Geoscience Reference
In-Depth Information
or
A
0
+
3
ω
2
a
2
− U
0
P
0
(sin
β
)+
A
1
P
1
(sin
β
)
3
ω
2
a
2
P
2
(sin
β
)+
∞
+
A
2
−
1
A
n
P
n
(sin
β
)=0
.
(2-108)
n
=3
This equation applies for all values of
β
only if the coecient of every
P
n
(sin
β
) is zero. Thus, we get
1
3
ω
2
a
2
,
A
0
=
U
0
−
A
1
=0
,
(2-109)
A
2
=
3
ω
2
a
2
,
A
3
=
A
4
=
...
=0
.
Substituting these relations into (2-100) gives
Q
0
i
u
E
Q
2
i
u
E
V
(
u, β
)=
U
0
−
3
ω
2
a
2
1
+
3
ω
2
a
2
Q
0
i
Q
2
i
P
2
(sin
β
)
.
(2-110)
b
E
b
E
This formula is basically the solution of Dirichlet's problem for the level
ellipsoid, but we can give it more convenient forms. It is a closed formula!
First, we determine the Legendre functions of the second kind,
Q
0
and
Q
2
.As
coth
−
1
(
ix
)=
1
i
i
tan
−
1
1
cot
−
1
x
=
−
x
,
(2-111)
we find by (1-80) with
z
=
iu/E
:
Q
0
i
u
E
=
i
tan
−
1
E
−
u
,
(2-112)
1+3
u
2
E
2
tan
−
1
E
.
Q
2
i
u
E
=
i
2
3
u
E
u
−
By introducing in (2-112) the abbreviations
1+3
u
2
E
2
tan
−
1
E
,
q
=
1
2
3
u
E
u
−
(2-113)
1+3
b
2
E
2
tan
−
1
E
q
0
=
1
2
b
E
b
−
3
and substituting them in equation (2-110), we obtain
tan
−
1
E
u
tan
−
1
E
b
V
(
u, β
)=
U
0
3
ω
2
a
2
q
q
0
P
2
(sin
β
)
.
1
+
3
ω
2
a
2
−
(2-114)
