110 
SIR B. C. BROD1E ON THE CALCULTJS OE CHEMICAL OPERATIONS. 
The congruence 
f{x , y, z . . .) =/(«, b, c .. .), mod {x—a) mod iy~-b) mod ( z—c ) . . . 
asserts that the residue f{a, b, c . . .) is derived from the function f(x, y, z . . .) by the 
substitutions specified in the modules. The problem now to be considered is in what 
way these substitutions are effected. The solution of this problem is afforded by means 
of the following theorem. 
The demonstration of Taylor’s theorem, in which it is assumed that f(x) may be 
developed in a series of ascending powers of x with integral indices, is entirely inde- 
pendent of the interpretation of the symbols, and is recognized as dependent solely upon 
those formal properties of symbols expressed in the equations xy=yx , x(y -\-z)=xy-f- xz, 
known as the commutative and distributive laws. Now in the first part of this Calculus 
(Part I. Sec. II. (5), ( 6 )) it has been fully demonstrated that the symbols of chemical 
operations, with the interpretation there assigned to them, satisfy these fundamental 
conditions. Precisely, therefore, as we are enabled through these properties of those 
symbols to work with them according to the processes of elementary algebra, so we are 
equally justified in applying to them the principles of the differential Calculus regarded 
from this point of view. I shall now prove that if f(x) satisfies the condition given in 
the congruence 
fix) ~fia), mod (x— a), 
f{x) =f(a) +f(a)(x - «) + 1 7 A /*(«) (x -af+ a)(x - a ) 3 + . . . + ^f n (a)(x -a)\ 
wher e f(a), f 2 (a), f 3 (a) . . . fja ') are the first, second, third, and nth derived functions of 
fix) in which a is substituted for x. 
For let f[x) be any function of x, and in that function let x undergo a variation of 
such a nature that x becomes x-\- Ax ; we then have, by Taylor’s theorem, 
fix + Ax) =fix) +fix) Ax+~fix) Ax 2 + ffx) A^ 3 +. . .+~f a ix)^x\ 
Now let Ax be that variation which x undergoes when a is substituted for x; we 
have then x-\- Ax=a, Ax—a—x, and fix + Ax) =f{ a) . Substituting in the above 
development /(a) for fix Ax) and a—x for Ax, we have 
fia)=fix)+f l ix)ia^x)+^ffx)ia-x) 2 + T ^f3( x )( a -x) 3 +---+fifn(xX a -x) n ’ 
whence, changing a into x and x into a, 
f(x)=fia)+f 1 (a)ix-a) + ^f 2 ia)ix-a) 2 + Y ^f 3 ia)(x-a) 3 +...+±f n (a)(x-a) n . 
Now among the properties of chemical functions demonstrated in Sec. I. it has been 
shown that every expression of the form (x—a) n , when n>l, is necessarily equal to zero. 
