12 ANALYTICAL AERIAL TRIANGULATION, SCHUT 
stipulates that the vectors from the projection centres of two photographs to corre- 
sponding image points in these photographs must lie in one plane: 
B XX; Xit1= 0 
The same equation is used in the method developed by Dr. H. H. Schmid at the U.S.A. 
Ballistic Research Laboratories. 
b. The British Ordnance Survey Method is a special 6-point case with a pre-deter- 
mined solution derived from the stipulation that when triangulating in the X-direction 
the Y-parallax between corresponding rays must be equal to zero. The Y-parallax is 
defined as the Y-difference of points on these rays at the height where their X-difference 
is equal to zero. This gives the condition equation 6b in reference [1] which can be 
written: 
1 
XZ, 1— Xi 
c. The Herget Method stipulates that the scalar minimum distance between cor- 
responding rays must be equal to zero: 
ga Xe Xam? 
1 
. . 0 x 0. I 
sin 0 Ba T 
In these equations, 
B is the vector which connects the two projection centres, 
X; and Xii are the vectors from projection centre to image point in photo- 
graphs i and i41, 
Vo, and VO, are unit vectors in the direction of X, and X;,, respectively, 
X, Z, X;,1and Z,,, are the X- and Z-components of X, and X;,, with respect to a 
right-handed orthogonal coordinate system, and 
0 is the angle between X, and X; ,. 
It is obvious that the second and third equations are more complicated than the 
first because they possess, besides a scalar triple vector product, an additional factor 
which is a function of the photograph coordinates and the orientation of the photographs. 
In aerial triangulation, since corresponding rays cannot be parallel, these additional 
factors are never equal to zero. Therefore the second and third equations can be satisfied 
only by making the scalar triple vector product equal to zero. 
Obviously then, if no redundant observations are used, the additional factors do not 
affect the triangulation result. If redundant observations are used the method of least 
squares gives a result that depends only upon the weights and correlations, attached to 
the observations. The result does not depend upon the actual form of the equations by 
which the conditions are expressed. In particular, here, the result does not depend upon 
the use of any non-zero factor by which a condition equation is multiplied. 
Consequently the additional factors do not affect the triangulation result at all and 
they must be omitted from the condition equations as they are non-essentials which only 
add to the labour of computation. 
On omitting these factors the second equation reduces to the first one. The third 
equation still differs from the first one because of the use of unit vectors instead of 
vectors from projection centres to image points. The difference amounts to a factor which 
is constant for each pair of corresponding points, independent of the orientation of the 
photographs. As in the case of the above factors, this factor cannot affect the trian- 
gulation result. Also, the measurement of photograph coordinates gives the vectors from 
projection centres to image points in the first instance, while the conversion to unit 
vectors requires additional computation. Therefore the use of unit vectors should be 
omitted as a non-essential addition to the labour of computation.