SIE B. C. BRODIE ON THE CALCULUS OE CHEMICAL OPERATIONS. 
65 
mp +m'q +m"r + = 0 , 
= 0 , 
mp 2 -}-m"q 2 -{-7n"'r 2 -{- =0, 
It will be sufficient to consider the above question in relation to the fundamental 
equation (Part I. Section IV. (1)) 
xy=x+y, (1) 
which affords the simplest example of a chemical equation. If we proceed to deal with 
this equation with the same freedom as with a numerical equation, and to multiply each 
member of this equation, not by a numerical symbol m (which is permissible), but by a 
chemical symbol v, we have 
vxy=vx+vy, (2) 
a conclusion which is obviously false ; for, interpreting the equation consistently with 
the principles on which these symbols are constructed, we are informed by it that the 
unit of matter symbolized as vxy is identical, as regards “ weight ” or matter, with the 
aggregate of the two units symbolized as vx and vy ; and further, if we apply to equa- 
tion (2) the principle laid down in equation (1), we have vxy=v-\-xy, vx=v-\-x, vy=v+y, 
and, as a final conclusion, v — 2v. Again, taking the equation x?y=x-\-xy, which is true, 
dividing both members of the equation by the chemical symbol x, we are led to an 
absurd conclusion, namely, that 
xy=l+y, • (3) 
that is to say, that the matter symbolized as xy is always identical with the matter of 
an empty unit of space, symbolized as 1, and the matter symbolized as y. But it is only 
necessary to observe that the equations (2) and (3) to which we are thus led, 
vxy=vx+vy, 
xy=l+y, 
are not chemical equations at all, for they do not satisfy the fundamental conditions 
afforded by the system of indeterminate equations previously referred to, 
mp-\-m'q-\~m"r-\- ... 0 , 
In short, if we inconsiderately deal with these equations without reference to the 
principles on which they have been constructed, precisely as though they were numerical 
equations, we are not only involved in a tissue of absurdities and contradictions, but are 
compelled to introduce into the system of chemical equations, equations which do not 
belong to that system at all, and which could never arise from the expression by means 
of equations of chemical facts. 
