Geoscience Reference
In-Depth Information
For
r
=
R
, this becomes
∞
∂V
∂r
1
R
=
−
(
n
+1)
Y
n
(
ϑ, λ
)
.
(1-147)
n
=0
This is the equivalent of (1-143) in terms of spherical harmonics. From this
equation, we get an interesting by-product. Writing (1-147) as
∞
∂V
∂r
1
R
V
P
−
1
R
=
−
nY
n
(
ϑ, λ
)
(1-148)
n
=0
and comparing this with (1-143), we see that
2
π
π
∞
R
2
2
π
V
−
V
P
1
R
sin
ϑ
dϑ
dλ
=
−
nY
n
(
ϑ, λ
)
.
(1-149)
l
0
λ
=0
ϑ
=0
n
=0
This equation is formulated entirely in terms of quantities referred to the
spherical surface only. Furthermore, for any function prescribed on the sur-
face of a sphere, one can find a function in space that is harmonic outside
the sphere and assumes the values of the function prescribed on it. This is
done by solving Dirichlet's exterior problem. From these facts, we conclude
that (1-149) holds for any (reasonably) arbitrary function
V
defined on the
surface of a sphere. These developments will be used in Sect. 2.20.
1.15
Laplace's equation in ellipsoidal-harmonic
coordinates
Spherical harmonics are most frequently used in geodesy because they are
relatively simple and the earth is nearly spherical. Since the earth is more
nearly an ellipsoid of revolution, it might be expected that ellipsoidal har-
monics, which are defined in a way similar to that of the spherical harmonics,
would be even more suitable. The whole matter is a question of mathematical
convenience, since both spherical and ellipsoidal harmonics may be used for
any attracting body, regardless of its form. As ellipsoidal harmonics are more
complicated, however, they are used only in certain special cases which nev-
ertheless are important, namely, in problems involving rigorous computation
of normal gravity.
We introduce
ellipsoidal-harmonic coordinates u, ϑ, λ
(Fig.1.10).Ina
rectangular system, a point
P
has the coordinates
x, y, z
.Nowwepass
through
P
the surface of an ellipsoid of revolution whose center is the origin
O
, whose rotation axis coincides with the
z
-axis, and whose linear eccentric-
ity has the constant value
E
. The coordinate
u
is the semiminor axis of this
