SIE B. C. BRODIE ON THE CALCULUS OF CHEMICAL OPERATIONS. 
75 
But we may also proceed thus. Beducing the above equation to the “ normal ” form, 
we have 
2 + 2a”V'"‘ = 3a + aV 1 . 
Putting »=1, we have 
2 + 2a”=3 «+a n , 
whence 
a(2u m ~ l a"-i 1) ^ 
a- 1 ‘ 
Since this equation is always true, it is true where a=l. But the numerical value 
of the left-hand member of this equation, when a= 1, is 2(m— 1) — (n— 1) ; whence we 
have 
2m— w— 1=2, 
and, as before, 
2m=3+w; 
again, putting a=l, we have 
2+2r-=3+y 
and 
V m i — v n i 
T."'. — T 
— 1 . 
But the numerical value of 
when v=l 
whence we have, as before, 
2 m Y —n v 
We thus arrive by the consideration of the numerical value of chemical symbols at 
the results previously inferred by an apparently different process. But the truth is 
that the very same principles are really employed in both methods, which differ only in 
the stage at which those principles are introduced. 
The process here employed for the reduction of a chemical equation to the “ normal ” 
form, with the view, of rendering that equation amenable to algebraical treatment, is 
strictly in conformity with the spirit of algebraical methods. The ordinary artifice for 
the solution of a quadratic equation is based upon a somewhat similar principle. A 
particular form of the symbol 0 is there added to the function, by which its value and 
interpretation are not affected, but which enables us to express the function by means 
of factors. But an illustration, perhaps still more in point, is the process familiar to 
mathematicians, by which an equation is rendered “ homogeneous,” which is effected by 
the introduction of an arbitrary factor with which we work precisely as with one of 
the real factors of the symbols which appear in the equation, and which, at the 
conclusion of the algebraical operations, to which the equation is submitted, is put equal 
to 1 and suppressed. 
(7) It is, however, important to notice that although it is strictly true that if the 
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