198 HINRICHS— ATOMIC WEIGHT OF VANADIUM. [April 21, 



The absolute atomic zi'eight a is indicated by the laboratory 

 work; in case of doubt, the criterion just referred to has to be 

 made use of. 



The departure e is expressed in units of the third decimal 

 (thousandths) of the unit of atomic weights. Its exact determina- 

 tion is the main object of this paper. 



The atomic ratio i? is a function of the absolute atomic weights, 

 expressed by the quotient P/Q above given. 



If now the absolute atomic weight for any one given element in 

 this ratio be increased by o.i, that ratio will change or vary by an 

 amount readily calculated from the formula of R as given ; we use 

 throughout seven place logarithms which give the precise value 

 sought with the least trouble. This change or variation we denote 

 by the Greek capital delta A for the particular element of which the 

 atomic weight was increased by o.i in the atomic ratio. In 190 1 we 

 made this calculation only for one element in the ratio ; now it has 

 to be made for every element in the ratio. 



In the tables here followmg this matter will become quite readily 

 understood by simply repeating some of the calculations thus indi- 

 cated. For the reaction no. 98 this work is quite simple, for only 

 two elements are present, namely Va and O. In reaction 270, the 

 work required is about double in amount, because four elements are 

 in reaction, namely : Va, O, CI and Ag. 



The analytical ratio r has to be calculated for each single deter- 

 mination made; it is considerably simplified if the weighings are 

 given by the chemist to the hundredth of the milligram, are rounded 

 off to the tenth of the milligram, which is as far as the weighings 

 can be trusted ; see, for example, my demonstration of this fact for 

 the weighings of Richards made at Harvard-Berlin.* 



The analytical excess e is now obtained as it is r-R ; it is also 

 expressed in units of the fifth place. 



Now we have in hand all the quantities required for obtaining 



the departure e sought, by solving the equation of condition. While 



this indeterminate or diophantic equation is, of course, insoluble in 



general, we have nevertheless obtained two practical solutions of 



the same-"* of which the one properly named ex-cequo is the most 



Wlojiitciir Scioitifique, Juin, 1909, especially pp. 384-385- 



^Comptcs Rcndus. T. 149, p. 1074, 1909. ^vith a most instructive figure. 



