Earth’s figure and gravitation - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 348

- ( ∂ R / ∂q ) d R

U i + d U i

g

U i g

d s

q + dq

R dq

q

a a + d a

Figure 5.3 Two neighbouring equipotentials with orthometric separation δ s .

Consider two neighbouring internal equipotential surfaces, characterised by
mean radii r 0 and r 0 + δ r 0 , as shown in Figure 5.3.
At an arbitrary point, their orthometric (measured perpendicular to the surfaces)
separation is δ s .Take( r ,θ ,λ ) as spherical polar co-ordinates of the integration
point. We have

∂ R
∂ r 0 δ r 0

∂ R

∂θ δθ ,

δ R = + (5.143)

and from Figure 5.3, it can be deduced that

1
R δ s sinγ,

= δ s cosγ, δθ =− δ R (5.144)

with

1

R ∂ R

1 1

cosγ =

∂θ 2 , sinγ = ∂θ 2 ,

(5.145)

1 1

R 2 ∂ R R 2 ∂ R

∂θ

1 1

+ +

giving

R 2 ∂ R

2 ∂ R

∂ r 0 ∂ R

∂ r 0 δ r 0 =

1

or δ s

1 + δ s

δ r 0 =

∂θ 2 .

(5.146)

1

R 2 ∂ R

∂θ

1

+

Again from Figure 5.3, the element of surface area can be seen to be

R 2 ∂ R

2

1

R 2 sinθ d θ d λ dS = 1 + (5.147)

∂θ