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where
c
=
a
2
b
(2-150)
is the radius of curvature at the pole. The normal radius of curvature,
N
,
admits a simple geometrical interpretation (Fig. 2.11). It is, therefore, also
known as the “normal terminated by the minor axis” (Bomford 1962: p. 497).
2.9
Expansion of the normal potential in spherical
harmonics
We have found the gravitational potential of the normal figure of the earth
in terms of ellipsoidal harmonics in (2-124) as
V
=
GM
E
tan
−
1
E
u
+
1
3
ω
2
a
2
q
q
0
P
2
(sin
β
)
.
(2-151)
Now we wish to express this equation in terms of spherical coordinates
r, ϑ, λ
.
We first establish a relation between ellipsoidal-harmonic and spherical
coordinates. By comparing the rectangular coordinates in these two systems
according to Eqs. (1-26) and (1-151), we get
r
sin
ϑ
cos
λ
=
√
u
2
+
E
2
cos
β
cos
λ,
r
sin
ϑ
sin
λ
=
√
u
2
+
E
2
cos
β
sin
λ,
(2-152)
r
cos
ϑ
=
u
sin
β.
The longitude
λ
is the same in both systems. We easily find from these
equations
cot
ϑ
=
u
√
u
2
+
E
2
tan
β,
r
=
u
2
+
E
2
cos
2
β.
(2-153)
The direct transformation of (2-151) by expressing
u
and
β
in terms of
r
and
ϑ
by means of equations (2-153) is extremely laborious. However, the
problem can be solved easily in an indirect way.
We expand tan
−
1
(
E/u
) into the well-known power series
E
u
3
E
u
5
tan
−
1
E
u
=
E
1
3
+
1
5
u
−
−
... .
(2-154)
The substitution of this series into the first equation of formula (2-113), i.e.,
1+3
u
2
E
2
tan
−
1
E
,
q
=
1
2
u
−
3
u
(2-155)
E
