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where
cos
ψ
5+3ln
r
.
(2-303)
R
2
r
2
S
(
r, ψ
)=
2
R
l
+
R
r
−
3
Rl
−
R
cos
ψ
+
l
2
r
r
2
−
On the geoid itself, we have
r
=
R
, and denoting
T
(
R, ϑ, λ
)simplyby
T
,we
find
∆
gS
(
ψ
)
dσ ,
R
4
π
T
=
(2-304)
σ
where
3cos
ψ
ln
sin
ψ
2
(2-305)
1
sin(
ψ/
2)
−
6sin
ψ
+sin
2
ψ
2
S
(
ψ
)=
2
+1
−
5cos
ψ
−
is obtained from
S
(
r, ψ
) by setting
r
=
R
and
l
=2
R
sin
ψ
2
.
(2-306)
By Bruns' theorem,
N
=
T/γ
0
, we finally get
∆
gS
(
ψ
)
dσ .
R
4
πγ
0
N
=
(2-307)
σ
This formula was published by G.G. Stokes in 1849; it is, therefore, called
Stokes' formula
,or
Stokes' integral
. It is by far the most important formula
of physical geodesy because it performs
to determine the geoid from gravity
data
. Equation (2-304) is also called Stokes' formula, and
S
(
ψ
)isknownas
Stokes' function.
Using formula (2-302), which was derived by Pizzetti (1911) and later on
by Vening Meinesz (1928), we can compute the anomalous potential
T
at any
point outside the earth. Dividing
T
by the normal gravity at the given point
P
(Bruns' theorem), we obtain the separation
N
P
between the geopotential
surface
W
=
W
P
and the corresponding spheropotential surface
U
=
W
P
,
which, outside the earth, takes the place of the geoidal undulation
N
(see
Fig. 2.15 and the explanations at the end of the preceding section).
We mention again that these formulas are based on a spherical approx-
imation; quantities of the order of 3
10
−
3
N
are neglected. This results in
an error of probably less than 1 m in
N
, which can be neglected for many
practical purposes. Sagrebin (1956), Molodenskii et al. (1962: p. 53), Bjer-
hammar, and Lelgemann have developed higher approximations, which take
into account the flattening
f
of the reference ellipsoid; see Moritz (1980 a:
Sect. 39).
·
