520 VAN ORSTRAND AND WRIGHT: MINERAL ANALYSES 



If now y\ = most probable value of the unknown quantity Zi 

 y'2 = most probable value of the unknown quantity Z2 



y'n = most probable value of the unknown quantity Zy, 

 c? = 100 - (1/1 + 2/2+ . . . +2/J, 

 then we have from Gauss' method of correlatives 



y'n = Vn + Vn «'„ = - '~-l 



Vn d 



\ (14) 



Equations (1) to (14) express in concise mathematical form 

 the different least square equations for the adjustment of the 

 observational data given in a chemical analysis. They can be 

 applied directly to the discussion of a particular method of ad- 

 justment by a proper assignment of the weights involved. Let 

 us now examine the methods of Schaller and Wells in the light 

 of the above equations. 



Schaller begins his computation by dividing the observed weight 

 percentages by the atomic weights. The weights of the mol 

 nmnbers thus obtained are therefore in accordance with (12) 

 proportional to the squares of the ratio of the atomic weights to 

 the probable errors: 



Pi '■ P2'- • • • ' '• Pn = ^^ • ^ „2 



'1 '2 'n 



in which Ai, ^2, • • A „ are the atomic weights and ri, ra, . . . r^ 

 the probable errors of the original analysis. 



He then employs equation (5) with the computed mol num- 

 bers as the X values and the theoretical molecular ratios as the 

 y values thus adopting two systero.s of independent weights, a 

 procedure which is not necessarily justifiable. A third system of 

 weights is introduced in reducing these numbers to approximate 



