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.
98-99]
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.
10-11]
and
the relations between
roots
and coefficients in
[eqs.
98-100],
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
Einstein-Besso
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
semi-major
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).
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