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which is of the same type as (9-15). But since
C
depends only on the distance
ψ
and not on the azimuth
α
, the spherical harmonics cannot contain any
terms that explicitly depend on
α
. The only harmonics independent of
α
are
the zonal functions
R
n
0
(
ψ, α
)
≡
P
n
(cos
ψ
)
,
(9-24)
so that we are left with
C
(
ψ
)=
∞
c
n
P
n
(cos
ψ
)
.
(9-25)
n
=2
The
c
n
≡
c
n
0
are the only coecients that are not equal to zero. We also
use the equivalent expression in terms of fully normalized harmonics:
C
(
ψ
)=
∞
c
n
P
n
(cos
ψ
)
.
(9-26)
n
=2
The coecients in these series, according to Sects. 1.9 and 1.10, are given
by
2
π
π
c
n
=
2
n
+1
4
π
C
(
ψ
)
P
n
(cos
ψ
)sin
ψdψdα
α
=0
ψ
=0
(9-27)
π
=
2
n
+1
2
C
(
ψ
)
P
n
(cos
ψ
)sin
ψdψ
ψ
=0
and
c
n
√
2
n
+1
.
c
n
=
(9-28)
We now determine the relation between the coecients
c
n
of
C
(
ψ
)in
(9-25) and the coecients
a
nm
and
b
nm
of ∆
g
in (9-18). For this purpose
we need an expression for
C
(
ψ
)intermsof∆
g
, which is easily obtained by
writing (9-27) more explicitly. Take the two points
P
(
ϑ, λ
)and
P
(
ϑ
,λ
)of
Fig. 9.2. Their spherical distance
ψ
is given by
cos
ψ
=cos
ϑ
cos
ϑ
+sin
ϑ
sin
ϑ
cos(
λ
−
λ
)
.
(9-29)
Here
ψ
and the azimuth
α
are the polar coordinates of
P
(
ϑ
,λ
) with respect
to the pole
P
(
ϑ, λ
).
The symbol
M
in (9-6) denotes the average over the unit sphere. Two
steps are required to find it. First, we average over the spherical circle of
radius
ψ
(denoted in Fig. 9.2 by a broken line), keeping the pole
P
fixed and
letting
P
move along the circle so that the distance
PP
remains constant.
This gives
2
π
1
2
π
C
∗
=
∆
g
(
ϑ, λ
)∆
g
(
ϑ
,λ
)
dα ,
(9-30)
α
=0
