Earth’s figure and gravitation - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 349

so that

V≈ ρ( r 0 ) δ s ρ d

δ r 0 dr 0 dS
= ρ( r 0 ) ∂ R

∂ r 0 R 2 sinθ d θ d λ dr 0 ,

(5.148)

after substitution for δ s /δ r 0 from (5.146). ρ( r 0 ) is to be interpreted as the density
on the equipotential with mean radius r 0 . Similarly, since the intensity of gravity is
inversely proportional to the orthometric separation of the equipotentials,

≈ δ U i = δ U i

δ r 0 δ r 0

g dS

δ s dS δ s dS

1

(1/ R 2 ) (∂ R /∂θ ) 2
∂ R /∂ r 0

+

dU i
dr 0

R 2 sinθ d θ d λ . = (5.149)

Together with the addition theorem (B.9),

n

1) k P n (cosθ) e ik λ P − n (cosθ ) e − ik λ

P n (cos Θ ) = ( − , (5.150)

k =− n

(5.148) and (5.149) can be used to reduce (5.138), including (5.140), to the single
scheme
4π GP n (cosθ) d

r 0 ρ( r 0 ) π

R n − 1 ∂ R

1

∂ r 0 P n (cosθ )sinθ d θ dr 0

0

2 P n (cosθ) π
0

P n (cosθ )sinθ R
r 0

dr

( r ) n − 1 d θ

+ 2 Ω

dr 0 P n (cosθ) π

(1/ R 2 ) (∂ R /∂θ ) 2
∂ R /∂ r 0

dU i 1

R n − 1

1 +

P n (cosθ )sinθ d θ +

0

2 GM

2 r 0 , n

⎩

2

−

3 Ω

=− 1,

2 U i

2 r 0 ,

− +Ω n = 0,

= (5.151)

2 P 2 (cosθ), n

(2/3) Ω = 2,

0, n = 1,3,4,5, ··· .

In arriving at the relations (5.151), the following should be noted: all the terms in
the summation in the addition theorem for k 0 vanish on integration over azimuth;
and the second integral on the left of (5.138) over the volume 0 involves

integration over radius from r 0 to R , with R a function of θ , so the integration over
radius is to be completed before the integration over θ .

V −V i