239
q (yi.iki.i) -I- (91.2ai.2) ~l~ (yi.s«1.3) -T (<pi.»-i
1 n 1
where <px.! is the bearing at station I to orientation point x
91.2 is the bearing at station I to orientation point 2
etc.
ai,4 the observed direction at station I to the orientation
point i
n is the total number of observed directions at I, the
direction to P inclusive.
The bearing 91.P then becomes
<pi.p «I .p 0\.
This bearing is to be used for the adjustment of the coordinates
of P. The other bearings 911,p, 9111.P etc. are determined in a
similar way.
Correlation exists between the coordinates of all stations and
consequently also between the orienting bearings not only at the
station I but also at the other occupied stations e.g. 91.,- and 911.4.
Therefore theoretically the adjustment of taking the mean 01 of
all orienting angles <pi.i au of equal weights is a (very convenient)
practical simplification of an adjustment which should be one of
correlated observations with weight coefficients derived from the
weight coefficients of the coordinates.
The proper weight coefficients of the orientation bearings will be
derived and submitted to an investigation as to the conditions
under which they may be replaced by new weight coefficients
expressing equal variances and zero covariances.
Only if these conditions are satisfied the simple adjustment of
01 by taking the mean of all observed orienting angles would be
the most accurate indeed.
The bearing 91^ can be written as
Yi—Yi t
91.» arc tan x._Xj- (2)
After the expansion of this equation into a Taylor series the following
correction equation can be derived
<j>U an\Y\ -f- b\.iAX\ - auAYi b\,iAXj au vH i Öi (3)
where a and b are the so called direction coefficients
p" cos <t>U j p" sin <j>u
«(.i «I.» -~jand b,.i =—bu r-
»I.i dl.i
d\,{ being the distance from station I to point i. Similar equations
can be derived for the stations II etc.
Putting
Vi i «uAYi bj.iAXj T" ai.iAYi bnAXi -\- (4)
in (3) gives:
V\.i au <|>i.»- (5)