DOC. 14
EINSTEIN AND
BESSO MANUSCRIPT
381
[p. 10]
(Einstein) (continued)
is given as
27r
1/1
~/~_
in
[eq.
73],
which
should
be multiplied
by
1/2.
The
integral
is
evaluated
using
the
same
two contours
that
were
used
to
derive
[eq.
72].
The
integral
from
r1
to
r2
of 1/r2Jr - r1)(r2 - r)
is
mistakenly
set
equal
to
the
integral
of
the
same
expression along
these
contours
rather than
to
1/2
times these
contour integrals.
[50]In
order
to
evaluate
[eq.
68]
on [p.
9]
for the
angle
between
perihelion
and
aphelion
with
the
help
of
the
results of
the
contour integrations
in
[eqs.
69-74],
one
needs
the
relation
between
the
roots
and the coefficients of
the
fourth-order
polynomial
under
the square root signs
in
[eq.
68]
and
[eq.
69].
In
[eq.
75],
the various
coefficients of
this
polynomial
are
labeled
a,
b, c,
d,
and
e.
The coefficient
a
is set
equal to
1
in
[eq. 78].
[51]when
r
becomes of the order of
magnitude
of
r'
and
r", r3 and
r4
become
negligible
compared
to r2
and
r.
Hence,
r' and
r"
will be the
roots
of
the
quadratic equation
[eq. 76].
It
follows that
[eq. 77]
holds for their
sum.
[52]The
relation between
roots
and coefficients of
any
fourth-order
polynomial
can
be
read off
from
[eq.
78]. Neglecting
all
terms containing
one
of
the small roots
r'
or
r",
one
obtains
[eq.
79]
and
[eq.
80]
for the
sum
and
the
product
of
the
large
roots
r1
and
r2,
respectively.