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inertia;
B
22
would vanish only if the earth had complete rotational symmetry
or if a principal axis of inertia happened to coincide with the Greenwich
meridian.
The five harmonics
A
10
R
10
,A
11
R
11
,B
11
S
11
,A
21
R
21
,and
B
21
S
21
-
all first-degree harmonics and those of degree 2 and order 1 - which must,
thus, vanish in any spherical-harmonic expansion of the earth's potential,
are called
forbidden
or
inadmissible harmonics
.
Introducing the
moments of inertia
with respect to the
x
-,
y
-,
z
-axes by
the definitions
A
=
(
y
2
+
z
2
)
dM ,
B
=
(
z
2
+
x
2
)
dM ,
C
=
(
x
2
+
y
2
)
dM ,
(2-89)
and denoting the
xy
-product of inertia, which cannot be said to vanish, by
D
=
x
y
dM ,
(2-90)
we finally have
A
00
=
GM ,
A
10
=
A
11
=
B
11
=0
,
A
20
=
G
(
A
+
B
)
/
2
C
,
−
(2-91)
A
21
=
B
21
=0
,
A
22
=
4
G
(
B
−
A
)
,
B
22
=
2
GD.
Now let the
x
-and
y
-axes actually coincide with the corresponding prin-
cipal axes of inertia of the earth. This is only theoretically possible, since
the principal axes of inertia of the earth are only inaccurately known. Then
B
22
= 0; taking into account (2-78) and (2-79), we may write explicitly
r
3
1
2
C −
(
A
+
B
)
/
2
(1
−
3cos
2
ϑ
)+
V
=
GM
r
+
G
(2-92)
4
(
B − A
)sin
2
ϑ
cos 2
λ
+
O
(1
/r
4
)
.
3
