Gravity field of the earth - Physical Geodesy - page 48

surface normal

P

Fig. 2.5. Definition of mean curvature

Here the minus sign is only a convention. This is an expression for the mean
curvature of the level surface .
From the generalized Poisson equation

4 πG +2 ω 2 , ∆ W ≡ W xx + W yy + W zz = − (2-37)

we find

− 2 gJ + W zz = − 4 πG +2 ω 2 . (2-38)

Considering

∂g
∂z =

∂g
∂H ,

W z = − g, W zz = − − (2-39)

we finally obtain

∂g
∂H

2 ω 2 . = − 2 gJ +4 πG − (2-40)

This important equation, relating the vertical gradient of gravity ∂g/∂H to
the mean curvature of the level surface, is also due to Bruns (1878). It is
another beautiful example of the interrelation between the geometric and
dynamic concepts in geodesy.

Plumb lines
The curvature of the plumb line is needed for the reduction of astronomical
observations to the geoid. A plumb line may be defined as a curve whose
line element vector

d x =[ dx, dy, dz ] (2-41)

has the direction of the gravity vector

g =[ W x ,W y ,W z ] ; (2-42)