INTRODUCTION. XIII 



we have for the sum or difference of two quantities, A and 

 B: 



(4.) ' = \/a'^ + b' 



For the product of A multiplied by B the probable error is 



(5.) ' = \/{Mf + (B«)^ 



For the product of three quantities, ABC : 



(6.) ^ = -/(BCa)'' + {hfZbf + (ABf)* 



For a quotient, j, the probable error becomes 





(x)" + - 



A 



Given a proportion, A : B : : C : a;, the probable error of 

 the fourth term is as follows : 



(8.) 



\ (¥)' + (^-^^^ + (^')' 



This formula is used in nearly every atomic weight cal- 

 culation, and is, therefore, exceptionally important. Rarely 

 a more complicated case arises in a j^roportion of this kind : 



A:B::C + .r:D-fx 



In this proportion the unknown quantity occurs in two 

 terms. Its probable error is found by this expression, and 

 is always large : 



(9-) ' = -x/l-^TT^I (E'^^'^ + ^'^') + 



[A — B)'^ 



When several independent values have been calculated 

 for an atomic weight they are treated like means, and com- 

 bined according to formulae (2) and (3.) Each final result 



