WRIGHT AND VAN ORSTRAND : MINERAL ANALYSES 229 



when we assign equal weight to all the observation equations. 

 In Schaller's analysis, the factor 



\fy.r>j 049706 



^ 2009.61 ^-^^^^^^ 



which is only slightly different from the factor m obtained by the 

 least square method. To find the required value for S in the 

 above analysis by this method, we multiply the weight number 

 352.79 (column 4) by the factor m' = 0.049706 and obtain the 

 value 17.54 listed in column 6; the remaining values of column 

 6 are obtained from column 4 by multiplying by the same factor. 

 To impose the conditions (1) that the sum of the residuals 

 shall vanish and (2) that the sum of their squares shall be a 

 minimum, we write the observation equations in the general form 



y ^ a -\- mx, 



and adjust them by the usual least square method. There seems, 

 however, to be no justification for adopting such a procedure here. 

 Fourth, fifth and sixth ynethods. These methods are identical 

 with the three preceeding methods except that the adjustments 

 are made between the mol numbers and the molecular ratios. 

 Thus the mol numbers may be taken as the observed values {y) 

 and the molecular ratios as the theoretically correct values (x) 

 to be used in the observation equations (1). Adjustment can 

 then be made by any one of the methods described above. Thus 

 in method (4), the factor m of observation equations (1) is found 

 from equation (3) by considering the mol numbers (column 2) 

 the {y) values and the molecular ratios (column 3) the {x) values. 

 The final values are then obtained by multiplying the {x) values 

 (column 3) by the factor m thus computed. In method (5) the 

 same values of {x) and {y) are used and the proper value for m 

 computed from, equation (4) above. Similarly in method (6) 

 the proper value for m' is computed from equation (6). The 

 results obtained by these three methods are, however, theoreti- 

 cally inferior to those of the first three methods, for the reason that 

 the mol numbers are obtained from the weight percentages by 

 dividing them by the proper molecular (respectively atomic) 



