28 
Proceedings of Boyal Society of Edinburgh. [sess. 
constants. Let t v x v y v and z 1 be the values of t, x , y , and z when 
y is a maximum. Then 
mx 1 = ny 1 
x 1 = 
z 1 = a(\ + 
\ 
P— e -™\ + e -nt x 
m-n n—m 
A = x 1 + y 1 + z 1 . 
By eliminating t v y v and z x from these equations we obtain 
’HnF mA ’ 
and thence 
Vi 
When n = m these expressions become indeterminate, but we may 
write 
and the limit of this last expression for m — n is — , so that 
A 
when the two velocity constants are equal, x 1 = y 1 = ~. This 
6 
result may also be obtained directly from the expression for z 
when m = n. It may be noted that, whatever the values of the 
velocity constants may be, when the maximum rate of increase 
of z is reached, the value of z itself cannot be more than 
2A 
A — = 0’264A. This can be proved by showing that x 1 + y 1 \ 
or A 
U m\— /?n\— ) 
m— m _j_ | \n-m l 
n) \nj ) 
is a minimum when m = n. 
It will be observed that the expression for x Y becomes that for y Y 
when m and n are interchanged, and similarly, y 1 becomes x v If 
the velocity coefficients were thus interchanged, the maximum 
dz 
value of ^ would remain unaffected, the only change being that 
the concentrations in the first and second stages would now be 
reversed. 
