SIR B. C. BRODIE ON THE CALCULUS OE CHEMICAL OPERATIONS. 
Ill 
We may therefore at once resolve the above equation into the following system of 
equations : — 
1 • /O) -/(«) -/(«)(* ~ «) = °- 
II. ±/J[aXx-dr=0. 
HI. ^ /,(«)(*-«)■= 0. 
~fja)(x-a)"=0. 
The first of these equations is that which has been already given as indicating the 
condition by which f(x) is related to the residue f(a). The aggregate of the succeeding 
equations is identical with that equation ; and the equations separately considered inform 
us of the successive steps by which the result indicated in equation I., 
f{x) =f(a) +f l {a){x- a), 
is attained. 
This method of development is applicable to every chemical function ; but if f(x) be 
a chemical equation of the form f(x) = 0, it follows from the property of chemical 
equations demonstrated (Section I. (8)) that /(a) is also a chemical equation ; whence 
/(«)=<>. 
/■(«)(>— «)=o. 
It is to be observed that in this development of f(x) in ascending powers of the 
modulus (x—a), the symbols of the units of matter, which appear in the several events 
of which f(x) is constituted, are resolved into their components according to the classi- 
fication of those components given in Section III. (3), namely, as “ variables,” “ values,” 
and “ constants,” — the “variable ” being symbolized as x, the “value ” of that “ variable ” 
as a, and the “ constants ” being given in the coefficients 
/iO)> r^/ 2 («)> r^3/»W» 
This classification of the components of the units of matter thus satisfies the conditions 
required of an accurate method of classification, being both exclusive and exhaustive. 
A chemical function of two variables f(x, y) satisfying the condition 
f(x,y)=f(a, b), mod (x—a) mod (y— b) 
may be developed on precisely the same principles. 
Using a similar notation to that previously employed, let 
