SIR B. C. BRODIE ON THE CALCULUS OE CHEMICAL OPERATIONS. 
87 
This event therefore is an event which may be referred to either of two “ causes,” 
namely, to the substitution of a for x and of b for y, which “ causes ” represent two 
alternative hypotheses as to the ways of the occurrence of the event. 
But, further, not only is it true that the event may be referred to one or the other 
of these two causes, the occurrence of either of which is sufficient to account for the 
event, but the event must be so referred as there is no other substitution whatever which 
will produce the result in question. This may be proved experimentally by altering 
the arrangement of the letters in every permissible manner, and, as before, superposing 
the arrangements. 
But it is sufficient to observe that there are only two conceivable cases ; for Avhere A xy 
appears before the event, either A ay appears after the event or A xb so appears. In the 
former case the result is attained, if possible at all, by the substitution of a for x ; in 
the latter case, by the substitution of b for y. 
We might also, in demonstrating the above proposition, have reasoned thus : having 
proved that the event may occur by the substitution of a for x, since in the above equa- 
tion we may change the places of x and y, provided also we change the places of a and 
b, and the equation will be unaffected by this alteration. Hence, whatever is true of x 
and a is also true of y and b ; that is to say, whatever assertion may be made in reference 
to that equation in regard to x and a is also true if in that assertion we substitute y for x 
and b for a. 
Writing the above equation in the form u= 0, we have, from the distributive law 
(Part I. Sec. II. (6)}, 
u=A(x—a)(y—b), 
from which it is apparent that the substitution by which the event occurs, that is to say, 
the “ causes ” of the event, are indicated to us by the factors of this equation, and are 
coincident with those factors ; and, precisely as the factors in a numerical equation of 
this form indicate to us the two sole conditions under which the numerical identity 
asserted in the equation is possible, namely, the identity of x and a, or the identity of y 
and b, so in the case of this chemical equation the factors indicate to us the two sole 
conditions, or hypotheses, under which the chemical identity expressed in that equation 
is possible, namely, the substitution of a for x, or the substitution of b for y. For the 
only possible w r ay in which the “ weights,” or matter, symbolized as Axy -\- Aab can have 
been chemically converted into the “ weights,” or matter, symbolized as Axb-\-Aay is by 
the occurrence of one or other of these two “ substitutions,” either of which affords an 
adequate and sufficient cause of the metamorphosis. 
As the expression A (x—a){y—b) is the symbol of the event of the transformation of 
Axy and Aab into A ay and A bx, so the expression — A(x— a)(y— b) is the symbol of the 
transformation of A ay and A bx into Axy and Aab. This event will be termed, in rela- 
tion to the former event, the “ reverse ” of that event or the “ reverse event ; ” and while 
the symbol A (x~a)(y — b) is interpreted as the symbol of a “ substantive ” event in which 
