9-2 The conformal mappings “ellipsoid-of-revolution to plane” 267
Proof (first postulate).
First postulate (Λ 0 = 1):
Λ 0 =
ar cos φ 0
N 0 cos Φ 0
⇒ r =
N 0 cos Φ 0
a cos φ 0
.
(9.56)
End of Proof.
Proof (second postulate).
Second postulate (Λ
0 = 0 ⇔ (ln Λ)
0 = Λ
0 /Λ 0 ):
ln Λ = ln ar + ln cos φ − ln[N (Φ) cos Φ] ,
d ln Λ
dφ
= −
sin φ
cos φ
−
N
(Φ) cos Φ − N (Φ) sin Φ
N (Φ) cos Φ
dΦ
dφ
,
N (Φ) =
A 1
(1 − E 2 sin
2 Φ) 1/2 , N
(Φ) =
A 1 E
2 sin Φ cos Φ
(1 − E 2 sin
2 Φ) 3/2 ,
dΦ
dφ
=
1 − E
2 sin
2 Φ
1 − E 2
cos Φ
a cos φ
=
N (Φ)
M (Φ)
cos Φ
a cos φ
⇒
d ln Λ
dφ
= −
sin φ
cos φ
+
sin Φ
a cos φ
= − tan φ +
sin Φ
a cos φ
⇒
(ln Λ)
(φ 0 , Φ 0 ) = 0 ⇒ a sin φ 0 = sin Φ 0 .
(9.57)
End of Proof.
Proof (third postulate).
Third postulate (Λ
0 = 0 ⇔ (ln Λ)
0 = 0 ):
d
2 ln Λ
dφ
2 = −
1
cos 2 φ
+
a cos φ cos Φ
dΦ
dφ + a sin φ sin Φ
a 2 cos 2 φ
=
= −
1
cos 2 φ
1 −
1
a
sin φ sin Φ −
1
a
cos φ cos Φ
dΦ
dφ
,
d
2 ln Λ
dφ
2 (φ 0 , Φ 0 ) = 0
⇒
a cos φ 0 =
1 − E 2 sin
2 Φ 0
1 − E 2
cos Φ 0 .
(9.58)
End of Proof.