Reference Ellipsoid and the Geodetic Coordinate System - Geodesy: Introduction to Geodetic Datum and Geodetic Systems - page 214

Fig. 5.31 Correction for
the deflection of the vertical

sin R
sin z 1 :

sin q ﾼ sin ʼ

Substituting this equation into the above equation results in:

sin R
sin z 1 :

sin ð ʴ 1 Þﾼ cos z 1 sin ʼ

Since both ʴ and ʼ are small quantities, we can write:

ʴ 1 ﾼ ʼ sin R cot z 1 ,

with R ﾼ A ʸ (A denotes the geodetic azimuth from point A to point m); then:

ʴ 1 ﾼ ʼ sin A ð ʸ Þ cot z 1

ﾼ ʼ ð sin A cos ʸ cos A sin ʸ Þ cot z 1 :

Given ʾ ﾼ ʼ cos ʸ , ʷ ﾼ ʼ sin ʸ , we obtain:

ʴ 1 ﾼʾ ð sin A ʷ cos A Þ cot z 1

ð 5 : 55 Þ

ﾼʾ ð sin A ʷ cos A Þ tan ʱ 1 ,

where ʱ 1 is the vertical angle between the line of sight and the horizontal. It can be

seen that the correction for deflection of the vertical is primarily concerned with the
deflection of the vertical at the observation point and the zenith distance (vertical
angle) at the target point.
In Fig. 5.31 the horizontal circles perpendicular to the plumb line and the
ellipsoidal normal do not coincide, and the angle between them is ʼ . However,

they are considered to coincide because ʼ is a small quantity and its effect on the

horizontal circle reading can be neglected.