88 SIR B. C. BRODIE ON THE CALCULUS OF CHEMICAL OPERATIONS. 
A is constant, and which occurs by the two substitutions indicated ; the symbo 
— A(x — o)(y — b) is to be interpreted as the symbol of an event which is the “reverse” 
of an event in which A is constant, and which occurs by those two substitutions, — that is 
to say, in the latter case the event is defined by specifying the relation in which that 
event stands to another event, the conception of which is essential to its comprehension. 
This interpretation is a consequence of the identity (Part I. Sec. II. (4)) 
— A{x— a)(y— &)=+[- A(a?— a){y— ft)]. 
Since 
A{x—a){y—b)=—A(a—x)(y—b\ 
every event of this kind may be considered from these two points of view, namely, either 
as a “ substantive ” event or as the reverse of some other event ; and wherever in this 
Calculus the symbol of a chemical event occurs preceded by the negative sign, thus, — U, 
that symbol is always to be interpreted as though it were written +(— U), namely, as 
an event Which is the reverse of the event U. 
In the light of the preceding observations the reader will interpret the following iden- 
tities, which are demonstrated from the principles established, I. Sec. II. (4) (6), 
A{x-a)(y—b)=A(a-x)(b-y ), 
-A(x-a)(y-b)=A(a—x)(y-b), 
-\_—A(x-a)(y-bJ]=A(x-a)(y—b). 
In the previous reasoning the three properties of chemical symbols are utilized which 
were demonstrated in Part I. Sec. II., namely, that xy-\-ab=ab-\-xy, that xy=yx , and 
that A(x^ra)=Ax+Aa. If any one of these properties should not belong to such 
symbols, the reasoning altogether fails. 
For the application to a chemical equation of the preceding principles, the equation 
must be expressed by means of some specified set of prime factors. Now the only 
expression of this kind which we have as yet had occasion to consider is the expression 
of chemical equations by means of the prime factors ascertained in Part I., namely, the 
prime factors a, £, 6, . . ., which are the symbols of those “ weights ” which are 
undistributed in the total system of chemical phenomena. The term “ simple ” or 
undistributed weight, however, is a purely relative term, having been defined as a 
“ weight ” which in some special system of chemical events is not resolved into two or 
more weights or made up from such weights (Part I. Sec. I. 12). Hence an equation 
may be expressed, possibly even in numerous ways, by means of different sets of prime 
factors according to the system of events to which the event under consideration is 
referred — these prime factors being indeed the symbols of the “ simple weights ” among 
which the “ substitutions ” are conceived of as occurring which are the causes of the 
events, every such expression necessarily involves an assertion or an hypothesis as to the 
composition of some one or more among the units of ponderable matter, of which the 
transformations are considered, which constitutes the base of the symbolic system, 
