DOC.
14
EINSTEIN
AND
BESSO MANUSCRIPT
377
[p. 9]
(Besso)
[39]The
perihelion
motion
is
derived from
[eq.
58]
and
[eq.
59], expressing
conservation of
angular momentum
and
conservation of
energy,
respectively.
[Eq. 58]
is
[eq. 57]
on
[p. 8]
in
polar
coordinates. The notation
W is
introduced for ds/dt
(see
[eq. 105] on
[p. 15]
where W
is
explicitly
defined this
way).
Using
this
notation,
[eq. 55] on [p. 8]
is
rewritten
as [eq. 59].
[40]When
W is
eliminated from
[eq.
58]
with
the
help
of
[eq. 59]
and
terms
smaller than of
order A/r
are
neglected,
one
obtains
[eq. 61].
The notation F
is
introduced for the combination
of
constants
Bc20/E.
[41][Eq.
62]
is
obtained
by
inserting
the
expression
for
g44
in
[eq.
42] on
[p.
6]
and the
expression
for
W
=
ds/dt
in
[eq.
53] on [p. 8]
into the
equation
g244
=
E2
W2
(which
follows
directly
from
[eq.
59]).
Terms
smaller than of order
A2/r2
are
neglected.
[42]The constant
e
=
1
-
E2/c20
is
introduced. Since
E2/c20
is
of
the
form
1
+ O(A/r)
(see
[eq.
62]),
s
is
a quantity
of order A/r.
[43][Eq. 64]
is
obtained
by
taking
[eq. 62],
bringing
q2/c20 over
to
the left-hand
side,
putting
everything
else
on
the
right-hand side,
and
eliminating
the
constant
E
in
favor of
e
with
the
help
of
[eq. 63].
In the
term
(A2/r2)
(3/8
-
7/4 c20/E2
(1
+ £)),
the
term with
s can
be
neglected.
[44][Eq.
64]
is
written
in polar coordinates,
giving
[eq.
65]. Eliminating
dt
from
[eq. 65]
and
[eq.
66]
(which is
[eq.
61] squared), one
arrives
at
a
differential
equation relating
dQ
and
dr
(see
[eq.
67]).
To
find
the advance of the
perihelion, d(p
has
to be
integrated
between the values of
r
at perihelion
and
aphelion
(see
[eq.
68]).