Gravity field outside the earth - Physical Geodesy - page 241

see also Sect. 2.14 and Fig. 2.15.
The deflection of the vertical, which is the deviation of the actual plumb
line from the normal plumb line at P , is represented by its north-south and
east-west components,

∂N P
∂ ϕ

∂N P
∂λ

1
r

1
r cos ϕ

ξ P = − , = − ; (6-38) P

these equations correspond to (2-377). Since γ varies very little with latitude
and is independent of longitude, we have

T P
γ Q

=

∂N P
∂ ϕ

∂
∂ ϕ

1
γ Q

∂T P
∂ ϕ −

T P
γ 2

∂γ Q
∂ ϕ

1
γ Q

∂T P
∂ ϕ

=

= (6-39)

Q

and

∂N P
∂λ

1
γ Q

∂T P
∂λ

= . (6-40)

Substituting the results of (6-39) and (6-40) into (6-38) and comparing then
with (6-20) shows that

1
γ Q

1
γ Q

ξ P = − δg ϕ , = − δg λ . (6-41) P

We see that N P ,ξ P ,η P

are given by Eqs. (6-21) and (6-30), apart from

the factor 1 /γ Q . Hence, these equations are the extensions of Stokes' and

Vening Meinesz' formulas for points outside the earth and reduce to these
formulas for r = R , t =1.
Writing Eqs. (6-41) in the form

±

δg ϕ = − γξ, δg λ = − γη, (6-42)

we see that the horizontal components of δ g are directly related to the de-
flection of the vertical, which is the difference in direction of the vectors
g and

. The radial component δg r , however, represents the difference in
magnitude of these vectors, since as a spherical approximation

γ

− δg r = δg = g P − γ P , (6-43)

which is the scalar gravity disturbance (see Sect. 2.12).
Note that here the gravity disturbance δg is the basic quantity to be
computed, rather than the gravity anomaly ∆ g , because both g and γ refer
to the computation point P .