4 SMITHSONIAN MISCELLANEOUS COLLECTIONS VOL. 54 



N'ecessaril}^ this work omits many details relative to experimental 

 methods, and particulars as to the arrangement of special forms of appa- 

 ratus. For such details original memoirs must be consulted. Their 

 inclusion here would have rendered the work unwarrantably bulky. There 

 is such a thing as over-exhaustiveness of treatment, which is equally 

 objectionable with under-thoroughness. 



Of course, none of the results reached in this revision can be consid- 

 ered as final. Every one of them is liable to repeated corrections. To 

 my mind the real value of the work, great or little, lies in another direc- 

 tion. The data have been brought together and reduced to common 

 standards, and for each series of figures the probable error has been de- 

 termined. Thus far, however much my methods of combination may 

 be criticised, I feel that my labors will have been useful. The ground is 

 cleared, in a measure, for future experimenters ; it is possible to see more 

 distinctly what remains to be done; some clues are furnished as to the 

 relative merits of different series of results. 



On the mathematical side my method of recalculation has obvious 

 deficiencies. It is special, rather than general, and at some future time, 

 when a sufficiently large mass of evidence has accumulated, it must 

 give way to a more thorough mode of treatment. For example, the ratio 

 Ago : BaBr^ has been used for computing the atomic weight of barium, 

 the atomic weights of silver and bromine being supposed to be known. 

 But these atomic weights are subject to small errors, and they are super- 

 imposed upon that of the ratio itself in the process of calculation. Ob- 

 viously, the ratio should contribute to our knowledge of all three of the 

 atomic weights involved in it, its error being distributed into three parts 

 instead of appearing in one only. The errors may be in part compensa- 

 tory; but that is not certainly known. 



Suppose now that for every element we had a goodly number of atomic 

 weight ratios, connecting it with at least a dozen other elements, and all 

 measured with reasonable accuracy. These hundreds of ratios could 

 then be treated as equations of observation, reduced to linear form, and 

 combined by the general method of least squares into normal equations. 

 All errors would thus be distributed, never becoming cumulative ; and 

 the normal equations, solved once for all, would give the atomic weights 

 of all the elements simultaneously. The process would be laborious 

 but the result would be the closest possible approach to accuracy. The 

 data as yet are inadequate, although some small groups of ratios may 

 be handled in that way; but in time the method is sure to be applied, 

 and indeed to be the only general method applicable. Even if every ratio 

 was subject to some small constant error, this, balanced against the 

 similar errors of other ratios, would become accidental or unsystematic 



