191 1.] HIXRICHS— ATOMIC WEIGHT OF VANADIUM. 197 



minations of any series without introducing any false theoretical 

 notion, as is done by the calculation of the so-called probable error 

 of the mean. The actual mean value we do calculate and use. 



While the individual analytical ratios vary for the different indi- 

 vidual determinations in a series and even the means for the different 

 series, it is found, as a matter of fact, that they bear a close relation 

 to the atomic ratio. We call the excess of the analytical ratio over 

 the atomic ratio, the analytical excess and designate it by the symbol 

 e. That is: r^R -[- ^- The value of e is also expressed in units 

 of the fifth decimal. 



The Equation of Condition and the Solution Ex-^Eouo. 



In the true atomic weights of 1894 (p. 139 to p. 169, esp. p. 158) 

 the solution of the great problem is already shown to require an 

 application of the method of the variation of constants. 



In the absolute atomic weights of 1901, the change or variation 

 A of the atomic ratio for an increase of o.i in the atomic weight 

 is determined for each reaction and applied for several important 

 objects throughout the entire work. On pages 144-147 of that work 

 the final solution is really given but implicit only, and lacking the 

 equal distribution of the analytical excess among the elements 

 present in the reaction. 



The actual equation of condition was established in 1907 through 

 long and difficult work, both analytical and geometrical. The gen- 

 eral analytical deduction by means of Taylor's formula was in the 

 hands of eminent men abroad in the form shown in fac-simile 

 (reduced to J'^) as printed p. 61 of my " Cinquantenaire," 1910. 

 The most general construction, which permits the establishment of a 

 criterion for the absolute atomic weight, is printed on p. 60 of the 

 same " Cinquantenaire." 



Here we will present the final practical solution of the resulting 

 " insoluble " indeterminate or diophantic equation in the simplest 

 and most direct manner, suitable for common, current, practical 

 application. 



The true atomic weight, A, is the quantity sought, in the unit for 

 which carbon-diamond is 12 exactlv. 



