106 SIE B. C. BEODIE ON THE CALCULUS OF CHEMICAL OPEEATIONS. 
A xb from Axy by the substitution of b for y , and A ah from A xy by the substitution 
of a for x and of b for y. Similarly, in the case of a simple event of the third order, 
A(x— a)(y— b)(z— c)=0, 
we have as the symbols of the units which appear in that event 
Axyz, 
A ay z, 
A xb z, 
Axy c, 
Aab z, 
A aye, 
A xb c, 
Aab c, 
which contain the combinations of the letters x, y, z , a, b, c taken three and three 
together, and may severally be regarded as derived from the symbol Axyz by the sub- 
stitution in that expression, in all possible ways, of a for x, b for y, and c for z. 
Similar relations obviously prevail between the symbols of the units of matter which 
appear in simple events of the fourth or any higher order. These relations are at once 
perceived on the consideration of the general forms of chemical equations. In special 
instances they are to a certain extent veiled by the identification of the symbols of some 
among the simple weights, by the exchanges of which the event occurs, and also by the 
suppression of the chemical symbol 1. As regards the last point it may be observed 
that we are always at liberty to replace the chemical symbol 1, where it appears as the 
symbol of a simple weight, by a special symbol (say the symbol ■&), since this symbol, 
thus introduced, satisfies all conditions required of the prime factor of a chemical equa- 
tion. Before interpreting results this symbol is to be put equal to 1. Take, for example, 
the equation given, Sec. II. (12), 
«Mx-i) 3 =0; 
putting l=7zr we have as the equation to the event in question the homogeneous 
equation 
oc 2 x(x—vr) 3 =0, 
whence 
(c4 2 x)p£ 3 -f- 3(a 2 x)p/c7 2 = 3(« 2 x)^ 2 ot -j- (a 2 x)ro- 3 , 
the symbols of the units which appear in the event being 
(« 2 % 3 > 
(a 2 x)x 2 Gr, 
(a 2 ^)^w 2 , 
(a 2 x> 3 . 
