Geoscience Reference
In-Depth Information
For reducing astronomical observations (Sect. 5.12), we need only the
projection curvatures (2-50) and (2-51).
We mention finally that the various formulas for the curvature of level
surfaces and plumb lines are equivalent to the single vector equation
2
ω
2
)
n
+
gκ
n
1
,
grad
g
=(
−
2
gJ
+4
πG
−
(2-53)
where
n
is the unit vector along the plumb line (its unit tangent vector) and
n
1
is the unit vector along the principal normal to the plumb line. This may
be easily verified. Using the local
xyz
-system, we have
n
=[0
,
0
,
1]
,
n
1
=[cos
α,
sin
α,
0]
,
(2-54)
where
α
is the angle between the principal normal and the
x
-axis (Fig. 2.6).
The
z
-component of (2-53) yields Bruns' equation (2-40), and the horizontal
components yield
∂g
∂x
=
gκ
cos
α,
∂g
∂y
=
gκ
sin
α.
(2-55)
These are identical to (2-50) and (2-51), since
κ
1
=
κ
cos
α
and
κ
2
=
κ
sin
α
,
as differential geometry shows. Equation (2-53) is called the
generalized
Bruns equation
.
More about the curvature properties and the “inner geometry” of the
gravitational field will be found in topics by, e.g., Hotine (1969: Chaps. 4-
20), Marussi (1985) and Moritz and Hofmann-Wellenhof (1993: Chap. 3).
z
plumb
line
n
1
sin
®
y
®
1
n
1
x
Fig. 2.6. Generalized Bruns equation
