The reliability of the determination of s, is consequently increased with the number 
of measured points, which is self-evident. 
For n = 20 we find, for instance, 
So = 0- I 150, 
which means that the standard error of s, is about 17 per cent of s,. 
If the accuracy of the basic measurements is determined in this way, evidently 
the results are founded upon the method of the least squares and can be regarded 
as unique. All small errors of the measuring device are included in the results 
and also possible errors of the surface plate itself. The results are evidently more 
reliable than those which would have been obtained from repeated settings only. 
The procedure can be regarded to be absolutely objective, particularly if small 
quantities of the translation and rotations can be introduced before the basic 
measurements. Probably there are specific values of s, for each combination of 
measuring gauge, surface plate and operator. From the Diagram ! and the actual 
value of s, the standard error of the residual deflections of the surface can be deter- 
mined. From a comparison between the standard error of the residuals and the 
residuals themselves according to the expression (6) the significance of the residuals 
can be judged, whether they are due to the errors of the measurements only or to 
real deflections of the surface. A certain confidence-interval must then be used. 
Frequently the rule is used that three times the standard error is regarded as the 
maximum value of the measuring errors. The frequency of this value is about 
1 : 370 for a normal distribution of the errors, provided » > 30. 
Consequently, if the residual discrepancy in a certain point amounts to a 
value which is three times as large as the standard error to be expected in the point 
the deflection of the surface from the plane 1-2-3 is to be regarded as significant. 
Sometimes 2 and 2-5 times the standard error is used for the judgement of signific- 
ance. For the factor 2 the frequency of the maximum value is about 1 : 22. The 
laws used for error propagation can be tested in an easy way. 
We assume the measurements of points in the surface plate to be corrected 
with respect to three control points only according to Fig. 1. The residuals v are 
determined individually from the expression (27) and the mean square value is 
computed. This value shall then, according to (24a), amount to about 1, 3 So. A 
number of practical tests has proved this theoretical relation. 
5. THE MICROSCOPE METHOD FOR THE MEASUREMENT OF FLATNESS OF 
SURFACES 
In some cases the surface, the flatness of which is to be measured. is so sensitive 
that considerable deformations will be caused even at the slightest touch of the 
measuring device, for instance the measuring gauge. Glass plates have proved to 
be very sensitive for any touch of the measuring device. For investigations of the 
flatness of negative and diapositive glass plates in photogrammetry it became 
necessary to use a method for the measurements which did not require any touch 
at all. The use of a microscope proved to give the solution. The principles are 
very simple, see Fig. 4 
The small depth of focus of the microscope is used for the setting in arbitrary 
details (contrast differences) in the surface. The accuracy of the settings increases 
with the enlargement of the microscope. Since the range of the settings (the depth 
differences to be measured) becomes smaller for increasing enlargement the enlarge- 
ment must be chosen as a compromise between the two mentioned requirements