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SIR B. C. BRODIE ON THE CALCULUS OF CHEMICAL OPERATIONS. 
or matter. So that, as regards “ weight ” or matter, our conception of the “ weight ” or 
matter of a unit of hydrogen symbolized as a is precisely the same thing as, and cannot 
be discriminated from, our conception of the “weight” or matter of a unit of hydrogen 
and the “ weight ” or matter of an empty unit of space. Hence as long as “ weight ” 
or matter is the only subject under our consideration, we may assert with perfect truth 
that a.—a-\- 1, this assertion being strictly analogous to the arithmetical assertion that 
1 = 1 -(-0. In this peculiar property of the chemical symbol 1 is supplied to us the 
means of remedying the imperfection of chemical equations and of so changing their 
form, without affecting their interpretation, as to bring these equations within the 
recognized domain of algebra. For if in any chemical equation v=0 the sum of the 
numerical coefficients be not equal to zero, it is in our power to make that sum equal 
to zero, by the addition to the function v of the necessary number of numerical symbols 
affected by the proper sign ; and when this transformation is effected, we may operate 
upon the equation with confidence and security by every process of algebra — the result 
being on the one hand greatly to enlarge our powers, and on the other to prohibit 
inadmissible processes. 
Take for example the equation discussed, Part I. Sec. VII. (8), 
2a 2 y= 3a-j-a» 2 . (1) 
This equation is “ abnormal.” We should be tempted from the form of the equation 
to consider the result of dividing by the symbol a, which leads to an equation 
2«*=3+v 2 , 
an equation external to the system of chemical equations, the conditions previously 
referred to not being satisfied in that equation. But this equation, when rendered 
“ normal ” on the principles just laid down, is 
2-j“2a 2 f=3 a-^-av 1 ( 2 ) 
Now this equation, as regards any assertion made by it in relation to “weight,” has 
precisely the same meaning as the equation from which it is derived ; for it is perfectly 
immaterial whether we say that two units of ammonia are identical as regards “ weight ” 
with three units of hydrogen and one unit of nitrogen, or whether we say that two units 
of ammonia and two “ units of space ” are thus identical with three units of hydrogen 
and one unit of nitrogen. But as regards their algebraical properties the equations are 
fundamentally different ; for on the latter equation (2) the performance of the operation 
of division by the symbol a is impossible, and the result of the transformation of the 
equation is absolutely to prohibit this operation. But, on the other hand, we may now 
multiply this equation by any chemical symbol, and the resulting equation will necessarily 
be interpretable and a proper subject for our consideration. As, for example, if we 
multiply both sides of the equation by the symbol | 2 , the equation becomes 
2f + 2a 2 *£ 2 =3 af + a*' 2 ? 2 , 
an equation in which the requisite conditions are satisfied. 
