( 144 ) 



np* in the above equation, if we put 



2 g (arQxx -\- 2abQ.vy -\- b"Qyy -\- 2aeQxz -f- 2bcQyz -f- c"Qzz), 

 equal to k and if we solve \i out of the equation, we obtain 



H=\/ -, i. e. the formula of which I made use to determine 



Y n - k 



the mean error of the unity of weight out of a particular group of 



apparent errors. 



I arrived at about the same result by another consideration putting 

 to myself the problem to determine the mean value Af /t of a definite 

 apparent error /,-. In the relation : 



fi = k — <H & — h y — a z 

 I substituted for x, y, z respectively [«/], [,?/], [//] to obtain ƒ,■ in 

 the form of a linear expression of the results of measuring / which 

 are supposed to be quite independent of each other. 



Then : 



BPft = |l-2 (a; a { -f- b t ft -\- a Tl )|- + 



9i 



(at a + bi + Oi y) 2 - 

 9 J 



It would now be the only way of reduction of this equation to 

 make use of the well-known relations existing between the coefficients 

 «, (i, y and a, b, c and the numbers of weight Q, namely : 



« = g (a Q.v.v -\- b Qxy -f <■' Qxz) , 



P — 9 (a Qxy + b Qyy -f c Qyz) , 



y = g{a Qxz -f b Qyz 4- c Qzz) , 

 in order to prove that 



[ 



a{ a + b t fi -f ci yy - 



9A 



<*i «« f hi ft 4- 'ï 7i 



I propose however to deduce this equation directly from the 

 minimum condition : 



[g (I — ax — by — c^) 3 ] = minimum. 



If here too x, y, z are replaced by [«/],- [j?/] and [y/], then after 

 calculation and combination of the equal powers and products of the 

 quantities / an expression appears of the form 2 2 C' ;] l ;j . U , having 

 for the right set of coefficients a, ft y a minimum value. I observe 

 here that the coefficients a, ft y have to satisfy the minimum condi- 

 tion independently of the particular values which the measurements 

 furnished for the quantities /. Out of this observation ensues that 

 the partial derivatives of Cjj. with respect to each of the coefficients 

 a, ft y furnish zero by substitution of the right values of these 

 coefficients. 



