VAN ORSTRAND AND WRIGHT." MINERAL ANALYSES 523 



nates of this line are each too long by the amount 0.03, and at the same 

 time, since m < m' , these ordinates are too short by the amounts 

 given in column (5). The random errors of the system are the residuals 

 tabulated in column (4). Again, let us apply the same method to the 

 adjustment of the values of column (6). Proceeding in the same 

 manner as before, we find the constants, 



a = 0.128. 

 m = 0.049396, 



and the computed weight percentages of colunni (7). By comparing 

 these values of a and m with the preceding values, it is easy to show 

 that a constant error, + 0.10 and a systematic error — 0.005 y has 

 been added to each of the values of column (2). A correct application 

 and interpretation of our equations thus leads to a complete solution 

 of the problem. 



Wells objects to the use of formula (7) on the basis that the 

 computed sum does not equal 100 or the observed sum. This 

 objection is not necessarily valid for various reasons. 



1. The observed weight percentages contain small errors, 

 consequently in accordance with the principles of probability, the 

 observed sum is likewise in error. 



2. The composition of a mineral is unknowm. It is there- 

 fore more nearly representative of the observed facts to adjust 

 accurately such percentages as have been obtained, leaving the 

 remainder for future determinations. Adjustments of this kj.nd 

 are not on the same basis as the adjustments of the angles of a 

 triangle where it is known a priori that the sum of the angles is 

 equal to 180°. 



3. Adjustment on the basis of 100 does not necessarily lead 

 to correct results. To illustrate, by an extreme example, let 

 it be assumed that As has not been determined in the above 

 analysis. The computed values of the remaining elements as 

 obtained from equation (7) are given in column (9). It will be 

 noted that the values are sufficiently accurate to enable us to 

 decide correctly in regard to the three elements involved, whereas 

 mere expansion to 100 would in this case lead to an absurd re- 

 sult. Any method of averages will at tunes lead to incorrect 

 results when one or more of the elem.ents are present in very 

 small quantities. An obvious remedy consists in assigning large 

 weights to these quantities. 



