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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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