SIE B. C. BEODIE ON THE CALCULUS OE CHEMICAL OPEBATIONS. 
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So that each symbol may be regarded as derived from the symbol (a 2 x)x 3 by the substi- 
tution in all possible ways in that symbol of w for This relation by way of substi- 
tution is a universal bond of chemical functions, and prevails among things, among 
groups of things, and among events. It is desirable to indicate this relation by a name 
and by a symbol. 
(2) Definition: — 
Two chemical functions will be said to be congruous to one another in regard to 
a special substitution if the two functions are of such a nature that they assume the 
same value when that substitution is effected in them respectively. The value which 
a function assumes when a special substitution is effected in it will be termed the 
“ residue ” of that function in regard to that substitution, and the symbol of the substi- 
tution when it appears in this connexion will be termed a “modulus.” Further, the 
symbol = is to be interpreted “ is congruous to ” ; thus, for example, if f(x) be the 
symbol of a chemical function of the letter x, if x — a be the symbol of the modulus, 
and R the symbol of the residue, 
f(x)-= R, modulus (x—a). 
This expression is termed a chemical congruence, and is to be read f(x) “is congruous 
to ” R for the modulus (x - a). 
Similarly we have 
f(x, y, z, . . .) = R, mod (x—a) mod (y—b) mod (z— c) . . . , 
where R is the value which f(x,y, z, . . .) assumes when in that function a is substituted 
for x, b for y, c for z . . . 
(3) I proceed to determine the conditions satisfied by functions congruous to a given 
residue for a given modulus. 
Let f(x) = R, mod (x—a); 
for x in f(x) we will write a -j- x — a\ we have then 
f(cc)=f(a+x^,). 
Now since f(x) is a rational and integral function of x (for there are no other chemical 
functions), f(a-\-x—a) is a rational and integral function of (x—a). Putting then 
AoT-A^# — a)- {-A 2 (T‘ — $) 2 T A. n (x — a)' 1 
as the general expression for such a function of (x—a), where A 0 , A„ A 2 , . . . A„ are 
functions of a and free from x, and where n is a positive integer, we have 
f(a+x-a)=k 0 -\-A l (x—a)-\-A 2 (x—a)‘ l -\- . . . A n (x—a) n . 
Now it has been demonstrated (Sec. I. (4)) that every continued product of two or 
more chemical factors of the form (x—a)(y—b) ... is necessarily equal to zero. 
MDCCCLXXVII. Q 
