Geology Reference
In-Depth Information
No further eigenvalues are encountered in the recursions for higher terms in the
power series expansions.
For numerical integration, they-variables are replaced by a system of
z
-variables
defined by
=
y
i
(
r
)/
r
α
,
z
i
(
r
)
i
=
2,4,
=
y
i
(
r
)/
r
α
+
1
z
i
(
r
)
,
i
=
1,3,6,
=
y
i
(
r
)/
r
α
+
2
z
i
(
r
)
,
i
=
5.
(3.187)
In summary, there are three fundamental solutions, regular at the geocentre, gener-
ated by the three free constants:
A
1,1
,
A
6,1
,forα
=
n
−
2, and
A
4,0
,forα
=
n
.
The fundamental solution generated by the free constant
A
1,1
is
A
1,3
r
2
A
1,5
r
4
z
1
(
r
)
=
A
1,1
+
+
+···
,
(3.188)
A
2,2
r
2
A
2,4
r
4
z
2
(
r
)
=
2 (
n
−
1)μ
A
1,1
+
+
+···
,
(3.189)
1
n
A
1,1
A
3,3
r
2
A
3,5
r
4
z
3
(
r
)
=
+
+
+···
,
(3.190)
n
−
1
n
A
4,2
r
2
A
4,4
r
4
z
4
(
r
)
=
2μ
A
1,1
+
+
+···
,
(3.191)
4
π
G
ρ
0
n
A
5,4
r
2
A
5,6
r
4
z
5
(
r
)
=
A
1,1
+
+
+···
,
(3.192)
A
6,3
r
2
A
6,5
r
4
=
+
+···
.
z
6
(
r
)
(3.193)
The coe
cients
A
1,3
,
A
2,2
,
A
3,3
,
A
4,2
are given in terms of
A
1,1
by relation (3.183).
In turn,
A
5,4
is given by (3.185) and
A
6,3
is given by (3.186). The coe
cients
A
1,5
,
A
2,4
,
A
3,5
,
A
4,4
are given by the system (3.139) in terms of
A
1,3
,
A
3,3
,
A
6,3
,
A
5,4
for
ν
=
5, η
=
n
+
2. In turn,
A
5,6
,
A
6,5
are given by the system (3.143) in terms of
A
1,5
,
A
3,5
for ν
=
3.
The fundamental solution generated by the free constant
A
6,1
is
6, η
=
n
+
=−
n
ρ
0
p
1
(
n
)
A
6,1
r
2
+
A
1,5
r
4
z
1
(
r
)
+···
,
(3.194)
q
1
(
n
)
p
1
(
n
)
ρ
0
A
6,1
r
2
A
2,4
r
4
z
2
(
r
)
=−
+
+···
,
(3.195)
ρ
0
p
1
(
n
)
A
6,1
r
2
A
3,5
r
4
z
3
(
r
)
=
+
+···
,
(3.196)
A
4,4
r
4
z
4
(
r
)
=
+···
,
(3.197)
1
n
A
6,1
A
5,4
r
2
A
5,6
r
4
z
5
(
r
)
=
+
+
+···
,
(3.198)
z
6
(
r
)
=
A
6,1
+
A
6,3
r
2
+
A
6,5
r
4
+···
.
(3.199)
Search WWH ::
Custom Search