SIE B. C. BRODIE ON THE CALCULUS OF CHEMICAL OPERATIONS. 
73 
or subtraction from it of the symbol 1 . The application of this principle to the general 
equation 
xy=x+y (1) 
leads to the “ normal ” form of that equation, 
1 +xy—x+y ( 2 ) 
(4) We may also arrive at this last equation through the following considerations, 
which involve some important principles which will hereafter find a further application. 
Let A, x,y be chemical symbols, and let 
A xy + A ab = A xb + A ya ; 
then, from equation (1), 
Axy=A-\-x+y , Axa=A+x-\-a, 
Aab^A+a+b, Ayb =A-\-y-\-b. 
Now putting 
v = A xy + A ab — A xb— Ay a, 
and substituting for Axy, A ab, Axa, Ayb the above values, 
v=0. 
But we have, from the distributive property of chemical symbols (established in 
I. Sec. II. 6), 
v=A(x-a)(y-b), (1) 
whence 
A(x-a)(y-b)=0; 
that is to say, the product of any two chemical factors of the forms r — a,y — b is neces- 
sarily equal to nothing ; and the further application of the same reasoning leads to the 
conclusion that the continued product of any number (not less than two) of factors of 
this kind is also nothing ; that is, 
A {x-a)(y-b)(z-c) . . . =0 (2) 
Now, since these equations are always true, they are true when A, a , and b are 
respectively put equal to 1, whence 
(x—l)(y 1) = 0 
and 
1 +xy=x+y, 
which is the equation referred to. 
Now this equation, regarded as a numerical equation, is satisfied by the values 
#=1,^=1; and since this property is perfectly general, and every symbol of a chemical 
operation satisfies this condition, every chemical equation must necessarily be true when 
the prime factors by which it is expressed are severally or collectively put equal to 1. 
(5) Now if this principle be applied to an abnormal equation, we are led to the 
assertion (Part I. Sec. IV. (3)) 
1(1) == 2(1)=3(1) . . . =m(1)=0; 
or, suppressing the chemical symbol (1), 1=2 = 3 . . . =n= 0, which assertion is indeed 
