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of the net. Accordingly, the other triangles can be computed as in any other
height network (e.g., leveling net).
It is curious that it may be shown that such closures are mathematically
equivalent to the well-known relation
∂
2
N
∂x ∂y
=
∂
2
N
∂y ∂x
.
(5-124)
See also Sect. 4.5.
5.15
Reduction for the curvature of the plumb line
Motivation
The astronomical coordinates Φ and Λ, as observed on the surface of the
earth, are not rigorously equal to their corresponding values at the geoid
because the plumb line, the line of force, is not straight, or in other words,
because the level surfaces are not parallel. Thus, if we wish our astronomical
coordinates to refer to the geoid, we must reduce our observations accord-
ingly.
Examples of such cases are the following:
1. The gravimetric deflections have usually been computed by Vening
Meinesz' formula for the geoid, so that either the gravimetric deflec-
tions must be reduced upward to the ground point or the astronomical
observations must be reduced downward to the geoid, in order to make
the two quantities comparable.
2. If astronomical observations are used for the determination of the
geoid, the same reduction, in principle, must be applied.
Important remark
The principle of reduction of the plumb line is of fundamental theoretical
importance for understanding the geometry of the earth's gravity field. In
practice, it is usually disregarded if the topography is suciently flat, or
replaced by more sophisticated methods in mountainous areas, as we shall
see later (Sects. 8.12 and 8.13). The present section may be skimmed at first
reading, except for the
normal curvature of the plumb line
at its very end.
Principles
Consider the projection of the plumb line onto the meridian plane. According
to the well-known definition of the curvature of a plane curve, the angle
