6 
The usual and most convenient condition is that the sum of the squares of 
the residuals shall be as small as possible. This is the principle of the method 
of least squares, which is recommended in the resolutions of Commission II 
as quoted above. A priori weighted observations can be introduced according 
to circumstances, see for instance Helmert 1907. 
The minimized sum of squares of residuals is an indication of how well the 
assumed number of regular errors, used in the computations, compensate the 
observed discrepancies. The square root of the sum of squares after division 
by the number of redundant observations (the degrees of freedom) is the 
statistical expression for the basic quality of the individual observations after 
elimination of the regular sources of error. 
This expression is denoted the standard error of unit weight and indicates 
the basic geometrical quality of the instrument in which the measurements 
were made, and after elimination of the actual regular errors. The reliability 
of the determination of the standard error of unit weight is of great importance. 
The fundamental factor for this reliability is the number of redundant observa 
tions (discrepancies) in the adjustment operation. The more observations, the 
higher reliability. An expression for the reliability is the standard error of the 
standard error of unit weight, see Helmert 1907. 
The instrument tolerances to be derived will also, among other things, be 
dependent upon the degrees of freedom in the determination of the standard 
error of unit weight of the basic observations. In general, it is desirable to use 
many, at least ten, degrees of freedom in the instrument tests. 
As part of the adjustment procedure, the covariance matrix 1 (the matrix 
of the weight coefficients) of the parameters should always be determined. 
This matrix contains in the principal diagonal the weight numbers which can be 
used for the determination of the standard errors of the parameters i.e. of 
each regular error which has been determined through the adjustment opera 
tion. Such standard errors are determined as the product of the standard 
error of unit weight and the square root of the corresponding weight numbers. 
The standard error of any function of the parameters is finally found with the 
aid of the general law of error propagation, in which the correlation numbers 
(located outside the principal diagonal of the matrix of the weight coefficients) 
also must be used. In this way the standard error of the individual residuals 
after the adjustment can be computed because they are functions of the 
adjusted parameters. 
A rather complete and detailed example of the procedure described above 
has been given in Hallert 1963 a for the test of a comparator with the aid of 
measurements of grid points, the coordinates of which are given with high and 
known geometrical quality. 
*) The covariance matrix usually includes the square of the standard error of unit weight 
(the unit variance).