Gauss and UTM Conformal Projections and the Plane Rectangular Coordinate System - Geodesy: Introduction to Geodetic Datum and Geodetic Systems - page 278

9
=

∂ x dX

dq þ

l dm 1 l 2 dm 2 l 3 dm 3 l 4 dm 4

q ﾼ dq þ dq þ dq þ dq þ ... ∂

∂ x

3m 3 l 2 4m 4 l 3

l ﾼ

m 1 þ 2m 2 l þ þ þ ...

∂

: ð 6 : 21 Þ

∂ y

l dn 1 l 2 dn 2 l 3 dn 3 l 4 dn 4

;

q ﾼ dq þ dq þ dq þ dq þ ...

∂

∂ y

3n 3 l 2 4n 4 l 3

l ﾼ

n 1 þ 2n 2 l þ þ þ ...

∂

Inserting the first condition for Gauss projection, namely the general condition
for conformal projection ∂ x
∂ q ﾼ ∂ y l ﾼ ∂ y l and ∂ x q , gives: ∂ ∂ ∂

dX
dq þ

l dm 1 l 2 dm 2 l 3 dm 3 l 4 dm 4 3n 3 l 2 4n 4 l 3

dq þ dq þ dq þ dq þ ...ﾼ

n 1 þ 2n 2 l þ þ þ ... ,

l dn 1 l 2 dn 2 l 3 dn 3 l 4 dn 4

3m 3 l 2 4m 4 l 3

m 1 þ 2m 2 l þ þ þ ...ﾼ

dq dq dq dq þ ...

To make the right sides of the above two equations equal, the necessary and
sufficient condition is that the coefficients of the same powers of l are equal. Hence:

9
=

m 1 ﾼ n 2 ﾼ m 3 ﾼ n 4 ﾼﾼ 0

dX
dq

n 1 ﾼ

1
2

dn 1
dq

m 2 ﾼ

1
3

dm 2
dq

: ð 6 : 22 Þ

n 3 ﾼ

;

1
4

dn 3
dq

m 4 ﾼ

1
5

dm 4
dq

n 5 ﾼ

⋮

Since n 0 ﾼ m 1 ﾼ n 2 ﾼ m 3 ﾼ n 4 ﾼﾼ 0, ( 6.20 ) can be simplified as:

m 2 l 2 m 4 l 4 x ﾼ X þ þ þ ...

: ð 6 : 23 Þ

n 3 l 3 n 5 l 5 y ﾼ n 1 l þ þ þ ...

It can be seen from the above equation that the projections on the east and west
sides of the central meridian are symmetrical about the central meridian.