1897 - 98 .] Dr Walker on the Velocity of Graded Actions. 27 
these constants, say n , to be infinitely great. The expression then 
reduces to 
z = A(1 - e~ mt ) , 
identical with the value for a pure reaction having the velocity 
constant m. If m becomes infinitely great, we have 
z = A(1 - e ~ nt ) . 
When n — m the above expression for z becomes indeterminate, 
but if we make the substitution in the differential equation, we get 
on solving 
z = A(1 - e"™* -mt e ~ mt ) . 
It will be observed that when both actions are unimolecular, the 
expression for z is symmetrical with respect to m and n. So far, 
therefore, as the effect on z is concerned, it is a matter of indiffer- 
ence whether the action with the velocity constant m or that with 
the velocity constant n takes place first. 
A characteristic point of difference between reactions proceeding 
in one stage and reactions proceeding in more than one is the 
following. If all the material is in the state A at the beginning of 
the action, then in the first case the maximum rate of increase 
of z takes place when z = 0, whilst in the second case this only occurs 
when z has attained a finite value. Let us consider what happens 
with a reaction proceeding in two unimolecular stages. The rate 
dx 
at which x diminishes, viz., - is greatest at the beginning of 
dz 
the action ; the rate at which z increases, viz., ^ is at that time 
doc 
zero. We have then a steady diminution of - ^ and a steady rise 
dx dz 
, dz 
ot df 
Now| = 
dt 
dt dt 
so that y will increase at a gradually 
dx dz 
diminishing rate and reach a maximum when - -r, = 
dt dt' 
Since 
dz . 
dt 
is proportional to y , the rate of increase of z will reach 
a maximum at the same time, i.e. } 
_ dx dz , 
when - jt = -T 2 , or when 
dt dt 
mx — ny. From the equations given above it is easy to calculate 
the values of x, y, and z at this point in terms of the velocity 
