Geology Reference
In-Depth Information
By definition, the equivolumetric radius
d
is the radius of a sphere with the same
volume, so,
a
3
(1
d
3
−
f
)
=
.
(5.60)
In dimensionless variables, the equivolumetric radius
d
is unity, hence,
1
a
=
f
)
1/3
,
(5.61)
(1
−
immediately giving the expansion
1
3
f
2
9
f
2
a
=
1
+
+
+···
(5.62)
for the equatorial radius. The ellipsoidal surface obeys the equation
R
2
sin
2
R
2
cos
2
θ
θ
+
=
1.
(5.63)
a
2
c
2
Solving for
R
2
,wefindthat
a
2
R
2
(θ)
=
f
)
−
2
1
−
cos
2
θ
+
cos
2
θ(1
−
a
2
=
cos
2
cos
2
(1
3
f
2
1
−
θ
+
+
2
f
+
+···
)
a
2
=
θ
,
(5.64)
1
+
2
f
cos
2
θ
+
3
f
2
cos
2
giving
=
a
1
2
f
2
cos
2
θ
3
−
f
cos
2
cos
4
θ
−
θ
−
+···
R
(θ)
(5.65)
a
1
3
f
cos
2
2
f
2
cos
2
θsin
2
=
−
θ
−
θ
+···
(5.66)
a
1
3
f
cos
2
8
f
2
sin
2
2θ
+···
=
−
θ
−
.
(5.67)
Using the expansion (5.62) for
a
then yields
9
f
2
2
3
f
1
θ
1
1
33
2
27
2
3cos
2
cos
2
cos
4
=
+
−
+
−
θ
+
+···
R
(θ)
1
θ
f
2
35
P
4
2
3
fP
2
4
45
−
23
63
P
2
12
=
1
−
+
−
+
+···
.
(5.68)
Comparison with the general expression (5.8), given (5.9), (5.18) and (5.42), shows
that for the value δ
=−
27/7 the true reference surface becomes an ellipsoid.
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