Modern views on the determination of the figure of the earth - Physical Geodesy - page 330

earth's surface. On the other hand, the normal plumb line is physically (or
dynamically) defined by means of the external gravity field of an equipo-
tential ellipsoid. Hence also ϕ, λ are dynamically defined. The quantities
obtained by replacing ϕ, λ by ϕ, λ so that

ξ =Φ λ )cos ϕ,

− ϕ, η =(Λ − (8-138)

are called “dynamical deflections of the vertical” at the earth's surface. By
(8-136) and (8-137) we have

ξ = ξ + δϕ normal ,

η = η, (8-139)

since δλ normal = 0. For an azimuth α we accordingly have

ε = ξ cos α + η sin α. (8-140)

Compare ε and ε in Fig. 8.13 and note that in this figure δ denotes the
curvature of the normal plumb line for the azimuth α given by the analogous
formula

δ = δϕ normal cos α +( δλ normal cos ϕ )sin α = δϕ normal cos α. (8-141)

8.13 Astronomical leveling revisited

From Fig. 8.15 we take the well-known differential relation

dN = − ε 0 ds , (8-142)

where ε 0 denotes the deflection of the vertical at the geoid. Integration be-
tween two points A and B yields the difference between their geoidal heights:

B

N B − N A = − ε 0 ds , (8-143)

A

plumb line
"

ellipsoidal normal

geoid

0

"

dN

0

N

ds

ellipsoid

Fig. 8.15. Astronomical leveling according to Helmert