
Perihelion of
Mercury,"
sec.
II.
1b, eqs.
(9)(10)).
[71][Eq.
102]
is
one
of
the
relations needed
to
rewrite
(Ac0/F)2
as
2A/a(1

e2)
in
[eq. 101].
F
was
defined in
[eq. 61] on
[p.
9] as Bc20/E.
From
c0/E
=
y/1
+
f
e ^
1
(see
[p. 9], [eq.
63]
where
E
is
introduced)
it
follows that F
^ Bc0.
Inserting
c0

ds/dt into
this
expression
and
referring to
[p. 8], [eq.
57],
one sees
that F

2f, f
being
the
area
velocity.
In
the
term
giving
the deviation
from
tx
of
the
angle
between
perihelion
and
aphelion,
the
area
velocity
can
be
considered
a
constant. Hence,
f
is
equal
to
the
area
of the
elliptic
orbit of
a
planet
divided
by
the
period
of revolution T.
With
the
help
of the
figure
next to
[eq.
102], an expression
for
the
area
of
an
ellipse
is
found
(the
semimajor
axis of
the
ellipse
is
called
a,
the
eccentricity e).
Using Kepler's
third
law
(in
the form
explicitly
given
in
[eq.
374]
on [p.
53]),
one
can
write the
constant
A (introduced
in
[eq. 6] on [p.
1])
as
87r2a3/c02T2.
With
the
help
of
this
expression
for
A
and
[eq. 102]
for
F,
(Ac0/F)2 can
be
rewritten
as
2A/a(l

e2).