DOC. 14 EINSTEIN AND BESSO
MANUSCRIPT
435
[p.
37]
(Einstein)
[170]Page number
in
Besso's hand
(see
note 166).
[171][Eq.
222]
is
a
continuation of
the
calculations
in
[eqs.
219-221]
on [p. 36]. [Eq.
220]
for
1/R',
with
(X,
Y,
Z)
expressed
in
spherical
coordinates
with
the
help
of
[eq.
221],
is
substituted
into
[eq.
219].
The
integrand
in
[eq.
219]
then becomes
a sum
of four
terms,
corresponding
to
the four
terms
in
[eq.
220].
The
integral
over
the
first
term, corresponding to 1/R
in
[eq.
220],
is
easily seen to
vanish. The
integrals
over
the three
remaining
terms
are
given
in
[eq.
222]
(written
over
four lines). Only
the
term
corresponding to
y
Y
/
R3
in
[eq. 220] gives a nonvanishing
result
(see
[eq.
223]). Hence,
(1)g14
is
given
by
[eq.
224] (a
factor
o
is
omitted
in this
equation).
A
calculation
completely analogous
to
the
one
in
[eqs.
222-224] is
given
in
[eq.
225]
for
g24(1)
(the
second line of this
equation
should
be
inserted before the second
equality
sign). [Eq.
226]
explicitly states
what
was already
found
on [p.
36] (see note 169).
[172][Eq.
227]
is
identical
to
[eq.
137] on
[p.
20]
(see
the editorial
note,
"The Einstein-Besso
Manuscript
on
the Motion of
the
Perihelion of
Mercury,"
sec.
II.2b,
eq.
(15)) and
closely
resem-
bles
Einstein 1913c
(Doc.
17), p.
1261,
eq.
1d.
[Eq. 227] gives
the
equation
of motion of
a
point
mass
in
a
metric
field
of the
general
form
given
in
[eq.
215] on
[p.
36].
Close
to
the
center
of
a
rotating spherical
shell,
where the
components
of
g
are
given by [eqs.
224-226],
[eq.
228]
holds
for
curl
g,
and
[eq. 227]
reduces
to
[eq.
229].
With
the definition of
ox
in
[eq.
231],
[eq. 229]
can
be
rewritten
as [eq.
230]
(the
factor
2/3n
in
[eq. 231]
should
be
2/3).
[173]When instead of
one
spherical
shell
with
mass
M
and radius
R
a
series of nested shells
is considered, together forming
a
sphere
of uniform
density
p0, mass
M, and
radius
R,
the
quantity
M/R has
to
be replaced
by
/0R
dM/r with dM
=
p04rtr2dr.
The
integral is
evaluated
in
[eq.
233].
[Eq.
232]
is
obtained
by
substituting
the
right-hand
side of
[eq. 233]
for
M/R in
[eq.
231]. In
[eq.
234],
the numerical value for the
expression on
the
right-hand
side
of
[eq.
232]
is
computed
for
a
sphere
the size and
mass
of the Earth. The
following
numbers
are
used:
p0
=
5.6
g/cm3,
R
=
6.3
•
108
cm
(see note 120);
K
=
1/(3.6
•
103)2 m3/gs2 (see note 119).