68 SIE B. C. BEODIE ON THE CALCULUS OE CHEMICAL OPERATIONS. 
Similarly, in the above equation, let <p, <p 15 <p 2 , <p 3 . . . have a common measure a\ so 
thatp=£+s, £=£+£, r=t+l . . 
<p =a t+s b v 'c v * . . . =a* + *0, 
= . . . =a* + *0 1; 
q> 2 =a t+l b r 'c r * . . . =a t+l ® 2 , 
then 
w = ma i+s ® + m!a t+lc ® 1 + m"a t+l ® 2 + . . . 
Now if we divide this equation by a', we have 
ma 8 ® -f- m!a! c ® 1 -t- m"a l ® 2 + ... =0 ; 
and if this equation be true, 
... = 0 . 
But we have from the original equation 
. . . =0 ; 
whence, subtracting, 
+ . . . = 0 . 
That is to say, we are justified in dividing a chemical equation by any common 
measure of the symbols by which the equation is expressed of the form a ‘ if the sum of 
the numerical coefficients in that equation be equal to zero, but not otherwise. An 
equation in which this condition is satisfied will be termed a “ normal ” chemical 
equation, being an equation to which the rules of algebra are applicable. An equation 
in which this condition is not satisfied will be termed an “ abnormal ” chemical equa- 
tion, being an equation to which these rules cannot be applied. 
If we open a chemical treatise and examine the equations expressive of actual results, 
we cannot fail to be impressed by the circumstance that numerous equations satisfy 
this condition and are of the “ normal ” form. For example, 
2 oL X =a+u X \ (1) 
a 2 <P Z 3 H-3ai=3 aZ + a ^r, (2) 
aVS + f=a|+ a Vf, (3) 
2a 2 . + a^ 6 r=2 aVg+aW?, (4) 
aVr + 3aVr=3a|+aVr, (5) 
aVi 3 +2aa>=aVf 2 +a£+a<y 2 , (6) 
a 7 z% 6 -j- llau=0L 7 z. 6 ciJ-\-Q(x^-{-5(xay ! , (7) 
5a+o«y 2 I; 5 = 5a| + aw 2 (8) 
These equations have been selected at random. They indicate to us the following- 
facts : — 
