Geography Reference
In-Depth Information
cos
B
B
(42)
A
sin
dB
2
2
2
0
(1
e
sin
B
)
Differentiating the both sides of (42) yields
2
2
2
dB
A
(1
e
sin
B
) cos
(43)
d
cos
B
For the same reason, we introduce the folllowing new variable
(44)
t
sin
and then denote
2
2
2
dB
A
(1
e
sin
B
) cos
(45)
ft
()
d
cos
B
(45) can be expanded into a power series of sin. Using the chain rule of implicit function
differentiation, one similarly arrives at
dB
2
4
6
8
10
(46)
ft
( )
A C
sin
C
sin
C
sin
C
sin
C
sin
2
4
6
8
10
d
where
4
41
4108
58427
28547
2
4
6
8
10
C
e
e
e
e
e
2
3
15
945
9450
3465
92
6574
223469
2768558
4
6
8
10
C
e
e
e
e
4
45
945
14175
93555
3044
28901
21018157
6
8
10
(47)
C
e
e
e
6
945
1890
467775
24236
2086784
8
10
C
e
e
8
4725
66825
768272
93555
10
C
e
10
To get the inverse expansion of the authalic latitude, one integrates (46) and arrives at
B
c
sin 2
c
sin 4
c
sin 6
c
sin 8
c
sin10
(48)
2
4
6
8
10
