The geometry of the earth - Physical Geodesy - page 214

only consider the last two terms on the right-hand side (i.e., we do not take
into account changes of the point coordinates). Omitting all subscripts and
introducing the auxiliary quantities of (5-92), we immediately get

∆ α =cot z sin α ∆ ϕ +(sin ϕ − cos α cos ϕ cot z )∆ λ (5-95)

or, using ∆ ϕ = ξ and ∆ λ cos ϕ = η , yields

∆ α = ξ sin α cot z +sin ϕ ∆ λ − η cos α cot z. (5-96)

This equation may be rearranged to

∆ α =sin ϕ ∆ λ +( ξ sin α

− η cos α )cot z.

(5-97)

Alternatively, by using ∆ λ = η/ cos ϕ ,weget

∆ α = η tan ϕ +( ξ sin α − η cos α )cot z. (5-98)

In first-order triangulation, the lines of sight are usually almost horizontal
so that z . =90 ◦ ,cot z = 0. Therefore, the corresponding term can in general
be neglected and we get

∆ α = η tan ϕ =∆ λ sin ϕ. (5-99)

This is Laplace's equation in its usual simplified form. It is remarkable that
the differences ∆ α = A λ should be related in such a

simple way. Laplace's equation is fundamental for the classical astrogeodetic
computation of triangulations (Sect. 5.14).
For later reference we note that the total deflection of the vertical - that
is, the angle ϑ between the actual plumb line and the ellipsoidal normal - is
given by

− α and ∆ λ =Λ −

ϑ = ξ 2 + η 2 (5-100)
and that the deflection component ε in the direction of the azimuth α is

ε = ξ cos α + η sin α. (5-101)

It is clear that ϑ in (5-100) has nothing to do with the two different ϑ
used for spherical and ellipsoidal-harmonic coordinates (polar distances).
Returning to the reduction of astronomical to the corresponding ellip-
soidal quantities, we have (5-94) for the reduction of Φ , Λ ,H to ϕ, λ, h and,
finally, the formula

α = A − η tan ϕ (5-102)

reduces the astronomical azimuth A to the ellipsoidal azimuth α .
For the application of these formulas, we need the geoidal undulation N
and the deflection components ξ and η with respect to the reference ellipsoid
used. Two points should be noted: