If the precision of the known coordinates x: is described by the variance 
; ; li : 
covariance matrix G , the variance covariance matrix G of the constrained 
e 
coordinates Xe is derived by application of the error propagation law on (2.10). 
2 - 
cil ct I 0 cl! 0 : c ci? 
C Cod. 4 K f f (2.11) 
c? c?? Zit! I 0 622 0 ; 
C Cc j f =F f 
Comparing (2.11) and (2.5) one observes a strong similarity: in (2.11) only en is 
to be replaced by G. to get (2:5). 
This means amongst other things, that comparing the variance covariance 
matrices of the free coordinates and the constrained coordinates can be reduced 
to the comparison of si and o . This reduction of problem is the 
background of the statement in the introduction implying the possibility to predict 
the precision of the constrained adjustment from the results of the free 
adjustment. 
In addition, because of this similarity between (2.5) and (2.11), we will be able to 
deduce in section 4 an artificial variance covariance matrix for constrained 
coordinates from the artificial variance covariance matrix of free coordinates. 
3. COMBINED ADJUSTMENT 
Most adjustment software does not compute constrained coordinates and their 
precision as described in section 2. 
Sometimes a free adjustment is actually carried out, because of the testing, as 
stated in the introduction, but the second phase as such is generally not 
programmed. For reasons of software development the constrained adjustment is 
carried out as a combined adjustment, in other words an adjustment in one step. 
This approach is applied in the Dutch computer programms as well ( /4/,/10/ ). 
Next to the observations | from the free adjustment we now also deal with obser- 
vations xK the coordinates of the known points (control points). The free adjust- 
1 
ment correction equations (2.1) are therefore extented as: 
2 
+ AX 
1] = AX 2 
i (3.1) 
QO r^ Oo tz: 
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