Gauss and UTM Conformal Projections and the Plane Rectangular Coordinate System - Geodesy: Introduction to Geodetic Datum and Geodetic Systems - page 312

Inserting into ( 6.70 ) and replacing

R 1 2 and

1

R 2 2 with

1

R m 2 produces:

1

y 1 2
2R m 2 þ

y m 2
2R m 2 þ

y 2 2
2R m 2 þ

y 1 4
24R m 4 þ

y m 4
24R m 4 þ

y 2 4
24R m 4

S
6

D ﾼ 6 þ 4 4 :

We set

y 1 þ y 2 Δ y

2 ﾼ

y 2 y 1

y m ﾼ ,

2 2

to obtain

y m Δ

y
2

y m þ Δ

y
2

y 1 ﾼ , y 2 ﾼ

y 2
2 :
Since the term containing y 4 is minute, and hence replaced by y m 4

þ ʔ and y 1 2 y 2 2 2y m 2 þ ﾼ

y 1 4 y 2 4

þ

ﾼ ,

2

one obtains:

,

y m 2 y 2

24R m 2 þ

y m 4
24R m 4

2R m 2 þ Δ

D ﾼ S 1 þ ð 6 : 71 Þ

and the resulting equation:

y m 2 y 2

24R m 2 þ

y m 4
24R m 4 2R m 2 þ Δ

Δ S ﾼ D S ﾼ S : ð 6 : 72 Þ

The above is the formula for the distance correction of the Gauss projection.
When S 350 km, the equation has an accuracy of 0.001 m, or

better. Therefore, this formula is applicable to computations of the first-order
triangulation.
For the second-order triangulation (accurate to 0.01 m), we can leave out the last
term in the above equation, namely:

< 70 km and y m <

y m 2 y 2

24R m 2 2R m 2 þ Δ

Δ S ﾼ D S ﾼ S : ð 6 : 73 Þ

For the third-order triangulation (accurate to 0.1 m), one has only to consider the
first term:

y m 2

Δ S ﾼ D S ﾼ

2R m 2 S

: ð 6 : 74 Þ