6 ANALYTICAL AERIAL TRIANGULATION, THOMPSON 
that have not been explicitly mentioned by Schut. Using Schut’s notation, the condition 
equation is, 
B*XX,-Xj1^70 
but this equation does not in any way decide for us what our unknows are to be. The 
problem is perhaps made more explicit if we express this equation in a matrix form. It 
was pointed out in a recent paper (Thompson 1959 b) that the above vector equation can 
be written in the form, 
(X, Yi Zi 41) 0 B. By X, =0 
B, 0 —Bx Y; 
(B. By 0 E Z; 
and, if the rotations of the two pictures are represented by orthogonal matrices R; and 
R;,,, this equation becomes 
(9j 1 Yırı %i41) R7, 0 -—B, By\R; X 29 
B, 0 —Bx V; 
—B, By 0 \ +, 
where the small letters indicate observed coordinates (plate coordinates). 
This equation contains apparently nine unknowns, viz. the three base components 
and the three independent elements of R; and R,; , ,. Now the coplanarity condition allows 
us to find only five unknowns and these five may be selected in a number of ways, of 
which two only seem to be of practical use. We may regard the base as the X-axis of the 
coordinate system and make the base components (1 0 0). The unknowns remaining are 
then the six independent elements of the two orthogonal matrices. But of these six it is 
possible to find only the difference of two of them, leaving five to be found. It is this 
selection that was made in the paper quoted above (Thompson 1956) and now adopted 
by the Ordnance Survey. 
I do not now consider the selection the most efficient. It requires the construction of 
two orthogonal matrices at each iteration though this has been simplified by the adoption 
of the Rodrigues parameters as unknowns. (Thompson 1959a). A more satisfactory selec- 
tion is that used by Schut and proposed in Thompson 1959b which leads, following a 
suggestion by Mr. Howell of the Ordnance Survey, to some economy in arithmetic. 
In this selection, R; is supposed known. It can be taken either as the unit matrix or 
as the matrix determined from the previous model, there being some slight advantage in 
taking it as the unit matrix. The equation being homogeneous in the base components, 
we can divide out by By which is not small and write f, — By/By, fg- B,|By. Also 
7; — 2,,, — f and we can divide out by this. The equation now becomes, 
(Fy Yip RE, / 0 —fz Py XA =0 
Bz 0 —1 V, 
s 1 0 1 
in which there are five unknowns; and there is only one orthogonal matrix to be evaluated 
at each iteration. 
Now the raw materials of an iterative process are the residuals of the condition 
equation after a trial solution; and these residuals have to be evaluated at every point 
used (say six in all). Some attention should therefore be paid to the ease by which the 
residuals can be calculated; and the above equation is remarkably convenient in this 
respect. The unknowns appear in adjacent matrices which can be multiplied together 
once and for all for the whole model at each iteration. This involves only twelve multi-