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+
15
1024
E
6
+O
3
(
E
8
)
sin 4
Δ
+
35
3072
E
6
+O
4
(
E
8
)
45
256
E
4
+
sin 6
Δ
+O
5
(
E
8
)
.
End of Corollary.
The hard work of the series expansion of the kernel representing the meridian arc length has
finally led us to Lemma
8.8
, where an elegant version of the meridian arc length up to the order
O(
E
12
) has been achieved.
Lemma 8.8 (Meridian arc length, forward computation).
f
(
Δ
)=
A
1
Δ
0
(1
− E
2
)(1
− E
2
cos
2
Δ
)
−
3
/
2
d
Δ
=
A
1
(1
− E
2
)
π/
2
E
2
sin
2
Φ
)
−
3
/
2
d
Φ
(1
−
Φ
⇒
f
(
Φ
)=
A
1
E
0
π
Φ
2
−
−
E
2
sin 2
Φ
−
E
4
sin 4
Φ
−
E
6
sin 6
Φ
−
E
8
sin 8
Φ
−
(8.39)
E
10
sin 10
Φ
+O(
E
12
)
,
subject to
−
1
4
E
2
64
E
4
3
256
E
6
5
16384
E
8
175
65536
E
10
,
441
E
0
=1
−
−
−
−
−
3
8
E
2
32
E
4
3
1024
E
6
45
4096
E
8
105
131072
E
10
,
2205
E
2
=
−
−
−
−
−
+
15
256
E
4
+
45
1024
E
6
+
525
16384
E
8
+
1575
65536
E
10
,
E
4
=
(8.40)
3072
E
6
35
12288
E
8
175
262144
E
10
,
3675
E
6
=
−
−
−
131072
E
8
+
2205
315
524288
E
10
,
E
8
=
+
1310720
E
10
.
693
E
10
=
−
End of Lemma.
8-22 Conformal Mapping
Let us postulate a
conformal mapping
of the ellipsoid-of-revolution onto a tangential plane at the
North Pole by means of the canonical measure of conformality, i.e.
Λ
1
=
Λ
2
. Such a conformal
mapping is illustrated by a vertical section of Fig.
8.4
.
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