Gravity field of the earth - Physical Geodesy - page 115

north pole

¸ '- ¸

90°- ' 90°- ' '

®

P

Ã

d ¾

Fig. 2.19. Relation between geographical and polar coordinates on the sphere

We now introduce the azimuth α , as shown in Fig. 2.16. From the spherical
triangle of Fig. 2.19 we get, using well-known formulas of spherical trigonom-
etry,

sin ψ cos α =cos ϕ sin ϕ − sin ϕ cos ϕ cos( λ − λ ) ,

(2-382)

sin ψ sin α =cos ϕ sin( λ −

λ ) .

Substituting these relations into the preceding equations, we find the simple
expressions

∂ψ
∂ϕ =

∂ψ
∂λ =

− cos α, − cos ϕ sin α, (2-383)

so that
∂S ( ψ )
∂ϕ

dS ( ψ )
dψ

∂S ( ψ )
∂λ

dS ( ψ )
dψ

= − cos α, = − cos ϕ sin α. (2-384)

These are substituted into (2-379) and the corresponding formula for ∂N/∂λ
and from equations (2-377) we finally obtain

2 π π/ 2

1
4 πγ 0

∆ g ( ϕ ,λ ) dS ( ψ )
dψ

cos α cos ϕ dϕ dλ , ξ ( ϕ, λ )=

λ =0 ϕ = −π/ 2

2 π π/ 2

1
4 πγ 0

∆ g ( ϕ ,λ ) dS ( ψ )
dψ

sin α cos ϕ dϕ dλ η ( ϕ, λ )=

λ =0 ϕ = −π/ 2

(2-385)

or, written in the usual abbreviated form,

∆ g dS ( ψ )
dψ

1
4 πγ 0

ξ = cos αdσ,

σ

∆ g dS ( ψ )
dψ

(2-386)

1
4 πγ 0

η = sin αdσ.

σ