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One can image that these procedures are quite complicated. Mathematica output shows that
the expression of the sixth order derivative is up to 40 pages long! Therefore, it is
unimaginable to derive the so long expression by hand. These procedures, however, will
become much easier and be significantly simplified by means of Mathematica. As a result,
the more simple and accurate expansions yield.
3.1.1. The inverse expansion of the rectifying Latitude
Differentiation to the both sides of (1) yields
2
dX
a
(1
e
)
(22)
dB
2
2
3/ 2
(1
e
sin
B
)
From (4) and (2), one knows
2
Xa

(1
eK
)
(23)
0
Inserting (23) into (22) yields
dB
2
2
3/ 2
Kd 
(1
e
sin
B
)
(24)
0
To expand (24) into a power series of sin, we introduce the following new variable
t
sin
(25)
therefore
d
dt
1
cos
(26)
and then denote
dB
2
2
3/ 2
ft
()
 
(1
e
sin
B
)
(27)
Kd
0
Making use of the chain rule of implicit differentiation
df
dB d
df
d
df
dB d
df
d

f
,
f
,

(28)
t
t
dB d
dt
d
dt
dB d
dt
d
dt
It is easy to expand (27) into a power series of sin
1
1
1
2
3
(10)
10
ft
( )
f
(0)
f
(0)
t
f

(0)
t
f

(0)
t
 
f
(0)
t
(29)
t
t
t
t
2!
3!
10!
Omitting the detailed procedure, one arrives at
 
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