Geology Reference
In-Depth Information
for η
=
α
+
ν
−
1, with
I
representing the unit matrix, and
⎝
⎠
2λβ
+
1
−
β
−
n
1
λβ
0
−
2δ
4μβ
n
1
δ
−
n
1
M
n
=
,
(3.140)
1
0
0
−
1/μ
δ
λβ
−
3
while the vectors
x
ν
and
b
ν
are
⎝
⎠
A
1,ν
A
2,ν
−
1
A
3,ν
A
4,ν
−
1
x
ν
=
,
(3.141)
and
⎝
⎠
0
4γ
+
ω
2
A
1,ν
−
2
2
−
+
2
Ω
+
(
n
1
γ
+
2
m
ω
Ω
)
A
3,ν
−
2
−
A
6,ν
−
2
b
ν
=
ρ
0
, (3.142)
0
ω
2
m
ω
Ω
/
n
1
A
3,ν
−
2
2
(γ
+
2
m
ω
Ω
/
n
1
)
A
1,ν
−
2
−
−
−
A
5,ν
−
1
for ν
=
0,1,2,....
Similarly, substitution of the expansions (3.138) in equations (3.134) and (3.135),
and equating the coe
cients of like powers of the radius, yields the system
N
n
+
η
I
y
ν
=
c
ν
,
(3.143)
⎝
⎠
,
1
−
1
N
n
=
(3.144)
−
n
1
2
with the vectors
y
ν
and
c
ν
given by
⎝
⎠
4π
G
ρ
0
⎝
⎠
,
A
5,ν
A
6,ν
−
1
A
1,ν
−
1
y
ν
=
and
c
ν
=
(3.145)
−
n
1
A
3,ν
−
1
again for ν
=
0,1,2,....For cases where the systems (3.139) or (3.143) are homo-
geneous, we will refer to the values of η permitting non-trivial solutions as
eigenvalues, although by conventional usage they are of opposite sign to the true
eigenvalues of the coe
cient matrices.
For ν
=
0, the systems (3.139) and (3.143) degenerate to
⎝
⎠
α
+
2λβ
−
n
1
λβ
⎝
⎠
=
0 d
⎝
⎠
−
2δ
n
1
δ
A
1,0
A
3,0
α
−
A
5,0
=
0,
(3.146)
1
α
−
1
n
1
−
δ
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