Modern views on the determination of the figure of the earth - Physical Geodesy - page 310

since Q 0 and Q 00 lie on the same ellipsoidal meridian. Furthermore, even
in extreme cases the distance between Q 0 and Q 00 can never exceed a few
centimeters. For this reason, we may also set

ϕ = ϕ (8-82)

without introducing a perceptible error. Hence, we can identify ϕ and λ with
ϕ and λ , which are the ellipsoidal coordinates of P according to Helmert's
projection (Sect. 5.5). Therefore, we may replace the above equations for ϕ ∗
and λ ∗ by

h
R

ϕ ∗ = ϕ + f ∗

sin 2 ϕ,

(8-83)

λ ∗ = λ.

Introducing the deflection components according to Helmert's projection,
defined as

ξ Helmert =Φ − ϕ,

(8-84)

η Helmert =(Λ − λ )cos ϕ,

we see that they are related to ξ and η by the equations

h
R

ξ Helmert = ξ + f ∗ sin 2 ϕ,

(8-85)

η Helmert = η.

Therefore, ξ and ξ Helmert differ by the normal reduction for the curvature of
the plumb line,

h
R

δϕ normal = f ∗ − sin 2 ϕ. (8-86)

The deflection components ξ Helmert and η Helmert are used in astrogeodetic com-
putations; ξ and η are those obtained gravimetrically from formulas such as
(8-77) and (8-88) below.
These relations are mathematically quite analogous to the corresponding
equations (5-138) for the conventional method using the geoid, but now,
with the use of the normal curvature, the once formidable obstacle of the
correction for plumb-line curvature practically belongs to the past.

Remark on accuracy
With Molodensky's theory, the accuracy problem mentioned at the end of
Sect. 2.21 even aggravates, because in a mountainous terrain it is almost
impossible to compute the Molodensky corrections with an accuracy of 0 . 03
(say), so that these observations cannot be directly used for precise horizontal
positions.