Geography Reference
In-Depth Information
Proof ((
16.48
):
l
2
).
d
α
2
=
−
2
A
1
A
2
cos
i
(
A
2
cos
2
i
−
A
1
)sin
α
cos
α
d
2
L
,
(
A
1
cos
2
α
+
A
2
cos
2
i
sin
2
α
)
2
(16.63)
2
l
2
:=
d
2
L
d
α
2
(
α
0
)
.
End of Proof.
Proof ((
16.52
)-(
16.54
)).
In a first step, (
16.44
) is specified by
Δx
(meta-equator) =
=
α
1
Δq
+
β
1
Δl
+
α
2
(
Δq
2
Δl
2
)+
β
2
2
ΔqΔl
+O
x
3
,
−
(16.64)
Δy
(meta-equator) =
=
β
1
Δq − α
1
Δl
+
β
2
(
Δq
2
− Δl
2
)
− α
2
2
ΔqΔl
+O
y
3
=0
.
Implementation of (
16.50
) constitutes the second step:
Δx
(meta-equator) =
=
α
1
q
1
Δb
+
α
1
q
2
Δb
2
+
β
1
Δl
+
α
2
(
q
1
Δb
2
− Δl
2
)+
β
2
2
q
1
ΔbΔl
+O
x
3
,
(16.65)
Δy
(meta-equator) =
=
β
1
q
1
Δb
+
β
1
q
2
Δb
2
− Δl
2
)
− α
2
2
q
1
ΔbΔl
+O
y
3
=0
.
In a third step, the boundary condition in the above form is represented
in the meta-longitude dependence by means of (
16.47
)-(
16.49
)
[where in a fourth step we identify (
16.45
)by(
16.46
)]:
− α
1
Δl
+
β
2
(
q
1
Δb
2
Δx
(meta-equator) =
=
α
1
(
q
1
b
1
Δα
+
q
1
b
2
Δα
2
+
q
2
b
1
Δα
2
)+
β
1
(
l
1
Δα
+
l
2
Δα
2
)+
+
α
2
(
q
1
b
1
Δα
2
l
1
Δα
2
)+
β
2
2
q
1
b
1
l
1
Δα
2
+O
x
3
=
=
s
1
Δα
+
s
2
Δα
2
,
−
(16.66)
Δy
(meta-equator) =
α
1
(
l
1
Δα
+
l
2
Δα
2
)+
β
1
(
q
1
b
1
Δα
+
q
1
b
2
Δα
2
+
q
2
b
1
Δα
2
)
−
−
α
2
2
q
1
b
1
l
1
Δα
2
+
β
2
(
q
1
b
1
Δα
2
l
1
Δα
2
)+O
y
3
=0
.
−
−
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