Earth deformations - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 233

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