Geology Reference
In-Depth Information
In order to allow for an unknown departure from ellipsoidal shape, we rewrite
(5.67) as
=
a
1
3
sin
2
2θ
+···
−
f
cos
2
8
f
2
R
(θ)
θ
−
+
κ
.
(5.69)
Then, (5.65) takes the form
a
1
3
cos
2
θ
f
cos
2
2
f
2
cos
4
R
(θ)
=
−
θ
−
+
4κ
θ
−
+···
.
(5.70)
The expansion (5.62) for
a
will then have an extra second-order term, giving
1
3
f
2
9
+
+···
.
a
=
1
+
+
(5.71)
With this expansion of
a
,wethenfindthat
9
f
2
2
33
2
+
36
f
2
cos
2
3
f
1
θ
1
1
9
f
2
−
3cos
2
R
(θ)
=
1
+
−
+
+
θ
27
2
+
36
f
2
cos
4
+
θ
+···
f
2
f
2
−
1
6
f
2
23
24
f
2
P
2
2
3
fP
2
4
45
1
63
=
1
−
+
+
−
+
12
32
f
2
P
4
1
35
+
+
+···
.
(5.72)
Comparison with (5.42) shows that
27
7
−
72
7
f
2
.
δ
=−
(5.73)
Expression (5.42) then has the alternative form,
m
2
1
23
4
+
6
f
2
P
2
3
8
f
2
P
4
f
2
4
45
5
7
9
7
R
2
(θ)
=−
+
−
+
.
(5.74)
The expression for the true reference surface, (5.8), then becomes
2
3
fP
2
R
(θ)
=
1
−
45
f
2
1
23
4
+
6
f
2
P
2
−
3
8
f
2
P
4
(5.75)
4
5
7
9
7
−
+
+
+···
,
showing that the expansion (5.71) for the dimensionless equatorial radius takes the
form
1
3
f
2
8
15
κ
+···
,
9
f
2
a
=
1
+
+
+
(5.76)
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