Reference Ellipsoid and the Geodetic Coordinate System - Geodesy: Introduction to Geodetic Datum and Geodetic Systems - page 246

,

e 2
2 þ

e 4
8

e 4
16 cos 2 m

e 4
16 cos 2 m sin

l ﾼ ʻ sin m ˃ þ ˃ cos 2M ð þ ˃ Þ

Or

h i ,

l 00

00 0 00 0 sin

ﾼ ʻ sin m ʱ ˃ þ ʲ ˃ cos 2M ð þ ˃ Þ ð 5 : 92 Þ

Where

0
@

1
A e 2

9
=

k 0 2
16

e 2
8

1
2 þ

0

ʱ ﾼ

:

e 2

16 k 0 2

0 00

;

ʲ ﾼ ˁ

k 0 2

ﾼ e 2 cos 2 m

Equations ( 5.89 )or( 5.90 ) and ( 5.92 ) are the formulae for projections of the side
lengths and longitude differences in the Bessel's solution of geodetic problems.
These expressions are expanded in the power series of e 02 and e 2 , not in the power
series of S. The accuracy of the solution is related to the expanded terms rather than
the distance. The formulae can be applied to the solution of the geodetic problem
over any distances (particularly long distances). The main deficiency of Bessel's
formula is that, while computing

given l, iteration is
needed and the auxiliary quantities m and M should also be calculated beforehand.

˃ given S and computing ʻ

5.6.4 Computations of Bessel's Direct Solution
of the Geodetic Problem

Steps for Solution

Project the Ellipsoidal Elements onto a Spherical Surface

1. Given B 1 , find u 1 :

p

1 tan u 1 ﾼ e 2 tan B 1 : ð 5 : 93 Þ

2. Calculate the auxiliary quantities m and M: