SIR B. C. BRODIE ON THE CALCULUS OE CHEMICAL OPERATIONS. 
115 
Now, since these equations are always true, they are true when a— 1, 5=1, c— 1 ; 
whence 
/(i, i, i, i . . .)=o,] 
/i(i, 1 , i, i • . 0(^-1) =o. 
fo. i(i, 1, 1, 1 . • 0(y-i)=o, 
/o.oaCl, 1, 1, 1 . . 0(^-l) = O. 
Hence it follows that if in any chemical equation u = 0 the prime factors by which 
the equation is expressed be severally and simultaneously put equal to 1 (which is the 
only numerical value of chemical symbols), that equation vanishes, and also if we dif- 
ferentiate that equation once in regard to any one (and every one) of these factors, and 
in the result of that differentiation put all the prime factors severally and simultaneously 
equal to 1, that differential coefficient also vanishes*. A chemical equation, therefore, 
may be defined as an equation which possesses these properties. 
(7) Further, these considerations supply us with a general and comprehensive theory 
of the mode of occurrence of chemical events. It has been shown in the first part of 
this Calculus that in every chemical equation the symbols of the units of matter are 
expressed by the smallest possible number of prime factors when expressed by the factors 
a, f , 6, (3, co, v, <p, . . . So that every chemical equation regarded as a member of the 
general system is of the form 
/(«>£, *9 <p> • • 0=°- 
Now every such equation necessarily vanishes when the prime factors os, • • • are 
severally and simultaneously put equal to 1, and also admits of being developed in 
ascending powers of themoduli(a— 1), (£— 1), (0 — 1),(^— 1), . . . Hence every chemical 
event, without exception, may be considered to occur by the transferences of the simple 
weights w(ol), w(£), w(%), . . ., and admits of being resolved into a set of “ consti- 
tuent simple events ” severally occurring in various ways by these same transferences ; 
and if we are to refer all chemical events to one set of causes, so as to bring them 
under one law, these transferences are the only set of causes to which these phenomena 
can be referred. Thus, as we are led to contemplate the resolution of the units of 
matter into the “ system of simple weights ” w(a), w(£), w(Q), w(^), ... as the ultimate 
* This latter property of a chemical equation has already been demonstrated in Part I. Section V., and 
forms indeed the basis of this Calculus. It here reappears in another form. The condition /(l, 1 . . .)=0 is 
(as is obvious) a necessary property of every chemical equation when brought to the normal form in the way 
indicated in Section I. (3). But it may be noticed that the analysis of a chemical event is quite independent 
of the prior reduction to this form, which is effected in the course of the development itself. Take, for 
example, the equation given in I. Section IY. (1), xy—se+y, the fundamental chemical equation. The equation 
in this form does not inform us of the process by which the transformation in question is effected. But put- 
ting f(x,y)=xy-x-y and developing we have /( 1, 1) = -1,/ 1 . 0 (1, l)=0,/ oa (l, 1)=0,/ 1 . 1 (1, 1)=1, and 
f( x i V) = — 1 + (# — l)(y— 1) or f(x, y)-\-l = (x—l)(y—l), the equation /(#, y) + 1 = 0 being the equation 
xy=x+y reduced to the normal form, namely l+xy=x+y. 
MDCCCLXXVII. R 
