164 E. LANCASTER JONES 
polygons will be absent. The increments (pg), (PQ) are of course 
algebraical quantities, so that (pg) = —(qp), (PQ) =—(QP). 
If we assume that the polygonal circuits such as OPQR are always 
traversed in the same direction, e.g., clockwise, then (PQ) occurs posi- 
tively in OPQR, but negatively in SQPT. 
For such a link as (PQ) there will result a normal equation 
(PQ) = (£9) — mr + ma, (x) 
where m, and m, are the multiplier correlates attached to the polygons 
OPQR and SQPT respectively. 
Similarly for every other link, except on the boundary of the net- 
work where one of the m’s will be absent. 
Since the rigorous equations continue to be satisfied, we can re- 
place every term such as (PQ) in them by its equivalent in the above 
equation. We thus get a series of equations, one for each independent 
polygon, of the form (see Figure 1). 
(0p) — mM, +m,+ (pq) Se Mm, (qr) 
— m, + my, + (ro) — m +m, = 0, 
where the polygons x, y and z as well as s also touch polygon 7 at 
OP, QR and RO respectively. 
Thus, for a four-sided polygon 7, having contacts with four others 
x, S, yand z, we have an equation 
4m, — Mz — My — My — M, = (op) se (pq) oF (qr) + (ro) = d,, 
where d, is the observed or computed excess around the polygon. 
Similarly, for an m,-sided polygon 7, which has contacts with adja- 
cent polygons x, s, y,- - - , we have an equation 
N,N, — Mz— Ms — My — ::: = d,. (2) 
There is one such equation for each independent polygon of the 
network, so that, for the set of unknown multipliers or correlates m, 
we have an equal number of linear equations such as (2). From these 
we can obtain uniquely each of the correlates m1, mo, ---, and by 
using equations (1) we can then obtain quite simply each adjusted 
link increment (PQ). The whole procedure thus hinges on the solu- 
tion of the set of correlate equations (2). As has been stated and 
demonstrated above, these can be written down for any network by 
inspection. 
§4. SOLUTION OF THE CORRELATE EQUATIONS. APPROXIMATIONS 
The series of correlate equations of the type 
N;m, — Um, = d; 
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