The geometry of the earth - Physical Geodesy - page 229

the leveling increment dn , which is the orthometric reduction d (OC). Thus,

N A = N B − N A − N B − OC AB , (5-143)

so that we can immediately apply Eq. (4-46):

B B

g − γ 0

dn + g B −

γ 0 g A − γ 0

N B − N A = − εds − H B − H A , (5-144)

γ 0 γ 0 γ 0

A A

where γ 0 is our usual constant γ 45 ◦ ; the deflection components ε are com-
puted from the observed ground values Φ and Λ by (5-116) and (5-115).
These ideas go back to Helmert, but they are hardly used anymore.

Curvature of the normal plumb line
If, instead of the actual gravity g , the normal gravity γ is applied for the
computation of the plumb-line curvature, we find, using

1+ f ∗ sin 2 ϕ − ,

2
a h ···

γ = γ a (5-145)

that

∂γ
∂x

1
R

∂γ
∂ϕ

= 2 γ R = 2 γ

R

=

f ∗ sin ϕ cos ϕ f ∗ sin ϕ cos ϕ,

(5-146)

∂γ
∂y

1
R cos ϕ

∂γ
∂λ =0 .

=

Hence, the integrand (1 /γ )( ∂γ/∂x ) in (5-128) does not depend on h ,sothat
the integration can be performed immediately. We find

f ∗
R

0 . 17 h [km] sin 2 ϕ, δϕ normal = − h sin 2 ϕ = −

(5-147)

δλ normal =0 .

The curvature of the normal plumb line in the east-west direction is zero,
owing to the rotational symmetry of the ellipsoid of revolution. The normal
reduction (5-147) is very simple and practically important , see especially
Sect. 8.13.

5.16 Best-fitting ellipsoids and the mean

earth ellipsoid

We define the mean earth ellipsoid physically as that ellipsoid of revolution
which shares with the earth the mass M ,thepotential W 0 , the difference