DOC.
14
EINSTEIN
AND
BESSO MANUSCRIPT
405
[p. 22]
(Besso)
[104][Eq.
146] is
[eq.
143]
on [p.
21]
for the
area velocity
with
2f
in
polar
coordinates
(0
should
be 0);
[eq. 147] is
obtained
by turning
[eq.
145]
on [p.
21]
for the Hamiltonian
into
an
expression
for
q2
and
writing
q2
in
polar
coordinates.
Eliminating
dt
from
[eq. 146]
and
[eq.
147], one
arrives
at
[eq. 148],
relating
d0
and
dr
for
a planetary
orbit
in
the
plane
of rotation of
the
sun.
[105]Since
the
field
of
a
static
sun
to first
order does
not
give
rise
to
a
perihelion
motion
in
the
"Entwurf"
theory,
the
integration
of
d(p
in
[eq.
148]
between
perihelion
and
aphelion
for
o
=
0
should
give x.
From
[eq. 149]
and
[eq.
150]
it
follows that
this is
indeed the
case.
For
o
=
0,
the
right-hand
side of
[eq. 148]
reduces
to
[eq. 149],
which
can
be
rewritten
as
C
1
y/2c0(c0
-
E)
r(r
-r1)(r2
-
r).
The
quantities
r1
and
r2 are
the
roots
of
the
quadratic
expression
left
under
the
square root
sign
in
[eq.
149]
after minus the coefficient of
r2 is
factored
out. With the
help
of
[eq. 72] on
[p. 10]
(with
r
-
r2 changed
to
r2
-
r
and with
2x
changed
to
r),
one
finds
that the
integral
of d0
between
r1
and
r2
is
equal
to
(C/y/2co(c0
-
E))
•
(x/ r1r2). The
product
of the
roots
r1r2 is
equal to
minus the
constant term in
the
quadratic expression
mentioned above
(see
[eq. 150]).
It
follows that the
angle
between
perihelion
and
aphelion
for
o
=
0
is
indeed
equal to
x.
[106]In
[eq.
151],
the
sum
r1
+
r2
is set equal
to
the coefficient of
r
in
the
quadratic
expression
left under the
square
root
sign
in
[eq.
149]
after minus the coefficient of
r2 is
factored
out
(see
note
105).
The
quantity
r
was probably
intended
to be
defined
as
r1r2/(r1
+
r2).
[Eq. 152],
however, sets
r
equal to
an
expression
for
2c30
times this
quantity.
In
the bottom
right
corner
of
[p.
22],
the
dimension of
r
is
checked
(M
=
mass,
L
=
length,
T
=
Z
=
time).