wells: interpretation of mineral analyses 419 



mination is assumed to be free from error and the remaining atom 

 numbers show discrepancies from the requu-ement of theory. 

 Schaller's method is a simple method of distributing the dis- 

 crepancies. It occurred to Wright and Van Orstrand that the 

 method of least squares would give a still better distribution of 

 discrepancies. In the illustration worked out by them, however, 

 (first method) it is obvious that the tj values (observed data) are 

 weighted by multiplication by the respective molecular weights 

 {x values). It would be perfectly possible to weight the observa- 

 tion equations in any arbitrary manner. It would seem, however, 

 that if the discrepancies are assumed to be random ones the weight- 

 ing should be based upon the magnitude of the discrepancies rather 

 than that upon the molecular weights involved. In view of what 

 has been said it can be seen that the weighting should probably 

 also vary with the mineral, the number of atoms, etc., so that the 

 chief difficulty would be in getting chemists to agree upon a 

 system of weighting. 



There is some objection from a chemical point of view to any 

 method involving a distribution of discrepancies. It must be 

 borne in mind that a mineral may be a mixture, a solid solution, 

 or a molecular species. All these possibilities occur in nature 

 and some species occur in a state of remarkable purity. Yet it 

 is hardly to be expected that natural products formed from, or 

 open to attack by, migrating solutions of various kinds can be 

 wholly free from inclusions of foreign matter. Under these circum- 

 stances a clear differentiation between fact and hypothesis should 

 be preserved. An analysis is a more or less imperfect expression 

 of certain facts, viz., the composition of the substance in question; 

 that this composition may be expressed in a chemical formula is 

 an hypothesis which may find only approximate verification in 

 the case of a mineral. As our knowledge widens minerals are 

 being found more and more to be solid solutions to a slight extent ; 

 these interesting relationships are brought out by independent 

 derivations of the atom numbers but masked by a distribution of 

 ''errors." Methods involving a distribution of errors are there- 

 fore strictly applicable only to very pure compounds. 



But in reverting wholly to a comparison of percentages we lose 



