94 
SIR B. C. BRODIE ON THE CALCULUS OF CHEMICAL OPERATIONS. 
mentally the substitutions which are the causes of the event, superposing the one 
arrangement upon the other, thus — 
Axyz-\-Aabz-\-Aayc-{-Axbc , I. 
Axyc-\-Aabc-\-Aayz-\-Ajjcbz, II. 
it is apparent that where z appears in the arrangement (I.) before the event c appears 
in the arrangement (II.) after the event, and where c appears in the arrangement (I.) 
before the event z appears in the arrangement (II.) after the event, and the arrangements 
differ in that respect, and in that respect alone. That is to say, according to the defi- 
nition, Section III. (1), the event occurs by the substitution of c for z. But since 
A(x-a)(y-b)(z-c)=A(x-a)(z-c)(y-b), 
the above reasoning in regard to z and c is equally correct with regard to y and b , that 
is to say, the event also occurs by the “ substitution ” of b for y ; and similarly the event 
also occurs by the “ substitution ” of a for x ; and there are no other two letters in regard 
to which a similar assertion can be made. 
(12) Every actual example of such an event in which x, y, z and a , b, c are all different 
involves the construction of no less than eight different chemical substances connected 
by the relation of identity expressed in the above equation. The problem of the reali- 
zation of such an event is greatly simplified where two or more among the letters x , y , z 
or a, b , c become identical. The simplest case, from this point of view (the case, that is, 
which involves the construction of the smallest number of different substances), is that in 
which x=y=z, and also a=b=c. The equation then becomes 
A(x—a) 3 =0, 
and 
Ax 3 -f- 3 A a 2 x = 3 A ax 2 + A a 3 . 
Example : — A unit of chloroform and three units of chloride of methyl are identical 
with three units of mono-methyl chloride and a unit of marsh-gas ; thus 
(a 2 *)* 3 + 3(a 2 x) Z = 3(a 2 x)* 2 + (a 2 *), 
whence 
“Mx— 1) 3 =0- 
The event therefore is fully defined by saying that the above phenomenon is an event 
in which a 2 x (the symbol of the unit of marsh-gas) is “ constant,” and which occurs by 
the transference of and occurs by that transference in three ways. 
Example : — A unit of the trichlorhydrine of glycerine and three units of monochlorhy- 
drine are identical with a unit of glycerine and three units of dichlorhydrine ; thus 
whence 
aV (x— f) 3 =o. 
In this event aV, the unit of the (so called) hydride of propyl, is constant. The event 
occurs by the substitution of £ for and by that substitution in three ways. 
