524 VAN ORSTRAND AND WRIGHT: MINERAL ANALYSES 



4. Even though one might admit that adjustment on the 

 basis of 100 is legitunate, there is no vahd reason for assuming 

 that mere expansion or contraction to 100 on the basis of pro- 

 portionahty is necessarily a correct solution of the problem. 

 This solution is only one of m.any which can be obtained by the 

 substitution of different weights in equation (14); an entirely 

 different set of solutions can be obtained by substitution of the 

 weights in the group of equations (9), (10), and (11). 



Colunm (12) gives the computed weight percentages when 

 the theoretical weight percentages are substituted in equation 

 (9) instead of the theoretical weight numbers. The method 

 is theoretically correct, but slightly less precise than the first 

 method, on account of using smaller coefficients. Tests for 

 systematic errors are readily made for the reason that the theo- 

 retical value of 771 is always equal to unity. 



Conclusion. — In the adjustment of the data of a mineral 

 analysis it is always best to deal directly with the actual errors 

 of observation. Any method of adjustment which involves 

 division or multiplication of the actual errors amounts to the 

 adoption of a complicated system of weights; this is generally 

 unjustifiable, and always renders the final result more difficult 

 of correct interpretation. 



By treating the problem from the standpoint of the least 

 square adjustment of an empirical formula, we not only adopt 

 a method in current use in the more exact sciences, but we obtain 

 thereby a generality which enables us to form a correct opinion 

 of the various methods of adjustment, and also to discover 

 the nature of the errors involved, and thus to obtain a complete 

 solution of the problem. In the present paper it has been shown 

 that all of the methods which have been proposed are least square 

 methods and differ chiefly in the m.anner of assigning weights to 

 the data of observation. 



Rigorous methods of calculation are given above by means 

 of which weights may be properly taken into account. To 

 apply these methods of computation, the chemist may either 

 assign weights, using his own judgment as a guide, or he may 

 determine the probable error of each percentage involved and 



