DOC.
14
EINSTEIN
AND
BESSO MANUSCRIPT
415
[p. 27]
(Besso)
[126][P.
27]
is
the
verso
of
[p. 26].
[127]The point
of Besso's calculation
on [p.
27]
seems
to
be to
check that the
expression
for
r1
+
r2
(the
sum
of
the
two large
roots
of
the
polynomial
in
the
expression
for the
angle
between
perihelion
and
aphelion
of
an
orbit
in
the
field
of
a
static
sun)
found
on
the basis of
[eq. 86]
given
by
Einstein
on [p. 11]
reduces
to the
expression
Besso found
on [p. 14] ([eq.
99]
and
[p.
15],
[eq.
112], see
note 69)
on
the basis of
[eq.
68],
which Besso himself derived
on [p. 9]
and which
is
essentially equivalent
to
Einstein's
[eq.
86] (see
note
58).
This
means
that
[p. 27]
would have
to
be
later than
[p.
14].
Related calculations
by
Besso
can
be
found
on [p.
25]
and
[p. 33]
(see note
113).
The calculation
on [p.
27] proceeds
as
follows.
Combining
[eqs.
75-78]
on [p.
10]
and
[eq.
85]
on [p.
11], one
can
write
r1
+
r2
as
a
combination of coefficients
of
the
polynomial
under the
square root sign
in
[eq. 86]:
r1
+
r2
=
-(coefficient of
r3)/(coefficient
of
r4)
+ (coefficient
of
r)/(coefficient
of
r2). Reading
off these
coefficients from
[eq.
86]
and
inserting
them
into
this
relation,
one
arrives at
an
expression
for
r1
+
r2 that
is
almost identical
to
[eq.
181].
The
only
difference
occurs
in
the denominator of
the second
term.
The
coefficient of
r2 in
[eq. 86]
is
A2(7/4c20
-
3/8
E2)
-
B2c20.
The factor in
parentheses
has
the
wrong
sign
(see
note
58). Correcting
this and
dividing
by
c20
(which
cancels
against
c20
in the
coefficient of
r),
one
arrives
at:
(A2E2/c20)
(3/8
-
7/4c20/E2)
-
B2.
Defining
e
=
3/8
-
7/4c02/E2
and
y
=
A2e
(see
the
line
at
the
top
of
[p.
27]),
one can
rewrite
this
as yE2/c20
-
B2.
So, yc20E2
in
[eq.
181] presumably
should
be
y
E2/c20.
[Eq.
181]
for
r1
+
r2
is
rewritten until
it
has the form
A
2e2
2
(1
-
...).
When
it is
assumed that the factor
in
parentheses
differs
only
very
little from
1,
the
expression
reduces
to
[eq. 99] on [p. 14],
r1
+
r2
=
A/e,
which
is what
Besso
probably
wanted
to
check. Note from
the definition of
E
in
[eq. 63] on [p. 9]
(not
to be confused
with
the
definition of
E
on [p.
27])
that
c02/E2
=
1
+
E.