DOC.
14 EINSTEIN AND BESSO MANUSCRIPT
457
[p.
46] (Besso)
[215][P.
46]
is
the
verso
of
[p.
45].
[216]With
the
help
of the
two figures at
the head of
[p.
46],
the
(x, y, z)-coordinates
of
a
planet orbiting
the
sun are
expressed
in
terms
of
(r,
0,
0,
i).
The coordinates
r
and 0
are
the
polar
coordinates of the
planet
in
the
plane
of
its
orbit;
the
angles i
and 0 fix the
orientation of
this
plane (see
the
figure on
the left
and
the
figure adapted
from
it in
the editorial
note,
"The
Einstein-Besso
Manuscript on
the Motion of
the
Perihelion of
Mercury," sec.
II.3).
The
figure on
the
right
shows the
projection
of
an arbitrary point
of
the
orbit
on
the
xy-plane.
From
this
figure,
the
expressions
for
x
and
y in
[eq.
315]
can
be
read
off.
Consider
the two line
segments
labeled
by
their
lengths
('r
cos
0"
and
"r
sin
0
cosi").
The
expression
for
x
in
[eq.
315] is
obtained
by
subtracting
the
lengths
of the
projections
of these
two line
segments onto
the x-axis;
the
expression
for
y is
obtained
by
adding
the
lengths
of their
projections onto
the
y-axis.
The
expression
for
z
can
be
read off from the
figure on
the
left.
The
figure on
the
left
can
also
be
used
to
read off the
expressions
in
[eq.
316]
for
the
components
of
a
unit
vector
perpendicular
to
the
plane
of the orbit.
[217]It is
assumed that the
angle
between the
plane
of
the
orbit
and
the
ecliptic
is fixed
(i
=
0),
that the
angular velocity
of the
planet
in its
circular orbit
is
a
constant
(0
=
0),
and that
the
precession
of
the
nodes takes
place at
a
uniform
velocity (0
=
0).
These
assumptions are
used
in
differentiating
[eq. 315] to
obtain
the
expressions
for
(x,
y, z)
and (x,
y, z)
in
[eqs.
317-318].
As
is
indicated
next to
[eq.
318], terms
with
02
are
neglected.
Over
one period
of revolution 0
changes
by
2jr,
whereas
0
changes
only by a tiny
fraction of
a
second of
arc.
Hence, 6
£
0.