228 WRIGHT AND VAN ORSTRAND : MINERAL ANALYSES 



^ _ x^yl-\^x^y2+ . . . + a^n^n 



7/2] — ■ 



0^1' + 0:22+ . . . +Xn^ 



yi' + y^'^ . . . +yr? 

 7TI2 — ""' • 



Xiyi + X2y2 + . . . + XnVn 



In these equations the weights of (x) and (y) and of all obser- 

 vation equations are assumed to be unity; mi is the value of m 

 in equations (1) when (x) is assumed to be correct and (y) to 

 contain the errors of observation; m^ is the value of m in equations 

 (1) when (y) is assumed to be correct and (x) to contain slight 

 errors. On applying this method to the above analysis we find 

 that the values obtained are identical with those in column 5 to 

 the second decimal place. 



Third method. The results obtained by this method are only 

 approximately correct but they are of sufficient accuracy to 

 be satisfactory for most analyses and are, moreover, readily com- 

 puted, the computation consisting simply in reducing the weight 

 numbers (x) (Column 4) proportionately, so that their sum (col- 

 umn 6) is equal to that of the given analysis (column 1), the as- 

 sumption being that when the two analyses have the same sum 

 (either the actual sum of the given analysis or 100.00), we have a 

 common basis for comparison. The differences (column 6a) 

 between the observed weight percentages (column 1) and those 

 computed by this method (column 6) are then an approximate, 

 but satisfactory measure of the agreement of observation with 

 theory. This method is, for general purposes, the simplest and 

 best. Mathematically it can be stated from a somewhat differ- 

 ent viewpoint, altho the computations are the same. If we 

 assume that the sum of the weighted residuals (piVi -\- P2V2 -{- . . 

 • + PnVn) in equations (2) is zero, the resulting equation gives 



PiXi + P2X2 -h . . . + PnXn 



an equation which reduces to 



m' = ^■ + ^^^+ • • • +y.^ (6) 



Xi -f- X-2 -|- . . . -f- Xn 



