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and the second eccentricity
=
√
a
2
b
2
e
=
E
b
−
.
(2-138)
b
The prime on
e
does not denote differentiation, but merely distinguishes the
second eccentricity from the first eccentricity which is defined as
e
=
E/a
.
Removing the constant terms by noting that
1=cos
2
β
+sin
2
β,
(2-139)
we obtain
GM
a
a
2
sin
2
β
+
b
2
cos
2
β
·
γ
=
(2-140)
1+
m
3
sin
2
β
+
1
cos
2
β
.
e
q
0
q
0
e
q
0
q
0
m
6
·
−
m
−
At the equator (
β
= 0), we find
1
;
e
q
0
q
0
γ
a
=
GM
ab
m
6
−
m
−
(2-141)
90
◦
), normal gravity is given by
at the poles (
β
=
±
a
2
1+
m
.
e
q
0
q
0
γ
b
=
GM
(2-142)
3
Normal gravity at the equator,
γ
a
, and normal gravity at the pole,
γ
b
,satisfy
the relation
1+
e
q
0
2
q
0
,
=
ω
2
b
γ
a
a
−
b
+
γ
b
−
γ
a
(2-143)
a
γ
a
which should be verified by substitution. This is the rigorous form of an im-
portant approximate formula published by Clairaut in 1738. It is, therefore,
called Clairaut's theorem. Its significance will become clear in Sect. 2.10.
By comparing expression (2-141) for
γ
a
and expression (2-142) for
γ
b
with the quantities within parentheses in formula (2-140), we see that
γ
can
be written in the symmetrical form
γ
=
aγ
b
sin
2
β
+
bγ
a
cos
2
β
a
2
sin
2
β
+
b
2
cos
2
β
.
(2-144)
We finally introduce the ellipsoidal latitude on the ellipsoid,
ϕ
,whichisthe
angle between the normal to the ellipsoid and the equatorial plane (Fig. 2.11).
Using the formula from ellipsoidal geometry,
