SIR B. C. BRODIE ON THE CALCULUS OF CHEMICAL OPEHATIONS. 805 



ndif 



j)a +Pih ^p^c+p^d + =0, 



i>=0,i?i = 0, ^2=0,^)3=0 



(8) Lastly, a composite symbol can only be expressed in one manner by means of prime 

 factors. That is to say, a compound weight can only be assumed to be composed of one 

 set of simple weights. This proposition may be proved in the same manner as the 

 con'esponding numerical proposition. 



This assertion does not imply that we cannot make more than one hypothesis as to 

 the expression of any given composite symbol by means of prime factors, that is, as to 

 the simple weights of which a given compound weight is composed, but only that two or 

 more such hypotheses cannot simultaneously be true. 



There is a close analogy between the symbols of simple weights in chemistry and the 

 symbols of prime numbers in arithmetic, but owing to the condition imposed on che- 

 mical symbols, given in the equation x-\-y=-xij, a chemical symbol which has only one 

 factor is also incapable of partition. 



The prime symbols of chemistry may be indiiferently defined by either property, the 

 one being a consequence of the other, and constitute a new and peculiar order of 

 symbols. There is, however, one numerical symbol of the class, namely, the symbol 1, 

 which has only one factor and one part, and like the primes of chemistry is incapable 

 of division or partition. 



(9) An integral compound weight has been defined (Sec. I. Def. 9) as a weight which 



is composed of an integral number of simple weights. If (p be the symbol of such a 



weight, a,b,c, , as before, the symbols of simple weights, and «, Wj, Wgj • • • • integral 



numbers, 



<p=:a"b"'c'^ 



This symbol is termed an integral composite symbol. It is identical in form with the 

 symbol of an integral number expressed by means of its prime factors. 



(10) It remains to consider the method by which we may arrive at the expression of 

 chemical symbols by means of an integral number of prime factors in a given system of 

 equations, if such an expression be possible, and further may select from the various 

 forms of symbols which satisfy this condition that form in which the symbols are 

 expressed by the smallest possible number of such factors. In this form the symbol is 

 said to be expressed in the simplest possible manner by means of prime factors, it being 

 the only symbolic expression which is at once both necessary and sufficient to satisfy the 

 conditions of the problem. 



To these conditions it is to be added that the prime factors thus chosen are to be the 

 symbols of real weights, it being possible to find symbolic expressions which satisfy the 

 requirements of the equation, but which do not admit of interpretation, the weights of 

 which they are the symbols being affected with the negative sign. Such expressions will 

 here be rejected. 



Now, first let the system of equations in which it is required to express the chemical 



MDCCCLIVI. 5 E 



