SIR B. C. BRODIE ON THE CALCULUS OF CHEMICAL OPERATIONS. 109 
Similarly, considering the congruence 
f(x , y, z) =/(«, b, c), mod (x—a) mod {y—b) mod (z—c), 
we have 
f(x,y, z)=f(a, b, c)+f. 0 . 0 (a, b, c)(x-a)+f 0 . 1 . 0 (a, b, c)(y—b)+f 0 . 0 . ,(a, b, c){z—c ), 
and so on, a similar relation holding good in the general case 
f(x,y , z,v,w,...) =f(a, b,c,d,e,...),mod(x—a)m.od(y—b)m.od(z—c)mod(v—d)mod(w—e)... 
(4) The symbol =, which is here adopted as the symbol of “ chemical congruence,” 
is used in mathematical investigations of the properties of numbers as the symbol of 
numerical congruence. Thus the expression x = a, mod^? indicates to us that the 
integral number x satisfies the condition given in the equation x=a+py, where a, p, y 
are integers as well as x, so that the difference x — a is divisible without remainder by the 
modulus^?. Now the simplest form of a chemical congruence, f(x) =/'(«), mod (x— a), 
is, in the most exact sense, an “ algebraical congruence,” for fix) satisfies the condition 
fix) =f(a) -f- Q(# — a), where f(x), f(a), and Q are respectively rational and integral 
functions of x and a , so that f(x) —f(a) is divisible without remainder by the modulus 
x—a. In the case of the chemical congruence to two moduli, such f(x, y)^: f (a, b), 
mod (x— a) mod (y— b), where f(x, y) satisfies the condition/^, y)=f(a, b)-\-A. l (x— a) 
fi-B i(y—b), an analogous condition prevails. In this case if we divide the difference 
f(x, y)-f(a, b) by the modulus (x—a) we have as the remainder of the division 
f(a, y)—f(a> b), which is divisible without remainder by the modulus y—b. It hence 
appears that a “ chemical congruence ” for two or more moduli is but a wider kind of 
numerical congruence, being a comprehensive form in which all such “ congruences ” 
are included. The same fundamental idea of congruity is applicable in either case; 
and as no confusion is likely to arise from the use of the symbol = as the symbol of 
the chemical relation referred to, which is, indeed, the fundamental relation of the 
science of chemistry, I have not hesitated to adopt it. 
(5) From the various illustrations already given of “ simple chemical events,” taken 
together with the explanation given in Section II. of the aggregation of events, the 
following definition of the “ theoretical analysis ” of a compound chemical event 
occurring by any number of substitutions will be readily appreciated. 
Definition : — The theoretical analysis of a chemical event occurring by any number 
of specified substitutions, namely, by the substitution of a for x, b for y, c for z, and so 
on, will be here said to be effected when all the different chemical events, occurring in 
any way whatever by these substitutions, are enumerated, the aggregate of which con- 
stitutes the event in question ; and the real analysis of such an event will be said to be 
completely effected when all these events are severally realized as independent pheno- 
mena, and will be said to be partially effected when two or more of such events or 
aggregates of such events are so realized. 
This analytical problem is presented to us in every chemical congruence. 
Q 2 
