Earth’s figure and gravitation - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 330

and

α m
r 4 +

3 mr dP 2

1

r ∂ U

1

∂θ = d θ .

(5.26)

On the level reference surface U = U 0 and r = R . By binomial expansion,

m 2 R 1 1)

∂ U
∂ r =

2
3 ( P 2

1 − + 3α P 2 − −

m 2 2 R 1 − 1)

(5.27)

2
3 R 1 ( P 2 −

3 R 1 − − 12α R 1 P 2 − +··· ,

and

m

m 2 1 4α R 1 dP 2

1

r ∂ U

1
3

∂θ =

α +

+ −

d θ +··· .

3 R 1

(5.28)

By binomial expansion, the intensity of gravity (5.24) becomes

⎝ ⎠

2 r 2 ∂ U

2

∂ U
∂ r

2

∂ U
∂ r

1

|∇ U |= 1 + +···

∂θ

2 r 2 ∂ U

2

= ∂ U

1

∂ r +

+···

∂θ

2 m 2

2 dP 2
d θ

2

∂ U
∂ r +

1 1

3

= α + +··· . (5.29)

The intensity of gravity, |∇ U | , on the equipotential described by (5.19) is then

m 2 R 1 1)

2
3 ( P 2

g = 1 − + 3α P 2 − −

m 2 2 R 2

2
3 R 1 ( P 2

3 R 1 − − − 12α R 1 P 2 − − 1)

(5.30)

1
2 α

dP 2
d θ

2

1
3 α +

1
18

2

+ + +··· .

Substituting for R 1 and α from (5.18) and (5.22), respectively, and retaining terms
to first order in m ,

2 5 P 2

m
3

2 f
m

g = 1 − − − +··· . (5.31)

Evaluated at the equator, (5.31) gives

9
2 −

m
3

f
m

g e = 1 − +··· , (5.32)