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cos
α
(
Λ, Φ
;
c
3
,c
4
)=
cos[
Λ
∗
(
Λ
;
c
3
)
−
−
Ω
]
=
cos
2
[
Λ
∗
(
Λ
;
c
3
)
E
2
)
2
tan
2
[
Φ
∗
(
Φ
;
c
4
)]
,
(H.70)
−
Ω
]+(1
−
sin
α
(
Λ, Φ
;
c
3
,c
4
)=
E
2
)tan[
Φ
∗
(
Φ
;
c
4
)]
(1
−
cos
2
[
Λ
∗
(
Λ
;
c
3
)
=
E
2
)
2
tan
2
[
Φ
∗
(
Φ
;
c
4
)]
,
(H.71)
−
Ω
]+(1
−
r
2
(
Λ, Φ
;
c
3
,c
4
)=
=
A
1
cos
2
[
Λ
∗
(
Λ
;
c
3
)
E
2
)
2
sin
2
[
Φ
∗
(
Φ
;
c
4
)]
Ω
]cos
2
[
Φ
∗
(
Φ
;
c
4
)] + (1
−
−
/
cos
2
[
Λ
∗
(
Λ
;
c
3
)
E
2
)
2
sin
2
[
Φ
∗
(
Φ
;
c
4
)]
3
/
2
Ω
]cos
2
[
Φ
∗
(
Φ
;
c
4
)] + (1
×
×
[
t
1
(
Λ, Φ
;
c
3
,c
4
)+
t
2
(
Λ, Φ
;
c
3
,c
4
)+
t
3
(
Λ, Φ
;
c
3
,c
4
)+
t
4
(
Λ, Φ
;
c
3
,c
4
)]
,
−
−
(H.72)
Λ
∗
=
c
3
Λ,
(H.73)
sin
Φ
∗
=
(H.74)
=
c
4
sin
Φ
1+
2
20
c
4
+11
c
4
)+(O
E
6
)
,
c
4
)+
1
3
E
2
sin
2
Φ
(1
15
E
4
sin
4
Φ
(9
−
−
c
1
c
2
c
3
c
4
=1
.
(H.75)
End of Lemma.
Corollary H.6 (The equiareal mapping of the biaxial ellipsoid with respect to a transverse frame
of reference and a change of scale, special case
Ω
=3
π/
2 (the Hammer projection of
2
A
1
,A
2
E
)).
(i)
The mapping of the right biaxial ellipsoid
A
1
∗
,A
2
∗
subject to
A
1
∗
=
A
1
,A
2
∗
=
A
2
onto the transverse tangent plane specialized by
Ω
=3
π/
2being
normal to
E
3
and with respect to a change of scale is equiareal if
E
A
1
,Λ
2
with respect to left biaxial ellipsoid
E
x
=
c
1
r
(
Λ, Φ
;
c
3
,c
4
)cos
α
(
Λ, Φ
;
c
3
,c
4
)
,
(H.76)
y
=
c
2
r
(
Λ, Φ
;
c
3
,c
4
)sin
α
(
Λ, Φ
;
·
c
3
,c
4
)
,
subject to
sin[
Λ
∗
(
Λ
;
c
3
)]
cos
α
(
Λ, Φ
;
c
3
,c
4
)=
sin
2
[
Λ
∗
(
Λ
;
c
3
)] + (1
− E
2
)
2
tan
2
[
Φ
∗
(
Φ
;
c
4
)]
,
(H.77)
E
2
)tan[
Φ
∗
(
Φ
;
c
4
)]
(1
−
sin
α
(
Λ, Φ
;
c
3
,c
4
)=
sin
2
[
Λ
∗
(
Λ
;
c
3
)] + (1
,
(H.78)
E
2
)
2
tan
2
[
Φ
∗
(
Φ
;
c
4
)]
−
r
2
(
Λ, Φ
;
c
3
,c
4
)=
A
1
sin
2
[
Λ
∗
(
Λ
;
c
3
)] cos
2
[
Φ
∗
(
Φ
;
c
4
)] + (1
E
2
)
2
sin
2
[
Φ
∗
(
Φ
;
c
4
)]
−
/
sin
2
[
Λ
∗
(
Λ
;
c
3
)] cos
2
[
Φ
∗
(
Φ
;
c
4
)] + (1
E
2
)
2
sin
2
[
Φ
∗
(
Φ
;
c
4
)]
3
/
2
−
×
(H.79)
×
[
t
1
(
Λ, Φ
;
c
3
,c
4
)+
t
2
(
Λ, Φ
;
c
3
,c
4
)+
t
3
(
Λ, Φ
;
c
3
,c
4
)+
t
4
(
Λ, Φ
;
c
3
,c
4
)]
,
Λ
∗
=
c
3
Λ,
(H.80)
sin
Φ
∗
=
c
4
sin
Φ
1+
2
20
c
4
+11
c
4
)+(O
E
6
)
,
(H.81)
c
4
)+
1
3
E
2
sin
2
Φ
(1
15
E
4
sin
4
Φ
(9
−
−
c
1
c
2
c
3
c
4
=1
.
(H.82)
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