Earth’s figure and gravitation - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 337

giving the extra second-order term as

8
15 κ.

= (5.77)

With substitution for δ from (5.73), the coe cients of the second- and fourth-

degree terms in the gravitational potential, (5.46) and (5.47), become

1 1 7

8 +

3 f 2 f

1
3

2 f
m

8
63 m

f
m

α =− − − · − , (5.78)

m

1 7

8 f 2 f

4
7

f
m

1
5

β = − + . (5.79)

m

The geocentric latitude, φ , is the complement of the co-latitude, θ; hence, cosθ =
sinφ and sinθ = cosφ . Equation (5.66) for the figure is transformed to

a 1

3 cos 2

f cos 2

2 f 2

θsin 2 R = − θ − + 4κ θ +···

= a 1

3 sin 2

− f sin 2

φ −

2 f 2

φ cos 2 φ +···

+

4κ

a 1

3 sin 2 2φ +···

f sin 2 φ −

8 f 2

= − + κ , (5.80)

showing that, with the figure expressed in terms of geocentric latitude, κ again
represents a perturbation to the second-order term of an ellipsoid.
Returning to dimensional variables, it is usual in geodesy to write the gravita-
tional potential on the reference surface as

1

J 2 a
R

2

J 4 a
R

4

GM
R

V =− − P 2 (cosθ) − P 4 (cosθ) +··· . (5.81)

We then have

a
d

2 a

d

4

J 2 ,β m 2 α m = = J 4 . (5.82)

For the true reference surface, (5.76) gives
a
d =

1
3 f

2 8

15 κ ··· . 9 f 2

+ + +

1 (5.83)

Using this relation to expand the ratio a / d in equations (5.82), correct to second
order in small quantities, (5.78) and (5.79) respectively produce

1
3 m

2
3 f

2
21 mf

1 8

21 κ +··· , 3 f 2

J 2 =− + + − + (5.84)

4
7 mf

4 32

35 κ +··· . 5 f 2

J 4 = − − (5.85)