Gauss and UTM Conformal Projections and the Plane Rectangular Coordinate System - Geodesy: Introduction to Geodetic Datum and Geodetic Systems - page 279

To determine the coefficients of n 1 , m 2 , n 3 , m 4 , n 5 , ... it is necessary to find the

dX
dq . It follows from the differential formula for meridian arc length

derivative

MdB and ( 6.3 ) dB

r
M that:

dX ﾼ

dq ﾼ

dX
dq ﾼ

dX
dB

dB
dq ﾼ

n 1 ﾼ r :

Hence, it follows that:

c
V cos B

n 1 ﾼ r ﾼ N cos B ﾼ : ð 6 : 24 Þ

We obtain dn 1
dq

by:

dB
dq ,

dB
dq ﾼ

dn 1
dq ﾼ

dr
dq ﾼ

dr
dB

dB
dq ﾼ

d
dB

c
V cos B

c
V 2

dV
dB cos B

c
V sin B

with

1
2

1
2 d

e 02 cos 2 B

2 1 dB e 02 cos 2 B

dV
dB ﾼ

d
dB

1

e 02 cos 2 B 1 þ ﾼ þ

1
V e 02 cos B V e 02 cos 2 B sin B

1

ﾼ ð sin B Þﾼ

cos B :

Some of the formulae in this chapter are quite long, so to simplify writing and
improve readability, we have intentionally introduce some symbols as follows:

e 0 cos B

ʷ ﾼ

:

t ﾼ tan B

Hence, with:

dV
dB ﾼ

1
V ʷ

2 t,

we get: