Distance equation
Following the three steps above the distance between two points i and j can be related to their eastings and northings by the equation
Differentiating with respect to the parameters gives:
In the four terms on the left-hand side of the equation, the coefficient may be re-expressed as a trigonometrical function of the direction of point j from point i. δEj is the correction to the provisional value of Ej , etc. On the right-hand side of the equation {(Ej−Ei)2 (Nj−Ni)2}12 is the distance lij derived from provisional values of the parameters. The right-hand side could now be written as lij(o−c) where the notation indicates that it is the observed value of the observation minus the computed value of the observation.With a change of sign and re-ordering the terms, the equation may now be more simply expressed in matrix notation as:
where aij is the bearing of j from i and is found from provisional values of coordinates of i and j:
If the units of distance are out of sympathy with the units of the coordinates, and the difference in scale is not known, it can be added to and solved for in the observation equation. This situation might occur when computing on the projection, or when observations have been made with an EDM instrument with a scale error, or even with a stretched tape.
Distance with scale bias equation:
The distance with scale bias equation is:
where s is the scale bias in parts per million (ppm).
The final equation above may now be replaced with:
