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9
=
1
2
B
1
þ
B
m
ﾼ
ð
B
2
Þ
:
1
2
A
1
þ
;
180
o
A
m
ﾼ
ð
A
2
Þ
A
S/2
, for the very small flattening of the ellipsoid,
although the difference between them is not large. We can then derive the estimat-
ing equation of B
m
Apparently, B
m
6ﾼ
B
S/2
, A
m
6ﾼ
A
S/2
so as to convert the derivatives based on
B
S/2
, A
S/2
to the derivatives based on B
m
, A
m
, derivations omitted. B
2
and A
2
are the
unknowns in the solution of direct geodetic problems, so the exact values of B
m
and
A
m
are unknown and need to be obtained using a successive approximation.
Equation (
5.71
) is the formula for a direct solution of the geodetic problem,
based on which one can derive the corresponding formula for an inverse solution of
the geodetic problem. These formulae are improvements of the Legendre series and
are applicable to the solution of short-distance geodetic problems, known as the
Gauss mid-latitude formula (see, e.g., Krakiwsky and Thomson 1974).
B
S/2
, A
m
5.6.3 Bessel's Formula for the Solution of the Geodetic
Problem
Overview
The series expansion of the solution of the geodetic problem is to express the
differences in geodetic longitude, latitude, and azimuth as a function of the geode-
sic distance S. It is evident that to achieve the desired accuracy, the longer the
distance, the more complex the formula structure becomes and may even become
unsolvable. Hence, such a formula is not suitable for solving long-distance geodetic
problems.
From the spherical trigonometry, we are clear that the formulae for spherical
triangles are all expressed by the trigonometric function of angles, where the
accuracy of solving spherical triangles is independent of the spherical distance. In
addition, the flattening of the Earth ellipsoid is very small, and when the ellipsoidal
elements (longitude, latitude, side length, and azimuth) are converted into the
corresponding elements on the spherical surface and represented by angles, the
corresponding corrections expressed by angles will be quite small and independent
of distances. Hence, the general approach to solving the long-distance geodetic
problem is to use a spherical surface as a bridge, which means to establish relations
of projection between the ellipsoidal elements and the corresponding spherical
elements under certain projection conditions, to establish the precise relations
between the elements on the spherical surface using the formula for spherical
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