Reference Ellipsoid and the Geodetic Coordinate System - Geodesy: Introduction to Geodetic Datum and Geodetic Systems - page 228

0. 3 00 and m ˆ ﾼ 0. 3 00 , corresponding to a terrestrial distance of

0.02s ﾼ

p

6m. Correcting
the deflection of the vertical to within 1 00 with gravimetric data, the mean square
error of the point becomes

9 m, and the mean square error of the point is 9m ﾼ 12 :

p ﾼ

1 00 4 00 , corresponding to a distance of 1 :

42 m. Taking the two aspects into consideration, the effect is 44 m.

5.5.6 Relationship Between Astronomical Azimuth
and Geodetic Azimuth (Laplace Azimuth Formula)

One of the applications of an astronomical survey is to determine geodetic azimuths
in order to define the orientation of the control network and control the accumula-
tion of azimuth errors. Hence, the observed astronomical azimuth should be
reduced to the geodetic azimuth.
From Fig. 5.31 , the astronomical azimuth of the AM direction is:

ʱ ﾼ ʸ 1 þ R 1 :

The geodetic azimuth of the AM direction is:

A ﾼ ʸ þ R :

The two equations subtracted from each other give:

ʱ A ﾼ ʸ 1 ʸ ð Þþ ð R 1 R Þ ,

where R 1 R is the correction for deflection of the vertical of the observed

direction, namely ( 5.55 )
To obtain the expression of ʸʸ 1 in Fig. 5.31 , applying Napier's rules

in the right-angled spherical triangle gives:

ʸ 1 ʸ ,inP

tan 90 ∘ ʻ

sin ˆ ﾼ tan ð ʸ 1 ʸ Þ ð ð L Þ Þ ﾼ ʸ 1 ʸ ð Þ cot ð ʻ L Þ

Hence, we get:

ʸ 1 ʸ ﾼ ʻ ð L Þ sin ˆ ,

and

A ﾼ ʱ ʻ ð L Þ sin ˆ ʾ ð sin A ʷ cos A Þ cot z 1 :

The last term at the right-hand side of the above equation is known as the term of
correction for the deflection of the vertical. In general, its value is only a few
hundredths of a second or even less. In the first-order astronomical survey, the mean