Geology Reference
In-Depth Information
cient matrix
A
given by
A
=
with the modified coe
A
+
2
I
,
(3.313)
where
I
is the 6
6 unit matrix. This completes the first three terms in the power
series expansions of the three fundamental solutions for
n
> 1.
For
n
×
1, in terms of
z
-variables, the three fundamental solutions are given
by the expansions (3.214) through (3.219), (3.221) through (3.226), and (3.228)
through (3.233). The first two terms in the expansions of the first fundamental
solution, generated by the free constant
A
1,0
, and the second fundamental solution,
generated by the free constant
A
6,0
, are obtained directly, as are the first terms of
the third fundamental solution, generated by the free constant
A
4,1
. The third terms
in the first and second fundamental solutions, and the second terms of the third
fundamental solution, are given by the systems (3.139) for ν
=
=
4, η
=
n
+
2
=
3,
and (3.143) for ν
=
5, η
=
n
+
3
=
4, which combine to yield the 6
×
6 system
⎝
⎠
⎝
⎠
0
A
1,4
A
2,3
A
3,4
A
4,3
A
5,5
A
6,4
4γ
+
ω
2
A
1,2
+
2
−
+Ω
(2γ
+
2
m
ω
Ω
)
A
3,2
−
A
6,2
0
ω
A
3,2
A
·
=
ρ
0
. (3.314)
2
(γ
+
m
ω
Ω
)
A
1,2
−
−
m
ω
Ω
−
A
5,3
0
0
The third terms in the third fundamental solution are given by the system (3.139)
for ν
=
6, η
=
n
+
4
=
5, and (3.143) for ν
=
7, η
=
n
+
5
=
6, which combine to
×
yield the 6
6 system
⎝
⎠
⎝
⎠
0
A
1,6
A
2,5
A
3,6
A
4,5
A
5,7
A
6,6
4γ
+
ω
2
A
1,4
2
−
+
2
Ω
+
(2γ
+
2
m
ω
Ω
)
A
3,4
−
A
6,4
0
A
·
ω
A
3,4
−
A
5,5
=
ρ
0
. (3.315)
2
(
γ
+
m
ω
Ω
)
A
1,4
−
−
m
ω
Ω
0
0
This completes the first three terms in the power series expansions of the three
fundamental solutions for
n
=
1.
0, the governing spheroidal system degenerates to fourth order and, apart
from the arbitrary constant determining the reference level for gravitational poten-
tial, there is only one fundamental solution regular at the geocentre, generated
by the free constant
A
1,1
.Intermsof
z
-variables, it is given by the expansions
(3.251) through (3.254). The first three terms of these expansions are obtained
directly.
For
n
=
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