114 SIR B. C. BRODIE ON THE CALCULUS OF CHEMICAL OPERATIONS, 
Ax, Ay, A z, Av, . . . respectively a— x, b—y , c—z, d—v,..., and in the result change a 
into x and x into a, b into y and y into b, c into z and z into c, d into v and v into d , 
and so on. We hence arrive at the development of f(x,y, z, v, . . .) in ascending powers 
of the moduli x—a,y—b , z—c,v—d.... Proceeding as before we have 
f(x, y, z, v,w,.. . )=f(a , b, c, d, e , . . .)+/i(«» b, c > d,e,.. .)(x—a) 
b, c, d,e,.. .)(y—b)+f 0 . 0A (a, b, c, d,e,.. . ){z—c ) 
+/o.o.o.i(«, b, c, d,e,.. .)[v—d)+8cc., 
together with a system of equations of the form 
.)(x-af(y-iY{z-cy(«-dy. . .=0. 
Eeasoning as before, if f(x,y, z,v ,.. .) be a chemical equation, we have 
f(x,y, z, v, . . .)=0, 
f(a, b, c,d,...)= 0. 
f x [a, b, c,d,.. .)(x—a)= 0, 
f 0 .i(ct,b, c, d , . . .){y—b)= 0, 
/o.o.i (a,b,c,d,...)(z—c)= 0, 
/o.o.o.iK b, c,d,.. .)(v—d)= 0. 
In these last equations, taken together with the equations previously referred to of 
the second and higher orders, the theoretical analysis of the event f(x, y,z,v, . . .)= 0 is 
effected, these equations collectively representing, for all purposes, the equation 
f{x,y, z, v,...)=0. 
As a necessary preliminary to the analysis of a compound chemical event, the substi- 
tutions must be specified in reference to which the analysis is to be effected. We can 
then proceed by means of the preceding theorem to resolve the event f(x,y, z, . . .)=0 
into its constituents, which consist essentially of two groups, the event f (a, b,c,.. .)=0, 
which does not occur by the substitutions in question, and the system of simple events 
enumerated in the various terms of the development, occurring in all possible ways by 
these substitutions, an analysis which, as we have said, is absolutely exhaustive. 
(6) From these considerations we arrive at a more exact definition than has hitherto 
been open to us of a normal chemical equation, and of the position which it occupies in 
the general algebraical system. We have seen that in the case of any chemical equation 
f{x,y, z,...)= 0, 
f(a , b, c,d , . . .)=0, 
f x (a, b, c,d,.. .)(x—a)= 0, 
fo.i (a,b,c,d,.. ,)(y.— b)= 0, 
/o.o.i(«» b, c,d,.. .)(z-c)= 0, 
