hA 
By subtraction it is found : 
Ax1 — Axa — (ai — a) x? + {(d; — ba) + 2|yl1,2. (p1 + po) } 
: : + {(c1— co) + Inl1,2.(gı + qz) 
Likewise: m T6) 
Ay — Aya (n — P3)? -- ((431 — 9) — 2 |n|1,2- (a1 + aa} 
+ {On — 72) — lain a. (P, + 5) J 
d, b, c and f, q, r are parameters of transformation while Inlı,2 is also constant 
for all tie points. Therefore the above formulae (6) may be written as: 
A1 — A43 a1. 3? H- 4-2. x + y1—2 l 7 
Ayı — A ya = 71 —2.47 + K1—2.4 + p1_9 : à : : (7) 
The difference between the corrections to corresponding points in runs 1 and 2 
accordingly also follow a parabola. 
If the actual discrepancies (a; — 12) observed between adjacent runs do 
in fact follow a parabola, then formulae (7) becomes a condition equation in 
a, B and y. According to (2) one may write: 
01—2..47 ob fi-o-x + Y1—2 — AX1> l 8) 
712-4 + By 2-4 + M-2 = AYı 2 : : : : ( : 
The parameters of connection a, 8 and y may now be determined from the observed 
discrepancies A X (three being sufficient) and x, K and p from AY. 
There are (2 — 1) common edges in a block of » runs. Consequently (5 — 1) 
sets of parameters have to be solved in such a block. Following the substitutions to 
obtain formulae (7) from (6) we arrive at the simple relation between the para- 
meters of internal adjustment and the parameters of connection as set out below: 
041 —. do — 41—29 
(Io —- (13 um oS 
uus. Mh ton vor (9) 
Qj—1 — Ap = Q(n—1)—n 
Q4 d-d»4-43....4- a4 .14- a, — 0 
These form ; simultaneous equations of n unknowns. Actually there are only 
(2 — 1) equations whose right-hand members have been determined in an earlier 
stage. The last equation has been added to make the set complete. At first glance 
the condition represented by this equation may be somewhat obscure. Any set of 
values a^; (1 — 1 n) that satisfies the first (11 — 1) equations can be replaced by 
a set a; — a'; - K (K is a constant) which will also satisfy the same equations. 
Making the condition that the parameters a; should satisfy the last equation, we 
obtain : 
Sa’; 
i+ nn. K=00or K = ~— = 
Apparently only the absolute value of the parameters a; is affected by the additional 
equation. Whether in fact the true absolute value is obtained is beside the point. 
In this stage of the adjustment, only the discrepancies between runs will be elimi- 
nated and that depends solely on the value of a;_» or the difference a; — as, etc. 
It will often be found, however, that the overall distortion in the entire block will 
be greatly reduced by assigning the value zero to the right-hand member of the 
last expression. One can even make this plausible by assuming that the chances for 
a; to be positive or negative are equal, while for a great number of a; the mean will 
be zero .and therefore also the sum. 
Likewise it is found: 
fh ore = Try 
s 0nd T2--8 
(10) 
Pn=i = Pu —T(»—1l)—mn 
Pa pc... Pn—1 + Pn = 0