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(ii)
E
2
)
1
/
4
(1
− E
2
cos
2
A
∗
)
3
/
4
×
A
1
(1
−
r
=
sin
B
∗
(1
E
2
cos
2
A
∗
)+
E
2
cos
2
B
∗
sin
2
A
∗
−
×
−
−
E
2
)(1
−
2
E
2
cos
2
A
∗
+
E
4
cos
2
A
∗
E
sin
A
∗
E
sin
B
∗
sin
A
∗
1
−
√
1
−
arcsin
+
(H.30)
−
2
E
2
c
os
2
A
∗
+
E
4
cos
2
A
∗
+
(1
−
E
2
)(1
−
E
2
cos
2
A
∗
)+
1
/
2
2
E
2
cos
2
A
∗
+
E
4
cos
2
A
∗
E
sin
A
∗
E
sin
A
∗
+
1
−
arcsin
√
1
,
−
2
E
2
cos
2
A
∗
+
E
4
cos
2
A
∗
(iii)
r
=2
A
1
sin
Δ
∗
2
=
=
A
1
√
2
√
1
cos
Δ
∗
=
A
1
√
2
√
1
−
−
sin
B
∗
(H.31)
(if
E
=0)
.
End of Corollary.
The proof for the integrals is presented in Sect.
H-3
.
H-13 The Equiareal Mapping In Terms Of Ellipsoidal Longitude,
Ellipsoidal Latitude
Λ
∗
,Φ
∗
}→{
A
∗
,B
∗
}
into the mapping equations (
H.29
)-(
H.31
)
which generate an equiareal mapping onto the transverse tangential plane according to (
H.21
)
and (
H.16
), respectively, in particular (
H.14
). Let us decompose (
H.30
) term-wise, namely
We implement the transformation
{
r
2
(
A
∗
,B
∗
)=
A
1
√
1
E
2
(1
− E
2
cos
2
A
∗
)
3
/
2
(
t
1
+
t
2
+
t
3
+
t
4
)
.
−
=
(H.32)
Λ
∗
,Φ
∗
}
has been performed in Sect.
H-4
. Here, the result is presented in form of Lemma
H.3
and Corol-
lary
H.4
.
The elaborate computation of the factor and the four terms
t
1
,t
2
,t
3
and
t
4
as functions of
{
Lemma H.3 (Equiareal mapping of the biaxial ellipsoid onto the transverse plane).
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