Gauss and UTM Conformal Projections and the Plane Rectangular Coordinate System - Geodesy: Introduction to Geodetic Datum and Geodetic Systems - page 286

m 0 0 þ m 0 2 y 2 m 0 4 y 4 q ﾼ þ þ

: ð 6 : 32 Þ

n 0 1 y n 0 3 y 3 n 0 5 y 5 l ﾼ þ þ þ

Taking the partial derivative of ( 6.32 ) based on the first condition for the Gauss
projection yields:

9
=

dm 0 0
dx þ

y 2 dm 0 2 y 4 dm 0 4

∂ q

x ﾼ dx þ dx þ ∂

∂ q

2m 0 2 y 4m 0 4 y 3

y ﾼ

þ þ

∂

: ð 6 : 33 Þ

y dn 0 1 y 3 dn 0 3 y 5 dn 0 5

∂ l ;

x ﾼ dx þ dx þ dx þ ∂

∂ l

n 0 1 þ 3n 0 3 y 2 5n 0 5 y 4

y ﾼ

þ þ

∂

Inserting the general condition for conformal projection gives

dm 0 0

y 2 dm 0 2 y 4 dm 0 4 n 0 1 þ 3n 0 3 y 2 5n 0 5 y 4

dx þ dx þ dx þﾼ

þ þ

y dn 0 1 y 3 dn 0 3 y 5 dn 0 5 2m 0 2 y 4m 0 4 y 3

dx þ dx þ dx þﾼ

The above equation must satisfy the condition that the coefficients of the same
powers of y are equal in order to be valid; hence:

9
=

dm 0 0
dx

n 0 1 ﾼ

dn 0 1
dx

1
2

m 0 2 ﾼ

dm 0 2
dx

1
3

n 0 3 ﾼ

: ð 6 : 34 Þ

;

dn 0 3
dx

1
4

m 0 4 ﾼ

dm 0 4
dx

1
5

n 0 5 ﾼ

⋮

To obtain the above derivative, m 0 needs to be determined first. According to the
third condition for the Gauss projection, with y

ﾼ 0, x ﾼ X f . In such a case, we