Reference Ellipsoid and the Geodetic Coordinate System - Geodesy: Introduction to Geodetic Datum and Geodetic Systems - page 235

0 1 0 1 0 1 0 1

2

d 2 L
dS 2

dL
dS

S
2

1
2

S
2

@ A S=2 @ A þ @ A S=2 @ A

L 2 L S=2 ﾼ þ

0 1 0 1 0 1 0 1

3 4

d 3 L
dS 3

d 4 L
dS 4

1
6

S
2

1
24

S
2

@ A @ A @ A @ A

þ þ ,

S=2 S=2

0
@

1
A

0
@

1
A þ

0
@

1
A

0
@

1
A

2

d 2 L
dS 2

dL
dS

S
2

1
2

S
2

L 1 L S=2 ﾼ þ

S=2 S=2

0
@

1
A

0
@

1
A

0
@

1
A

0
@

1
A

3 4

d 3 L
dS 3

d 4 L
dS 4

1
6

S
2

1
24

S
2

þ þ:

S=2 S=2

Subtracting the above two equations from each other gives:

d 3 L
dS 3

dL
dS

1
24

S 3 l ﾼ L 2 L 1 ﾼ S þ þ:

S=2 S=2

In like manner, we can obtain the formulae for differences in geodetic latitude
and azimuth. Combining them with the above expression yields:

0
@

1
A 2

0
@

1
A 2

9
=

d 3 L
dS 3

dL
dS

1
24

S 3 l ﾼ L 2 L 1 ﾼ S þ þ

0
@

1
A 2

0
@

1
A 2

d 3 B
dS 3

dB
dS

1
24

S 3 b ﾼ B 2 B 1 ﾼ S þ þ

: ð 5 : 71 Þ

;

0
@

1
A 2

0
@

1
A 2

d 3 A
dS 3

dA
dS

1
24

180 o S 3

a ﾼ A 2 A 1 ﾼ S þ þ

where the subscript S/2 indicates that the derivatives of various orders in the bracket
will be taken according to the geodetic latitude B S/2 and geodetic azimuth A S/2 at the
midpoint of the geodesic P S/2 .In( 5.71 ), although only two terms are listed, the
accuracy actually reaches S 4 terms. Thus, it converges more rapidly than the
Legendre series. In ( 5.71 ), the geodetic latitude B S/2 and geodetic azimuth A S/2
are actually unknown, so the equation cannot be solved directly, and we need to
convert the derivatives of the equation. Assume that: