OIG VAN ORSTRAND AND WRIGHT: MINERAL ANALYSES 



the various chemical elements are given as they have been 

 found to exist in that mineral. In the analysis the chemist 

 has eliminated, to the best of his ability, the systematic errors 

 in his observational data and the figures in his analysis are sub- 

 ject only to the errors of observation over which he has no con- 

 trol or to systematic errors of which he has no knowledge. It 

 is with such errors that the observer, who studies and compares 

 chemical analyses as given in final form by the analyst, has to 

 contend. It may be possible, as will be shown later, to detect 

 and to compute systematic errors of a certain kind in an analysis 

 but it is, of course, not possible, in general, to free a chemical 

 analysis of its systematic errors by any purely mathematical 

 procedure. Nor is it the purpose of the present paper to dis- 

 cuss such methods but rather the methods which have to do 

 with the accidental errors of observation. In all data of meas- 

 urement such errors creep in because no instrument or method 

 is absolutely accurate and because no observer is capable of 

 making perfect observations. In the discussion of such data the 

 method of least squares is universally adopted. It has for its 

 object the adjustment and comparison of observations in which 

 the errors are accidental. The term accidental is here used in a 

 technical sense to unply that positive and negative errors (de- 

 partures from the true value) of equal numerical magnitude are 

 equally probable. The principle of least squares is based on the 

 Gaussian law of distribution, a law which has been abundantly 

 verified experimentally not only in the theory of errors but also 

 in other fields of science and has led to results of the greatest 

 importance. 



In the mathematical discussion below it is proved that all 

 the methods which have been suggested for the adjustment and 

 comparison of chemical analyses are least square methods and 

 differ chiefly in the manner of assigning weights to the observed 

 data. This fact enables us to fix definitely the significance of 

 each m.ethod and to ascertain its good and its weak features. 



The adjustment of least squares. To show that the methods, 

 which have been proposed are special cases of the general least 

 square solution let us put 



