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after replacing
f
and
f
by expressions (5.169) and (5.170) in the first-order terms,
respectively, and by
e
and
e
in the second-order terms. To first order, equation
(5.181) reduces to
5
2
m
,
de
+
2
e
=
(5.182)
which, on squaring, yields the second-order relation
25
d
2
e
2
4
dee
+
4
e
2
4
m
2
+
=
,
(5.183)
and, on multiplication through by
e
, yields another second-order relation
5
2
me
.
dee
+
2
e
2
=
(5.184)
These permit the reduction of (5.181) to the condition
5
2
m
10
6
7
me
4
de
+
21
m
2
7
e
2
2
e
−
−
+
−
=
0.
(5.185)
Similarly, replacing
f
and
f
by
e
and
e
in second-order terms, (5.161) gives
d
2
e
2
2
dee
−
4κ
−
+
16κ
=
0,
(5.186)
which reduces to
5
4
me
25
d
κ
+
16
m
2
4κ
+
−
=
0,
(5.187)
after substitution from (5.183) and (5.184).
There are then two ways of solving for
e
. We can take the homogeneous first-
order form of (5.172) and integrate it to find the solution that is regular at the
geocentre and satisfies the first-order condition (5.182) at the surface. This gives a
first-order estimate of
e
,su
cient for calculation of the second-order non-
homogeneous terms in (5.172). The full equation can be integrated to find the solu-
tion that is regular at the geocentre and satisfies the full second-order condition
(5.185). Alternatively, we can use the integral equation for η that resulted from the
Radau transformation. First, ηis calculated assuming
F
(η)
=
1. Then, the first-order
equation (5.173) is integrated to give
e
(0) exp
r
0
0
r
0
dr
0
e
(
r
0
)
=
.
(5.188)
It is shown later that η(
r
0
) is linear in
r
0
for small
r
0
, so the integral in the expo-
nent is proper. Substituting
e
η for
de
, the surface condition (5.185) becomes the
quadratic equation
1
4
(14
35
8
m
5
e
2
6
m
2
−
+
7η
+
6
m
)
e
+
+
=
0.
(5.189)
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