Gravity field of the earth - Physical Geodesy - page 103

P N

north pole

®

P

Ã

d ¾

d à sin

î d

Fig. 2.16. Polar coordinates on the unit sphere

Hence, we find

2 π π

R
4 πγ 0

N = ∆ g ( ψ, α ) S ( ψ )sin ψdψdα (2-310)

α =0 ψ =0

as an explicit form of (2-307). Performing the integration with respect to α
first, we obtain

1
2 π

∆ g ( ψ, α ) dα S ( ψ )sin ψdψ.

π 2 π

R
2 γ 0

N = (2-311)

ψ =0 α =0

The expression in brackets is the average of ∆ g along a parallel of spherical
radius ψ . We denote this average by ∆ g ( ψ ), so that

2 π

1
2 π

∆ g ( ψ )= ∆ g ( ψ, α ) dα . (2-312)

α =0

Thus, Stokes' formula may be written

π

R
γ 0

N = ∆ g ( ψ ) F ( ψ ) dψ , (2-313)

ψ =0

where we have introduced

1

2 S ( ψ )sin ψ = F ( ψ ) .

(2-314)

The functions S ( ψ )and F ( ψ ) are shown in Fig. 2.17. Alternatively, we may