Reference Ellipsoid and the Geodetic Coordinate System - Geodesy: Introduction to Geodetic Datum and Geodetic Systems - page 243

Fig. 5.43 The relationship
between u and ˃

q

1

1
2 k 2 sin 2 M

1
8 k 4 sin 4 M

k 2 sin 2 M þ ð þ ˃ Þ ﾼ 1 þ ð þ ˃ Þ ð þ ˃ Þþ ,

With

1
2

1
2 cos 2 M

sin 2 M

ð þ ˃ Þﾼ ð þ ˃ Þ ,

2 cos 2 M

þ ˃ þ

8 cos 4 M

þ ˃ ,

3
8

1 1

sin 4 M ð þ ˃ Þﾼ

⋮

and hence:

q

1

k 2
4

3
64 k 4

k 2 sin 2 M

þ ð þ ˃ Þ ﾼ 1 þ þ

cos 2 M

k 2
4 þ

k 4
16 þ

k 4
64 cos 4 M

þ ð þ ˃ Þ ð þ ˃ Þ

þ:

Substituting the above equation into ( 5.88 ) and integrating, with
ð ˃

1
2

cos 2 M ð þ ˃ Þ d ˃ ﾼ ﾽ sin 2 M ð þ ˃ Þ sin 2M ﾼ sin ˃ cos 2M ð þ ˃ Þ ,

0

ð ˃

1
2 sin 2˃ cos 4M

cos 4 M ð þ ˃ Þd˃ ﾼ ð þ 2˃ Þ,

0

one gets