(Ca,?Yi Ca<2*i)2 a2- (x3)
245
In order to derive the weight coefficient of an orientation bearing
for the computation of 0\ the law of propagation of errors is applied
to the simplified equation (4)
V\ i a\ ;AYi bi iAXi. (14)
This gives in symbolical notation:
Qvi.i «i.;<2y'i h.iQx1 Qaii (15)
or worked out
Qvl.ivl.i a2l.iQyjYi 2aI.ibl.iQxY ^I.iQxjXi
The covariance between the bearings i and j at I is given by
QvjVj al.iaI.iQxiYi («I.i&I.i OLl.jbii) <?XiYj h.ih.jQxiXi- 0-7)
The question now arises whether there are any specific orientation
bearings 9j.j of length dx i satisfying the centroid condition ca
c/, o in formula 13), for which the assumption of equal weights
and freedom of correlation, is truly valid. Then 0$ would attain
a minimum.
The is now left out of consideration. An investigation
shows that there is a unique solution for two orienting directions
only.
We put in this case, omitting the index I for convenience
a\Qyy 2«1J1 QXy b\ Qxx p (ïSa)
a\ Qyy 2«2^2 Qxy "t" ^2 Qxx P (I8t>)
and axa2QYY (aA a2^i) Qxy ^1^2Qxx 0 I1®0)
Equation 18c can also be written
cos 9j cos <?2QYy (cos 9i sin <p2 cos 92 sin <Pi) Qxy
sin 9j sin 92QXx 0 (x8d)
from which it is apparent that the freedom of correlation is depend-
enl only on the bearings and not on the lengths of 1
From the equations (18) it is derived that
Qxx r a\
Qyy C b\ b\
Qxy n A a2b2
Qyy b\ b\ (20)
Expressing the centroid conditions as
al ®2 2«i.p A (21)
and b1 b2 2 blP B (22)
it is found by solution of (19), (20), (21) and (22) that