Lh 
II. Now the observed discrepancies — (#3 — #2) and — (y1 — yz) are plotted 
as points in a graph.against x, and an attempt can be made to draw a parabola as a 
mean curve through these points. It is difficult to see whether the mean curve drawn 
is a true parabola, particularly when a fairly large scale is chosen for the graph. 
For this reason the parameters of the function a1 _2.4? + B1_2.% + y1_9 = AX >» 
are determined from five points by the method of least squares. The values for AX 
are taken from the provisional curve for five imaginary tie points at regular intervals. 
Their relative positions can easily be expressed in an arbitrary co-ordinate system, 
leading to abscissae x^— O, 1, 2, 3 and 4 respectively. 
The relation between the two systems is: 
X4 — x, 
X (17) 
in which x, is the abscissa of the first of these five points and A the interval ex- 
pressed in units of the x, y co-ordinates. This leads to very simple coefficients in 
the condition equations 
a. {XP +B. (FN) +v=D0Xi+e(j=1-5)"." ." . (18) 
and consequently also in the normal equations which are easily solved. 
The same applies to the condition equations 
(445 HK 1) tp AY Hey > $9 . .— (9) 
After solution of the normals the residuals ex, and ey, give an indication of how 
closely the true parabolae follow the provisional curves for AX and A Y. The 
departures of the plotted positions in the graph from the computed parabolae are 
equal to the residual discrepancies after internal adjustment. 
The principle, therefore, of this approach is that the computations are readily 
completed while already at this stage it can be seen whether the adjustment is going 
to be successful in terms of small residuals. It is also emphasized that the technique 
of the least squares in itself at this stage does not automatically give the best answer. 
À certain amount of judgement is required in drawing the initial curve, which opens 
up the possibility of drawing another curve in an attempt to obtain a more accept- 
able pattern of residuals along the entire length of the curve. 
Similarly the parameters a, B, etc, for the line of tie points between runs 2 
and 3 are determined. By taking the five imaginary points in the same cross-sections 
as the points for runs 1 and 2 the same relation exists as given in formulae (17). 
Consequently the coefficients of the observation and normal equations are the same 
and the solution becomes a routine computation. The usual checks are carried out 
to ensure that the solution of the normals is correct. 
III. After solution for the parameters of connection a, f, etc., the parameters 
a, b, c, etc., for the actual correction of the strips must be determined from equations 
(9) to (14). For this it is necessary that the ordinates of the tie points have the 
same absolute value in two adjacent runs (see formulae (6) ). In practice this will 
seldom be the case, as the runs are not exactly situated as equal distances apart. 
The transformation formulae (5) are, however, invariant to a co-ordinate transla- 
tion and it is therefore possible to adopt another centre line for the adjustment of a 
strip which does not coincide with the longitudinal axis as indicated by the nadir 
points. 
Suppose that in a block of three runs the tie points and pass points have ap- 
proximately the following y-values after connection of the runs at two points (see 
Fig. 3) 
By adopting the mean of 300 and 700 it is found y — 500 as the centre line for 
run 3. The ordinate of the tie points in run 3 is 200. Giving these points the same 
absolute ordinate one finds y — 900 (— 700 4- 200) as centre line for run 2. 
Similarly the centre line for run 1 works out to be y — 1,100. a, 8, y, etc., have 
— T» The values for 
x = 
  
: ; : x 
been determined in terms of the arbitrary ordinates x” = 
4 
n to be entered in the solution of equations (9) to (14) are then for the above 
case of three runs: