can consequently be regarded as a “mathematical model" and the results of the
measurements can be controlled by the condition that they shall describe a plane.
From the discrepancies of this condition a statistical value of the errors of the
measurements can be computed. Evidently it is desirable to use a great number
of measured points. Using the three parameters one translation and two rota-
tions in connection with the determination of the standard error of the measure-
ments the number of redundant observations will be n—3 where n is the number
of measured points.
FiG. 2. The distribution of the standard errors of the final elevations within the surface
to be tested, after orientation of the surface to contain the points 1, 2 and 3. The standard
error of unit weight of the elevation measurements 1.
The number of settings in each point must be the same as during ordinary
practical measurements of surfaces to be determined (for instance three settings).
The discrepancies between the averages of the settings and the corresponding
errorless elevations of the surface plate are adjusted according to the method of
the least squares. The remaining discrepancies after the adjustment are used for a
statistical determination of the standard error of unit weight of the measurements.
The procedure is founded upon the differential formula (1).
4.1. The Procedure
We assume 7 points in the surface plate to have been measured. The positions
of the points are defined Ly the co-ordinates x and y in an arbitrary co-ordinate
system.
The task is to determine such values of the corrections dhy, dn and dé that
the square sum of the residual discrepancies in all n points become a minimum.
For such purposes it is suitable to transform the plane co-ordinates x and y
into a co-ordinate system, the origin of which is located in the point of gravity of
the points. This co-ordinate system is denoted X, Y.