19 
whereto? dy Q , dc, dXo, dY 0 , dZ 0 , doo, dcp and dx are corrections to the corres 
ponding approximate values. Putting 
dx = x—Xq—c 
(X-X 0 ) 
(Z-Z 0 ) 
, (Y-Y 0 ) 
dy = y-y^-c 
5.5.3 
5.5.4 
dx and dy are then defined as measured value minus given value, the given 
value being computed from the co-ordinates of the test object and the approxi 
mate orientation elements. Adding linear terms from expressions 5.2.4, 5.2.5, 
5.3.1, 5.3.2 and 5.4.10 we obtain the following differential formulas: 
dx — dx o + 
A-A 0 
Z-Z 0 
c 
dc ——y yr dXo + 
Z — Z o 
X — A 0 
(Z - Z 0 ) 2 
c dZo — 
(A - A 0 ) (Y - Y 0 ) , , 
— —- c a to -r 
(Z — Z 0 j“ 
(X - Xo) 2 l 
(z - z 0 y J 
c dcp + 
Y — Y 0 A — A 0 
+ Ty 7y- c dx + — — c (<2 3 r 2 + fl 5 r 4 +....) + 
z — z 0 z — z 0 
+ p { (r 2 + 2 (x — x 0 ) 2 ) + 2/?2 (x — x 0 ) (j — J 0 ) 
5.5.5 
dy = dyo + 
1 + 
Y — Y(\ 
Z-Zn 
dc 
Y y 
dY 0 + — c dZ„ 
(y- 
- y») 2 l 
(z- 
- ZoH 
c dco + 
Z — Z 0 (Z-Z,)* 
(X-X„) (y-y 0 ) 
(Z - Z*)> 
A-Ao 
Z-Z 0 
c dx + 
c (ayr 2 + +....) + 
+ 2pi {x — *0) (y — Jo) + p2 (r 2 + 2 (j—Jo) 2 ) + 
+ (j — jo) dm + (x — x 0 ) dp 5.5.6 
where a 3 , a.-,, a 7 , ai, p\, p 2 , dm, dp are parameters to be determined together 
with the corrections to the orientation elements mentioned above. In this way 
it is possible to introduce more parameters in the mathematical model in order 
to determine regular errors and reduce the remaining irregular errors. 
If the angles eo, cp and * are too large to allow higher order terms to be neg 
lected, the expressions 5.1.1 and 5.1.2 can be linearized by numerical differen 
tiation. The partial differential of F (k) with respect to the variable k is com 
puted as