SIR B. C. BRODIE ON THE CALCULUS OE CHEMICAL OPERATIONS. 
95 
Example : — A unit of glycerine and three units of diacetine are identical with three 
units of monacetine and a unit of triacetine, 
aV£ 3 + 3 a V£ 5 =3aV£ 4 + aV£ 6 , 
which equation may be written thus, 
«V£ 3 + 3a 4 K 3 ^ 3 (ax 2 ^) 2 = 3 gs i x 3 i- 3 (oix 2 %) + a 4 x 3 ^ 3 (ux 2 ^) 3 , 
whence 
-aV£ 3 (a* 2 £-l) 3 . 
That is to say, this event is the reverse of an event in which aV£ 3 (the unit of glycerine) 
is constant, and which occurs by the transference of ux 2 %, and in three ways by that 
transference. 
If in the above equation x=.y and a=b=c, the equation becomes 
A(x — a)\z — a) = 0 , 
and 
A x 2 z + 2Aa 2 x + A a 2 z = A ax 2 + 2 A axz + Aa 3 . 
Example : — A unit of ethyldiacetamide and two units of acetamide and a unit of 
ethylamine are identical with a unit of diacetamide and two units of diethylmonace- 
tamide and a unit of ammonia [ref. Kekule, vol. i. p. 575]; thus 
ct G x 6 £ 2 v -{- 2 tt, 3 x 2 £v -j- a 4 x 2 v = k 4 x 4 £ 2 v -|- 2 u 6 x 4 £v -j- u 2 v, 
which equation may be put in the following form, 
a.‘ 2 v(c(x 2 g)' 2 (a 2 x 2 )-{-2ci 2 v(cix 2 g)-l-u 2 y(a 2 x 2 )=(x, 2 v(ccx 2 g) 2 -t-2a 2 v(cix 2 g)(ci 2 x 2 ) 2 -l-cc 2 v, 
whence 
u 2 y(ccx 2 % - 1) 2 («V - 1) = 0. 
In this event u 2 v (the unit of ammonia) is constant, and the event occurs by the trans- 
ference of o?x 2 and by the transference of ax 2 %, and by the latter transference in two 
ways. 
Example: — A unit of diethylamylamine, a unit of amylamine, and two units of 
ethylamine are identical with a unit of ammonia, a unit of diethylamine, and two units 
of ethylamylamine, 
a 1 l x 9 v + a~x 5 v -\-2ot 4 x 2 v= a 2 v -f- a 6 x 4 v -f- 2a g x 7 v. 
This equation may be written thus, 
ci 2 y(a 2 x 2 ) 2 (cc 5 x 5 )-{-C6 2 y(c6 5 x 5 )-l-2ci 2 y(a 2 x 2 ) = ci 2 v-l-cc 2 y(c6 2 x 2 ) 2 -{-2a 2 y(c6 2 x 2 )(a 5 x 5 ), 
whence 
A(aV — l) 2 (u 5 x 5 — 1)= 0. 
In this event a 2 v (the unit of ammonia) is constant. The event occurs either by the 
transference of a 5 x 5 or by the transference of a 2 x 2 , and by the latter transference in one 
of two ways. 
If a=b=c, the equation becomes 
and 
A(# — a)(y — a)(z — a) = 0, 
A xyz + Aa 2 x -f- Aa 2 y-\-Aa 2 z = A axy + Aaxz + Aayz -f- A a\ 
