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f
andκ. Solving (5.94) forκ, and substituting the solution in equation (5.93), yields
the quadratic equation in the flattening,
1
m
1
f
1
3
2
J
2
5
8
J
4
f
2
2
m
1
+
−
+
+
+
=
0.
(5.95)
Solving the quadratic, we find that
1
2
J
4
2
5
m
1
−
m
2
2
f
=
1
+
−
6
J
2
+
−
,
(5.96)
1
selecting the negative root as the only one admissible, since
f
is first order in small
quantities. To second order, we have
1
2
m
1
2
J
2
1
2
J
2
m
1
2
1
3
3
5
8
J
4
.
f
=
−
+
+
+
(5.97)
κ is then computed directly from (5.94). Neglecting the departure from ellipsoidal
shape, as is conventional in geodesy, permits determination of the flattening,
f
e
,
from
J
2
alone, through the quadratic
2
7
m
1
f
e
9
f
e
−
m
1
+
+
+
3
J
2
=
0.
(5.98)
Again, selecting the negative root as the only one admissible, we obtain the second-
order formula
1
28
m
1
2
J
2
1
7
m
1
m
1
2
11
3
3
4
J
2
1
f
e
=
−
+
+
−
.
(5.99)
The history of the development of geodetic reference systems is described by
Hofmann-Wellenhof and Moritz (2006, pp. 83-90). The Geodetic Reference Sys-
tem 1967 (G.R.S. 1967) of the International Association of Geodesy (IAG, 1971)
is based on adopted values of
J
2
,
a
,
GM
and
, as is its successor, the Geodetic
Reference System 1980 (G.R.S. 1980). Both of these systems of adopted values are
based on an ellipsoidal reference surface. The GRS 1980 was the basis of the World
Geodetic System 1984 (WGS 84), a Conventional Terrestrial Reference System
(CTRS). A further terrestrial reference frame called the International Terrestrial
Reference System (ITRS) (see Section 4.1) was developed by the International
Earth Rotation and Reference Systems Service (IERS). Table 5.1 shows a compar-
ison between the adopted values and results compiled by Rapp (1974). Also shown
are values of
f
,
f
e
, κ and other parameters computed from the three basic data sets.
Once
f
and κ have been computed, the mean equivolumetric radius can be cal-
culated using the series (5.83). Then
m
can be computed from its definition (5.7) or
Ω
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