Geography Reference
In-Depth Information
d
1
=
d
B
d
Q
(
Q
0
)
,d
2
=
1
d
r
B
d
Q
r
(
Q
0
)
.
The standard coecients are given by
d
1
=cos
B
0
,
d
2
B
d
Q
2
(
Q
0
)
, ··· ,d
r
=
1
2!
r
!
1
2
cos
2
B
0
tan
B
0
,
d
2
=
−
1
6
cos
3
B
0
[1
tan
2
B
0
]
,
d
3
=
−
−
(D.48)
1
24
cos
4
B
0
tan
B
0
[5
tan
2
B
0
]
,
d
4
=
−
1
120
cos
5
B
0
[5
18 tan
2
B
0
+tan
4
B
0
]
.
d
5
=
−
Second choice.
The first or
remove step
is materialized by a
Taylor series expansion
of
Q
(
B
) around
Q
0
(
B
0
)
and leads
directly
to
Q
= artanh(sin
B
) = ln tan
π
,
4
+
B
(D.49)
2
Q
=
Q
0
+
ΔQ
=
Q
0
+
q
∀
q
:=
ΔQ ,
B
=
B
0
+
ΔB
=
B
0
+
b
∀
b
:=
ΔB
⇒
q
=
c
1
b
+
c
2
b
2
+
∞
c
r
b
r
(D.50)
r
=3
∀
d
r
Q
d
B
r
(
B
0
)
.
Consequently, the second or
restore step
is based upon
standard series inversion
(
D.51
)
according to (
D.52
)and(
D.53
).
d
2
Q
d
B
2
(
B
0
)
,
c
1
=
d
Q
d
B
(
B
0
)
,c
2
=
1
,c
r
=
1
···
2!
r
!
b
=
d
1
q
+
d
2
q
2
+
∞
d
r
q
r
,
(D.51)
r
=3
cos
B
0
,c
2
=
1
1
tan
B
0
c
1
=
cos
B
0
,
2
c
3
=
1
6
1
1
24
tan
B
0
cos
B
0
[1 + 2 tan
2
B
0
]
,c
4
=
cos
B
0
[5 tan
2
B
0
]
,
(D.52)
1
120
1
cos
B
0
[5 + 28 tan
2
B
0
+24tan
4
B
0
]
,
c
5
=
N→∞
N→∞
c
r
b
r
d
r
q
r
,
q
=
⇔
b
=
r
=1
r
=1
d
1
=
1
c
2
c
1
,d
3
=
2
c
2
c
3
5
c
2
c
1
+
5
c
2
c
3
c
4
c
1
,d
2
=
−
c
1
−
c
1
,d
4
=
−
c
1
−
c
1
.
(D.53)
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