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where
X ij = X j − X i ,
Y ij
= Y j
Y i ,
(5-72)
Z ij
= Z j
Z i
have been introduced accordingly. The relation (5-71) may also be expressed
as
δs ij = X ij
s ij
δX i )+ Y ij
s ij
δY i )+ Z ij
s ij
( δX j
( δY j
( δZ j
δZ i )
(5-73)
if the differentials are replaced by differences.
Azimuths
Again the same principle applies: the measured azimuth A ij as a function of
the local level coordinates is given in (5-69). If n ij ,e ij ,u ij , the components
of x ij , are substituted by (5-65), the relation
tan A ij = e ij /n ij
(5-74)
X ij sin Λ i + Y ij cos Λ i
=
X ij sin Φ i cos Λ i
Y ij sin Φ i sin Λ i + Z ij cos Φ i
is obtained. After a lengthy derivation, the relation
δA ij = sin ϕ i cos λ i sin α ij sin λ i cos α ij
s ij sin z ij
( δX j
δX i )
+ sin ϕ i sin λ i sin α ij +cos λ i cos α ij
s ij sin z ij
( δY j
δY i )
(5-75)
cos ϕ i sin α ij
s ij sin z ij
( δZ j
δZ i )
+cot z ij sin α ij δ Φ i
+(sin ϕ i
cos α ij cos ϕ i cot z ij ) δ Λ i
is obtained. Approximate values are sucient in the coe cients , denoted by
ϕ, λ, α, z instead of Φ , Λ ,A,z .
Directions
Measured directions R ij
are related to azimuths A ij
by the orientation un-
known o i . The relation reads
R ij = A ij
o i ,
(5-76)
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