Geoscience Reference
In-Depth Information
dS
d
M
dB
˃
ﾼ
du
,
dS
d
N
cos B
cos u
dl
d
˃
ﾼ
ʻ
:
e
2
a 1
ð
Þ
a
a
With M
ﾼ
ﾼ
p
1
, N
ﾼ
W
, and (
5.77
) and (
5.74
), one obtains:
W
3
V
3
e
2
a
V
d
dS
ﾼ
˃
,
ð
5
:
83
Þ
1
V
d
dl
ﾼ
ʻ:
ð
5
:
84
Þ
1
V
With cos u
ﾼ
p
cos B,
e
2
1
and from:
ﾼ
e
0
2
cos
2
B
e
0
2
cos
2
uV
2
V
2
e
2
e
2
V
2
cos
2
u,
ﾼ
1
þ
ﾼ
1
þ
1
1
þ
with
1
1
V
ﾼ
p
e
2
cos
2
u
substituted into (
5.83
)and (
5.84
) we obtain:
a
p
dS
ﾼ
1
e
2
cos
2
u
d˃,
ð5:85Þ
p
1
dl
ﾼ
e
2
cos
2
u
d
ʻ:
ð
5
:
86
Þ
Equations (
5.85
) and (
5.86
) are the differential equations that define the rela-
tionship between the side length and longitude difference on the ellipsoid and the
corresponding side length and longitude difference on the auxiliary sphere. They
are known as the Bessel's differential equation. The relations between S and
˃
and
between l and
can be obtained by solving this set of differential equations.
Applying different integration methods will result in different formulae, which
distinguishes Bessel's formula for the solution of the geodetic problem from
many other formulae for solving the problem over long distances.
Bessel's formula applies to any distances because its integration is not in the
power series of the side length, longitude, or latitude (differences of longitude or
latitude), but in the power series of the eccentricity squared e
2
(or e
02
).
ʻ
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