4 THE ATOMIC WEIGHTS. 



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

 Ag 2 : BaBr 2 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 

 with reference to the entire mass of material, and would practically 

 vanish from the final means. 



Concerning this subject of constant and accidental errors, a word may 

 be said here. My own method of discussion eliminates the latter, which 

 are removable by ordinary averaging ; but the constant errors, vicious 

 and untractable, remain, at least partially. Still, where many ratios 

 are considered, even the systematic errors may in part compensate each 

 other, and do less harm than might be expected. They have, moreover, 

 a peculiarity which deserves some attention. 



In the discussion of instrumental observations, the systematic errors 

 are commonly constant, both as to direction and as to magnitude. They 

 are therefore independent of the accidental errors, and computation of 

 means leaves them untouched. But in the measurement of chemical 

 ratios the constant errors are most frequently due to an impurity in one 

 of the materials investigated. If different samples of a substance are 

 studied, although all may contain the same impurity, they are not likely 

 to contain it in the same amount ; and so the values found for the ratio 

 will vary. In other words, such errors may be constant in direction but 

 variable in magnitude. That variation appears in the probable error 

 computed for the series of observations, diminishes its weight when com- 

 bined with other series, and so, in part, corrects itself. It is not removed 

 from the result, but it is self-mitigated. The constant errors familiar to 

 the physicist and astronomer are obviously of a different order. 



