Therefore the complete relations between the coordinates of the 
fiducial marks in the frame and in the image are 
X = X 0 + 
J x cos<p + 3; sinw sin?> + h coso sin?> tan?>J 
1 X sin?) + y Sino COS(p + h COS«) cos<p COSw) 
3; coso — h sin«» ) 
c co Sor 
+ tan<p, c sinor 
F- F 0 + 
\— X sin?> + 3; sin«) COS?> + h COS«) COS?) 
[ x cos?> + y sin«) sin?> + h cos«) sin?) tan«)| 
I X sin?) + y sin«) COS?) + h COS«) COS<p COS«)! 
C Sinor 
+ 
31 cos«) — h sin«) 
: : : ; : -f tan«) | c cosor 
X Sill?) + 3; Sin«) COS if + h COS«) COS?) J 
Each of these expressions are expanded as a Taylor’s series with the 
derivatives taken at zero for the rotations and at h for the principal 
distance. As the deviations from these values are always small it is 
only necessary to consider the first order terms to obtain the following 
differential formulas: 
X^ 
xy 
dX = dX 0 
+ 
dr ~ 
h 
do) — 
y da 
+ 
xy 
y~ 
0 
II 
+ 
~h d * - 
h 
de» + 
X da 
+ 
— X image % frame < d4 — ^ image y frame • 
X 
h 
y_ 
h 
dc 
dc 
As there are redundant observations the system of equations is solv 
ed according to the method of least squares which, for the fiducial 
marks shown in fig. 1, gives: 
dX 2 + dX 4 dY + dY 3 
dX0 = , dY 0 = 
da = — 2 + b2) [a(dY! - dY 3 ) - b(dX 2 - dXJ] 
h 
dip = ( dx i — dX 2 + dX3 — dX 4 ) 
h 
dc0 = — (dY t — dY 2 + dY 3 — dY 4 ) 
dc = 2 \J+ b2) [a(£/Zl ~ dXs) + b(dY * ~ dY ^ ] 
471