Geology Reference
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Then
P
4
can be expressed as
35
5
3
8
.
32
sin
4
2φ
+
8
sin
2
φ
+
P
4
(cosθ)
=−
(5.54)
The gravity intensity, as a function of geocentric latitude, is then found to be
9
2
−
15
2
−
sin
2
φ
m
3
f
m
−
3
f
m
g
=
1
−
117
14
m
2
9
f
2
m
2
+
f
2
m
2
δ
f
m
+
74
35
16
15
−
sin
2
f
2
m
2
+
f
2
m
2
δ
141
14
f
m
+
48
7
φ
+
−
sin
2
2φ
f
2
m
2
−
f
2
m
2
δ
135
8
f
m
−
9
4
21
8
+
−
+···
,
(5.55)
correct to second order. To the same order, gravity at the equator is
9
2
−
117
14
m
2
9
f
2
m
2
+
f
2
m
2
δ
m
3
f
m
f
m
+
74
35
16
15
g
e
=
1
−
−
+···
.
(5.56)
Expressed in terms of g
e
,wethenhavethat
g
=
g
e
1
5
2
m
−
f
+
sin
2
15
17
14
mf
−
3
1
4
m
2
7
f
2
9
f
2
φ
+
−
−
δ
15
8
mf
sin
2
2φ
+···
1
7
4
f
2
24
f
2
+
+
+
δ
.
(5.57)
Using relation (5.37), gravity can be given in terms of geographic latitude as
g
=
g
e
1
5
2
m
−
f
+
sin
2
15
17
14
mf
−
3
1
4
m
2
7
f
2
9
f
2
+
−
−
δ
φ
5
8
mf
sin
2
2φ
+···
5
7
4
f
2
24
f
2
−
−
−
δ
.
(5.58)
Because of its mathematical simplicity and its closeness to the true reference
surface, the oblate spheroid, or ellipsoid of revolution, is generally used in geodesy
as the form of the Earth's figure, even though it cannot be an exact equilibrium
form in a body with non-uniform density, such as the Earth (Volterra, 1903). For
geophysical purposes, it is therefore necessary to allow for the departure of the
reference surface from ellipsoidal shape. The volume of the ellipsoid is
4
4
3
π
a
2
c
3
π
a
3
(1
V=
=
−
f
).
(5.59)
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