Earth deformations - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 227

to give

k π
0

k π
0

dP l
d θ

m
−

P n cosθ d θ +

m
−

P l P n sinθ d θ − 2π( k + m )δ 2π k δ

k π
0

m
−

P l P n sinθ d θ = 2π k δ

1) m 4 π k
2 n

m n

= ( −

1 δ

− k δ l ,

(3.88)

+

using the orthogonality relation (1.180). Now consider the fourth integral arising
on the right of (3.84),

π

dP l
d θ

dP n
d θ

cosθsinθ d θ. (3.89)

0

On integrating by parts, it becomes
dP l
d θ

⎣ ⎝ ⎠ ⎦

π

dP l
d θ

dP l
d θ

π
0 −

d
d θ

P n cosθsinθ sin 2 P n d θ sinθ cosθ − θ

0

⎣ ⎦

π

dP l
d θ

1 ) P l cosθsinθ − k 2 P l cotθ + sin 2 P n d θ,

=

l ( l +

θ (3.90)

0

using Legendre's equation (B.1). This combines with the fifth integral arising on
the right of (3.84), for m =− k ,togive

⎣ ⎦

π

dP l
d θ

1) sinθcosθ P l + sin 2 P n , l ( l + θ (3.91)

0

which is easily evaluated using the recurrence relations (B.11) and (B.12), along
with the orthogonality relation (1.180). The third integral on the right of (3.84)
is evaluated directly using the orthogonality relation (1.180). Finally, collecting
terms, we find for n ≥ 1,

1 ) u − m

+ v − n

im Ω

v − Cn =

n

n ( n

+

( n

1

Ω
n ( n

− 1)( n + 1)( n + m ) n ( n + 2)( n − m + 1)

t − m t − m + + . (3.92)

n − 1 n +

+ 1) 2 n − 1 2 n + 3

After replacing − m with m , the radial coe cient of the transverse spheroidal part

of the body force per unit volume becomes

1 ) u n + v

n

2 m ω Ω ρ 0
n ( n +

m
Fn

2 n

v = ω ρ 0 v −

( n

t n + 1

2 i ω Ω ρ 0
n ( n

− 1)( n + 1)( n − m ) n ( n + 2)( n + m + 1)

t n − 1 +

− . (3.93)

+ 1) 2 n − 1 2 n + 3