Earth deformations - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 243

A 1,4 , A 2,3 , A 3,4 , A 4,3 are given by the system (3.139) in terms of A 1,2 , A 3,2 , A 6,2 ,
A 5,3 for ν = 4, η = 3. In turn, A 5,5 , A 6,4 are given by the system (3.143) in terms of

A 1,4 , A 3,4 for ν = 4. For the regular solution generated by the free constant

A 4,1 , the extra terms A 1,6 , A 2,5 , A 3,6 , A 4,5 are given by the system (3.139) in terms
of A 1,4 , A 3,4 , A 6,4 , A 5,5 for ν =

5, η =

6, η = 5. In turn, the extra terms A 5,7 , A 6,6 are given

by the system (3.143) in terms of A 1,6 , A 3,6 for ν = 7, η = 6.

0, near the geocentre, the fourth-order system governing the radial spher-
oidal deformations, (3.110) through (3.113), takes the form

For n =

d y 1
dr =−

2 λ r y 1

+ βy 2 , (3.237)

r 2

− ρ 0 4γ +

2

d y 2
dr =

2 δ 4 μ r y 2 − ρ 0 y 6 − ω

2

2 Ω + y 1 − ρ 0 y 1 , (3.238)

d y 5
dr =

4π G ρ 0 y 1 + y 6 , (3.239)

d y 6
dr =−

2
r y 6 .

(3.240)

Substitution of the power series expansions (3.138) in these equations, and equat-
ing coe cients of like powers of the radius, yields the systems

⎝ ⎠ ⎝ ⎠ = ρ 0 ⎝ ⎠ ,

0

2λβ + 1 + η − β A 1,ν

A 2,ν − 1

4γ + ω

2 A 1,ν − 2

2

− 2δ 4μβ + η − + 2 Ω − A 6,ν − 2

(3.241)

and

⎝ ⎠ ⎝ ⎠ = ⎝ ⎠ ,

1 + η − 1

02

A 5,ν
A 6,ν − 1

4π G ρ 0 A 1,ν − 1
0

(3.242)

+ η

for ν = 0,1,2,....The first system has a singular coe cient matrix for η =− 3and

for η = 0, while the coe cient matrix of the second system is singular for η =− 2

and for η =− 1.

For ν = 0, η = α − 1, the system (3.241) gives (α + 2λβ) A 1,0 =− 2δ A 1,0 = 0or

A 1,0 = 0. The system (3.242) gives α A 5,0 = 0or A 5,0 = 0, except for α = 0.

1, η = α, the system (3.241) is homogeneous and has a non-trivial solu-
tion only for η = α =

For ν =

0. The solution is (3λ + 2μ) A 1,1 = A 2,0 . The system (3.242) is

homogeneous and non-singular for α = η = 0. No further

singular values of η are encountered in either system with increasing ν.
For ν =

0, giving A 5,1 = A 6,0 =

2, we have η = α + 1 = 1forα = 0, so the system (3.241) is homogen-

eous and non-singular, implying that A 1,2 = A 2,1 = 0. The system (3.242) is also

non-singular, giving A 5,2 = 2π G ρ 0 A 1,1 and A 6,1 = 0.