and 
'u 
r 
x 
r 
X 
V 
= 
Y 
- 
Y 
w 
_Z_ 
Z_ 
j 
j 
u 
r 
'x 
r 
Y 
V 
= 
Y 
- 
Y 
_w 
_Z_ 
# 
_Z. 
j j j 
and 
is a scale-change for sub-block j 
(a, b, c). three small rotations (with the scale-change included) 
J about axes through the sub-block centroid and parallel 
to the general system 
(c 
X» 
C z ) . 
Z 3 
three translations of the centroid of sub-block j 
parallel to the general system. 
Formula 4.36 represents three observation equations for one point, 
r, in one sub-block, j. 
(4) For all N points in the sub-block j, 3N observation equations may 
be written in terms of seven unknowns (s, a, b, c, c x > cy, c z)j» 
Seven normal equations are formed from the observation equations using 
the principle of least squares. Because of referring the transforma 
tion to the centroid of the sub-block, it turns out that four out of 
(3 8 ) 
the seven normal equations contain only one unknown each * 
Consequently, the values for s, c x , Cy, and c z are immediately found 
and only three simultaneous equations need to be solved to determine 
a, b , and c. 
(5) After determining the seven transformation coefficients, new and 
improved values for the coordinates of all points in sub-block j are 
computed. 
(6) 
The various steps mentioned above for a sub-block, j, are performed 
for all sub-blocks in the project. This marks the end of one iteration 
and the situation reduces to the one before adjustment started, with 
the difference that the sets of coordinates for the same point being 
closer to each other and to ground control. 
21