SIR B. C. BRODIE ON THE CALCULUS OE CHEMICAL OPERATIONS, 
67 
imperfect system, presenting but a remote resemblance to the extended methods of 
arithmetical algebra. Such a system is really (although not explicitly and avowedly) 
presented to us in our actual chemical notation. No algebraical operation has ever as 
yet been properly performed upon chemical symbols except the operations of addition 
and subtraction. The great convenience, the brevity and suggestiveness of the notation 
has led to the universal adoption of the apposition of letters as the expression of 
what is termed chemical combination. Thus HHO, H 2 O, H 2 O (for each of these 
expressions is in use) represent to us the molecule of water as discriminated from the 
aggregate of hydrogen and oxygen, H + H+O. But these expressions are employed 
under protest and are algebraically lifeless, for no use is made of them ; and, as far as 
any definite processes of reasoning are concerned in which such symbols are engaged, 
the molecule of water might with equal advantage have been expressed by the letter A. 
(2) We shall now proceed to consider how chemical equations through the application 
of the principles of this Calculus may be brought into a form adapted to algebraical 
treatment. 
First, it is to be observed that although the processes equivalent to multiplication 
and division are not universally applicable to chemical equations, yet neither, on the 
other hand, is it true that these processes are universally inapplicable. There are certain 
cases in which we may either multiply or divide a chemical equation by a chemical 
symbol, and the result shall be both interpretable and true. These cases are easily 
discriminated. Let us take the general equation given, Part I. Section V. (10), 
«;=m<p+m'<p 1 +m ,, <p 2 +m'"(p3+ ••• =0, 
and multiply the equation by the factor e\ where e is a chemical symbol and t a number, 
we have then 
me t <p+m'e t <p l +m''e t <p 2 -\-ni!''e t q> 3 + . . . =0; 
and then, in addition to the indeterminate equation 
mp -F m'q + m'V+ = 0 
we have another equation to be satisfied, namely 
mt -f- m’t -f rn u t + -f- =0, 
whence, by dividing by t, 
wi+ra'Fm"-fm w + =0. 
Now if this equation be true, we are led into no error by effecting the multiplication 
in question. That is to say, we are justified in multiplying a chemical equation by any 
chemical factor of the form e t and by any product of such factors, if the sum of the 
numerical coefficients in that equation be equal to zero, but not otherwise. 
MDCCCLXXVII. L 
