( 143 ) 



f— the apparent error calculated for this result, 

 n = the number of errors out of the group, 



k = a number depending on the weights of the results of measure- 

 ment and on the coefficients the unknown quantities, determined 

 by the equating, are associated with in the equations expressing 

 the connection between these unknown quantities and the results 

 of measurement of the group. 

 In what way k is dependent on the above-mentioned quantities 

 will become clear by an example for which I choose the case that 

 3 unknown quantities x, y, z are determined by N equations of the 

 form ax -j- by -\- cz = /, whilst to the quantities / appearing in this 

 equation and obtained by measurement the weights g are due. In 

 this case 



k = 2 g (a* Q xx + 2ab Q xy -f 6 3 Q yy + 2ac Q xz + 26c Q ys + e" Q zz ) 



where the summations in the formulae for k and (i include expres- 

 sions relating to the same results of measuring. In the above for- 

 mula the quantities Q. the well-known numbers of weight, can be 

 calculated by means of the coefficients of the normalequations. 



For the deduction of this formula we have the same considerations 

 which lead to the mean error of the unity of weight out af all 

 observations. If the real errors are indicated by h then wfi* = 2 qh 1 • 

 this sum is expressed in the apparent errors that can be calculated, 

 and in the errors Ax, Ay and Ac of the quantities x, y and z, 

 calculated out of the normal equations, by means of the relation 

 h= ƒ-{- a Ax + bAy -\- cAz, so that 



n^ = 2 of -f 2 (A,v 2 gfa + Ay 2 gfb + Lz 2 gfc) + 



+ (A*y 2ga* + 2 (Ax) (Ay) 2' gab + (Ay)* 2' gV + 



+ 2 (Ax) (Az) 2gac + 2 (Ay) (Ac) 2 gbc + (Az) 1 2 gc' 



If we were to use the whole material of errors, then the first 

 three of the unknown terms would fall out on account of [gfa'] = 0, 

 [gfb] = and [gfc] = 0. (Here and for the future I make use of [ 

 as sign of a summation extending over all observations). To take as 

 well as possible the unknown terms in the above into account we 

 replace them by their mean values in the supposition that the same 

 complex of observations repeats itself manifold times so that all 

 calculable quantities return unmodified in each repetition. In that 

 supposition Ax, Ay and Az have zero as mean values and the mean 

 values of their squares and products are in the above order : 



Qxx f*' . Qxy H* i Qyy ft" i Qx: ^ i Qyz H* and Q, z fi 1 



If w r e connect these mean values having f* 1 as factor with the term 



