Earth’s figure and gravitation - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 332

and

sin 2 φ = sin 2 (φ − γ) cosφsinγ) 2

= (sinφcosγ −

) 2

= (sinφ − γcosφ +···

sin 2

= φ − 2γcosφsinφ +···

f

sin 2

m sin 2 2φ +··· .

= φ − m · (5.37)

The first-order expression (5.33) for gravity in terms of geographic latitude is then
unaltered in form and is
g = g e 1

5
2 m

f sin 2 + − φ +··· . (5.38)

The first-order description of the figure contains Legendre polynomials up to
and including second degree. We can then expect R 2 (θ) in the figure expression
(5.8) to include Legendre polynomials of fourth and lesser degrees. Odd degree
polynomials are missing because R 2 (θ)isanevenfunctionofcosθ.For z = cosθ,

9 3 1

4 .

( P 2 ) 2

4 z 4 2 z 2

= − + (5.39)

Since

8 35 z 4

3 ,

1

30 z 2 P 4 = − + (5.40)

we have the identity

18
35 P 4

2
7 P 2

1
5 .

( P 2 ) 2 = + + (5.41)

Description of the volume V in (5.12), correct to terms of second order in m ,
requires the constant term in R 2 (θ)tobe( 4/45)( f 2 / m 2 ). The coe cients of the

remaining polynomials, of second and fourth degree, are related by the requirement
that the flattening given by (5.17) has a vanishing second-order term in m . Only one
coe

−

cient, δ, remains to be determined, and we have that

m 2 1

1 P 2 + δ P 4

f 2 4

45

5
2

1
6 δ

R 2 (θ) =− + − . (5.42)

To complete the second-order theory, we add the fourth-degree term in the gravit-
ational potential,

β m 2
R 5 P 4 (cosθ)

= β m 2 P 4 (cosθ) +··· (5.43)

to expression (5.19) for the equipotential, and the corresponding term,

5β m 2 P 4 (cosθ) − +··· , (5.44)