Time sequence and spectral analysis - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 117

with γ 0 0 and defines the new

parameterγ 0 . Reversing the order of equations and variables in the auxiliary system
(2.112), and taking complex conjugates, yields the system
a ∗ 1,1 , a ∗ 1,0 R 1 =

= f 0,0 r 1 . Again this includes the equation for m =

0,α ∗ 1 .

(2.125)

Multiplying this system by q 0 and adding it to the system (2.124) gives
f 0,0 +

q 0 a ∗ 1,1 , q 0 a ∗ 1,0 R 1 =

g 0 ,γ 0 +

q 0 α ∗ 1 . (2.126)

If this is to be a solution of the full equations (2.111) for m = 1, then

q 0 = g 1

− γ 0
α ∗ 1 ,

q 0 a ∗ 1,1 , f 1,0 = f 0,0 + (2.127)

q 0 a ∗ 1,0 . f 1,1 =

For m 2, we reverse the order of the equations and variables in the next auxiliary

system (2.121), and take complex conjugates, to get
a ∗ 2,2 , a ∗ 2,1 , a ∗ 2,0 R 2 =

=

0,0,α ∗ 2 .

(2.128)

If we multiply this system by q 1 and add it to the system
f 1,0 , f 1,1 ,0 R 2 =

(g 0 ,g 1 ,γ 1 ), (2.129)

with γ 1 = f 1,0 r 2 + f 1,1 r 1 , we obtain

f 1,0

q 1 a ∗ 2,1 , q 1 a ∗ 2,0 R 2

g 0 ,g 1 ,γ 1

q 1 α ∗ 2 . q 1 a ∗ 2,2 , f 1,1 + + = + (2.130)

For this to be a solution of
f 2,0 , f 2,1 , f 2,2 R 2

= (g 0 ,g 1 ,g 2 ), (2.131)

requires

g 2 − γ 1
α ∗ 2 ,

q 1 =

q 1 a ∗ 2,2 , f 2,0 = f 1,0 +

q 1 a ∗ 2,1 , f 2,1 = f 1,1 + (2.132)

q 1 a ∗ 2,0 . f 2,2 =