Geology Reference
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for the surface value of
e
, given the surface value of η resulting from the integration
in (5.179). The surface value of
e
is given by the root
B
+
√
B
2
−
4
C
e
(
d
)
=−
,
(5.190)
2
with
1
4
(14
35
8
m
5
6
m
2
B
=−
+
7η
+
6
m
),
C
=
+
.
(5.191)
The smaller root is chosen, since the larger root is presumably associated with a
figure that is analogous to the highly oblate Maclaurin ellipsoid found as a second
solution for homogeneous bodies. A full integration of the exponent in (5.188)
provides a value for
e
(0) in terms of the surface value found from condition (5.185).
At all interior points,
e
is then determined from (5.188). An improved table of
values of
F
(η) is then constructed from formula (5.180). The process of calculating
η and
e
is then repeated until the desired accuracy is obtained. One repetition gives
second-order accuracy.
It is possible to perform a Radau transformation on equation (5.166) for κ,but
a rapidly converging iterative scheme does not result because κ is itself a second-
order quantity and the homogeneous equation does not provide a good first approx-
imation. Instead,
f
and its derivative are replaced by
e
and its derivative in the
second-order terms, to obtain
r
0
ρ
¯ρ
κ
+
6
1
r
0
κ
+
(6ρ/¯ρ
−
20)κ
−
ρ/¯ρ)
4
2
3
e
2
4
r
0
e
2
2
r
0
=
(1
+
6η
+
3η
−
(7
+
5η)η,
(5.192)
subject to (5.187). Thus, with values of η and
e
available, the solution for κ, regular
at the geocentre, is straightforward once starting values have been obtained.
Near the geocentre the density may be expanded as
r
0
ρ
(0)
ρ(
r
0
)
=
ρ(0)
+
+···
.
(5.193)
Thus, for small
r
0
we have
3
4
r
0
ρ
(0)
=
ρ(0)
+
+···
,
¯ρ(
r
0
)
(5.194)
4
ρ
(0)
ρ
(
r
0
)
¯ρ(
r
0
)
=
1
1
+
ρ(0)
r
0
+···
,
(5.195)
and
4
ρ
(0)
ρ
(
r
0
)
¯ρ(
r
0
)
=−
1
1
−
ρ(0)
r
0
+···
.
(5.196)
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