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and
GM/R
2
=
γ
0
we then obtain the following expression for the first-degree
harmonic of
T
:
T
1
(
ϑ, λ
)=
γ
0
(
x
c
sin
ϑ
cos
λ
+
y
c
sin
ϑ
sin
λ
+
z
c
cos
ϑ
)
.
(2-339)
Dividing by
γ
0
, we find the first-degree harmonic of the geoidal height:
N
1
(
ϑ, λ
)=
x
c
sin
ϑ
cos
λ
+
y
c
sin
ϑ
sin
λ
+
z
c
cos
ϑ.
(2-340)
Introducing the vector
x
c
=[
x
c
,y
c
,z
c
]
(2-341)
and the unit vector of the direction (
ϑ, λ
),
e
=[sin
ϑ
cos
λ,
sin
ϑ
sin
λ,
cos
ϑ
]
,
(2-342)
(2-340) may be written as
N
1
(
ϑ, λ
)=
x
c
·
e
,
(2-343)
which is interpreted as the projection of the vector
x
c
onto the direction
(
ϑ, λ
).
Hence, if the two centers of gravity do not coincide, then we need only
add the first-degree terms (2-339) and (2-340) to the generalized Stokes
formula (2-333) and to its analogue for
N
, respectively, in order to get the
most general solution for Stokes' problem, the computation of
T
and
N
from
∆
g
. Equation (2-273) shows that
any
value of
T
1
(
ϑ, λ
) is compatible with a
given ∆
g
field because, for
n
=1,thequantity(
n
−
1)
T
1
is zero and so
T
1
,
whatever be its value, does not at all enter into ∆
g
.
Hence, the most general solution for
T
and
N
contains three arbitrary
constants
x
c
,y
c
,z
c
, which can, thus, be regarded as the constants of integra-
tion for Stokes' problem. In actual practice, one always sets
x
c
=
y
c
=
z
c
=0,
thus placing the center of the reference ellipsoid at the center of the earth.
This constitutes an essential advantage of the gravimetric determination of
the geoid over the astrogeodetic method, where the position of the reference
ellipsoid with respect to the center of the earth remains unknown.
Zero-degree terms in
N
and
∆
g
Let us first extend Bruns' formula (2-237) to an arbitrary reference ellipsoid.
Suppose
W
(
x, y, z
)=
W
0
,
(2-344)
U
(
x, y, z
)=
U
0
