804 SIE B. C. BKODIE OX TlIE CALCULUS OF CHEMICAL OPEEATIONS. 



Section V.— ON THE SYMBOLS OF SIMPLE WEIGHTS. 



(1) A simple weight has been defined as a weight which is not compound, and two 

 weights as simple in regard to one another which have no common component. 



It follows from this definition that the symbol of a simple weight cannot be expressed 

 by more than one factor, and also that the symbols of weights simple in regard to one 

 another cannot have a common factor. 



The symbol of a simple weight is termed a prime factor, and the symbols of weights 

 simple in regard to one another are said to be prime to one another. 



The symbols of simple weights have the following properties : — 



(2) The operations of algebraic subtraction and division cannot be performed between 

 such symbols. For let a and h be two symbols of simple weights, and, if possible, let 

 a—h-=ay. Then «=«i + J=o'ji, that is, a is the symbol of a compound weight, which 

 is contrary to the hypothesis. 



Or again, if possible, let i=c. Then a—b=-c, and a=.b-\'C=^bc, which is, as before, 



contrary to the hypothesis. 



(3) If a and h be two symbols of weights simple in regard to one another, and if 



aa^=.ih^, then 



ai=bk, and l^=ak; 



for since o«i=Wp a+«j=J+Ji, and «^=S + ^i — a; and since by hypothesis b is the 

 Symbol of a weight simple in regard to a, no part of « is a constituent of b, therefore 

 a must be a constituent of b^, so that bi=a-\-k and ai = b-{-k. Whence also b^^ak, 

 and a^^=ak. 



(4) Hence also if «i be the symbol of a weight simple in regard to the weights a and b, 

 «i is the symbol of a weight simple in regard to the weight ab. 



For otherwise, if possible, let a^ and ab have a common component Jc, so that a^=ck, 

 and ab = C]k; then, since by hypothesis^ is a factor of ffj, ^isby hypothesis prime to a. 

 Therefore k is a factor of b, which is also contrary to the hypothesis. 



"We may also argue thus. If possible, let k be the factor common to Cj and ab. Then 

 ab^=k(l, and (Z=a+5— ^, But by hypothesis no part of ^ is a constituent of a; there- 

 fore A is a constituent of b, and b=^k-\-c=ikc, which is contrary to the hypothesis. 



(5) Hence if a is prime to b, a'' is prime to ¥, and no part of gb is a constituent 

 of^a. 



(6) Also, if «, b, c, d, be prime to Oj, b^, Cj, d^, , then a''b'''c^'d''' is 



prime to a''b'"G'"-d''' ; and also the operation of subtraction cannot be perfoimed 



between the weights ^«4-j;ji+p2<^+j>3<Z-)- and !?«i+2'i<^i+2'2^i+§'3(^i + 



(7) "Whence, li a, b, c, d, be symbols prime to one another, and if 



a''b'"<f'd''' =1, 



i'=0,i>i=0, j)2=0, 2>3=0 ; 



