Geoscience Reference
In-Depth Information
We may also directly express ∆
g
(
ϑ, λ
) as a series of Laplace surface spherical
harmonics:
∆
g
(
ϑ, λ
)=
∞
∆
g
n
(
ϑ, λ
)
.
(2-320)
n
=0
Comparing these two series yields
∆
g
n
(
ϑ, λ
)=
n
−
1
R
T
n
(
ϑ, λ
)or
T
n
=
1
∆
g
n
,
(2-321)
R
n
−
so that
T
=
∞
T
n
=
R
∞
∆
g
n
n
1
.
(2-322)
−
n
=0
n
=0
This equation shows again that there must not be a first-degree term in the
spherical-harmonic expansion of ∆
g
; otherwise the term ∆
g
n
/
(
n
−
1) would
be infinite for
n
= 1. As usual, we now assume that the harmonics of degrees
zero and one are missing. Therefore, we start the summation with
n
=2.
By Eq. (1-89), we may write
∆
gP
n
(cos
ψ
)
dσ ,
∆
g
n
=
2
n
+1
4
π
(2-323)
σ
so that the preceding formula becomes
∆
gP
n
(cos
ψ
)
dσ .
∞
R
4
π
2
n
+1
n
T
=
(2-324)
−
1
n
=2
σ
By interchanging the order of summation and integration, we get
∞
P
n
(cos
ψ
)
∆
gdσ.
R
4
π
2
n
+1
n
T
=
(2-325)
−
1
n
=2
σ
Comparing this with Stokes' formula (2-304), we find the
expression for
Stokes' function in terms of Legendre polynomials
(zonal harmonics):
S
(
ψ
)=
∞
2
n
+1
n
P
n
(cos
ψ
)
.
(2-326)
−
1
n
=2
In fact, the analytic expression (2-305) of Stokes' function could have
been derived somewhat more simply by direct summation of this series,
but we believe that the derivation given in the preceding section is more
instructive because it also throws sidelights on important related problems
such as the “bonus equation” (2-280).
